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Recent urbanization has coincided with the enrichment of geotagged data, such as street view and point-of-interest (POI). Region embedding enhanced by the richer data modalities has enabled researchers and city administrators to understand the built environment, socioeconomics, and the dynamics of cities better. While some efforts have been made to simultaneously use multi-modal inputs, existing methods can be improved by incorporating different measures of "proximity" in the same embedding space -leveraging not only the data that characterizes the regions (e.g., street view, local businesses pattern) but also those that depict the relationship between regions (e.g., trips, road network). To this end, we propose a novel approach to integrate multi-modal geotagged inputs as either node or edge features of a multi-graph based on their relations with the neighborhood region (e.g., tiles, census block, ZIP code region, etc.). We then learn the neighborhood representation based on a contrastive-sampling scheme from the multigraph. Specifically, we use street view images and POI features to characterize neighborhoods (nodes) and use human mobility to characterize the relationship between neighborhoods (directed edges). We show the effectiveness of the proposed methods with quantitative downstream tasks as well as qualitative analysis of the embedding space: The embedding we trained outperforms the ones using only unimodal data as regional inputs.

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Learning Neighborhood Representation from Multi-Modal Multi-Graph: Image, Text, Mobility Graph and Beyond Tianyuan Huang Stanford University Zhecheng Wang Stanford University Hao Sheng Stanford University Andrew Y Ng Stanford University Ram Rajagopal [email protected] Stanford University Learning Neighborhood Representation from Multi-Modal Multi-Graph: Image, Text, Mobility Graph and Beyond Recent urbanization has coincided with the enrichment of geotagged data, such as street view and point-of-interest (POI). Region embedding enhanced by the richer data modalities has enabled researchers and city administrators to understand the built environment, socioeconomics, and the dynamics of cities better. While some efforts have been made to simultaneously use multi-modal inputs, existing methods can be improved by incorporating different measures of "proximity" in the same embedding space -leveraging not only the data that characterizes the regions (e.g., street view, local businesses pattern) but also those that depict the relationship between regions (e.g., trips, road network). To this end, we propose a novel approach to integrate multi-modal geotagged inputs as either node or edge features of a multi-graph based on their relations with the neighborhood region (e.g., tiles, census block, ZIP code region, etc.). We then learn the neighborhood representation based on a contrastive-sampling scheme from the multigraph. Specifically, we use street view images and POI features to characterize neighborhoods (nodes) and use human mobility to characterize the relationship between neighborhoods (directed edges). We show the effectiveness of the proposed methods with quantitative downstream tasks as well as qualitative analysis of the embedding space: The embedding we trained outperforms the ones using only unimodal data as regional inputs. Introduction The world is full of connections between entities of different modalities, such as websites and urban neighborhoods. A website can be represented as a node containing multi-modal components like text, images, and videos; hyperlinks connected websites as directed edges. Similarly, an urban neighborhood can be regarded as a complex multi-modal node containing the natural and built environment, business activities, * Equal contribution Figure 1: Multi-modal multi-graph of urban neighborhoods in the City of Chicago. Each neighborhood is a container of multi-modal inputs, e.g., street views and POIs. The neighborhoods are considered connected if they are close spatially (e.g. A and B) or or if there are many human mobility trajectories in between (e.g. A and C). Notice that even A and C are spatially far away, the large number of trips in between indicates the strong relations which should be captured in the embedding space. and the people living there. Urban neighborhoods are interconnected implicitly with various types of relations such as geospatial proximity and human mobility trajectories between neighborhoods. With the vision of "smart city" being proposed in different parts of the world as well as the increasing availability of a great variety of data in cities, understanding the characteristics and dynamics of cities become essential, and more importantly, feasible with the help of stateof-the-art machine learning algorithms. Urban neighborhood embedding, or representing various urban features as vectors, is a preliminary task to many data-driven urban studies and applications such as spatiotemporal prediction, planning, and causal inference. Though abundant studies focus on representation learning for a single modality of data like images [Radford et al., 2015] and text [Mikolov et al., 2013], representing urban neighborhoods leveraging multi-modal data while maintaining their correlations is still a challenging task. Traditional data collection approaches like census are costly. For example, the 2020 U.S. census was estimated to cost 15.6 billion dollars [GAO, 2018]. Moreover, the data produced by census is usually aggregated at pre-defined geographic divisions (e.g., census tracts and counties) and can hardly be re-mapped into other customized geospatial units such as raster tiles or polygons, which limits the flexibility of using the data. There are recent attempts to extract or predict urban characteristics from widely-available urban-associated data using data-driven approaches, including both supervised and unsupervised learning. Supervised learning methods utilize geo-tagged data such as point-of-interest (POI) [Yuan et al., 2012], and street view imagery [Gebru et al., 2017] as inputs and output the inference of local socioeconomic attributes. However, supervised learning is task-specific: The representation learned is not necessarily transferrable to other tasks. Furthermore, developing supervised learning models with high-dimensional data like images requires a massive dataset with annotated labels of ground-truth socioeconomic attributes, which is not necessarily available for certain regions or at the desired geographic level (e.g., raster tiles). By contrast, unsupervised learning overcomes such limitations by developing a universal and versatile representation without task-specific ground-truth supervisions. Common urban features to use include POI [Fu et al., 2019], street views [Law and Neira, 2019], and taxi trips [Yao et al., 2018]. However, most of the existing unsupervised urban representation learning is still based on unimodal data, without fully leveraging various types of data both within and between neighborhoods. Urban neighborhoods are complex systems that can be modeled by a multi-modal multi-graph: Each urban neighborhood ("node") is a "container" which contains the built environment, business activities, and population inside the neighborhood. There are also relations ("edge") between neighborhoods, which can be characterized by geospatial proximity, mobility connections, or both. To obtain a comprehensive representation of urban neighborhoods, we model the neighborhoods in an urban area as multi-modal multi-graph (M3G) and develop an unsupervised representation learning framework to obtain the neighborhood embedding from the graph. Instead of learning the graph globally, we propose a contrastive sampling approach that samples triplet (anchor, positive, negative) according to the multi-graph edges, enabling scalable training with multi-city data. Our major contribution is three-fold: 1) We proposed a framework to learn neighborhood representation by jointly modeling both interand intra-neighborhood multi-modal data as a multi-graph. 2) We demonstrate this framework with real-world data in two U.S. metropolitan areas at the census-tract level, using street view images and POI features as intra-neighborhood characteristics, and geospatial proximity and mobility flow as inter-neighborhood relations. The neighborhood embeddings generated from our framework achieve state-of-the-art performance in all downstream prediction tasks. 3) We propose three qualitative evaluations for the neighborhood embedding space, showing that our model successfully integrates various data modalities in the embedding space. Related Work Spatiotemporal Representation Learning Spatiotemporal representation learning aims to produce region embedding using geo/temporal-tagged data under the First Law of Geography [Tobler, 1970] [Jean et al., 2019] starts the stream of imposing such prior to the embedding space through contrastive learning. Using geo-proximity as the single criterion to sample positive and negative tiles, this algorithm judiciously pushes the latter further away from the anchor point in the embedding space as compared with the former. Unfortunately, such framework can not be easily applied to multi-modal settings as a consistent and meaningful distance measure is required between any two samples across different modalities. Urban2Vec overcomes such drawbacks by introducing the neighborhood embedding. It is worth noticing the spatiotemporal relation between each sample can be viewed as a reciprocal relation denoted by an undirected edge. [Jiang, 2020] introduces the use of mobility, POI similarity or even the likeness of geo-tagged tweets [Zhang et al., 2017] as new metrics of proximity to define "edges". In this work, we generalize the contrastive learning approach to non-reciprocal relations such as mobility flow and propose a framework that can be easily extended to other graph-structured datasets with multi-modal edges and multi-modal nodes. Graph Embedding There are a lot of graph embedding methods (e.g., Deep-Walk [Perozzi et al., 2014], node2vec [Grover andLeskovec, 2016]) that generates embedding for a certain node in the graph. They can be applied to the mobility graph. For example, [Fu et al., 2019] incorporate such prior by directly impose an autocorrelation in the latent space. However, most of them are not able to model multi-modal edge (as in a multigraph), and their embedding space does not reflect the multiperspective proximity between nodes. To further incorporate information from both nodes (e.g. POI, street view) and edges (e.g. mobility, distance), [Jenkins et al., 2019] concatenate image embedding and graph embedding at each node. Our training strategy can be viewed as an extension of the contrastive sampling technique in Graph Neural Network setting ([Schroff et al., 2015;Qiu et al., 2020]): By sampling triplets according to multiple proximity measures, the embedding captures the multi-graph topological properties as well as the multi-modal features from each node. Urban Computing Urban Computing aims to tackle major issues in cities, such as traffic control, public health and economic development, by modeling and analyzing urban data. A lot of research have shown the possibility to infer this socioeconomic information from satellite image [Jean et al., 2016;Sheng et al., 2020], street view [Gebru et al., 2017], human mobility [Xu et al., 2018] and geo-tagged social network activities [Schwartz and Hochman, 2014]. Recent studies also demonstrate that similar tasks could benefit from multi-modal inputs: utilizes both POI data and taxi trip data to infer crime rate in Chicago. [Irvin et al., 2020] includes a fusion of auxiliary variables, such as elevation and air pressure, with a computer vision model on satellite images to improve the performance of forest loss driver classification. We hope the multi-graph framework proposed in this work will provide a much convenient and comprehensive tool for urban computing tasks with multi-modal data. Methods In the following section, we first mathematically define the problem of learning neighborhood embedding and give an overview of the construction of Multi-Modal Multi Graph (M3G). Then we introduce the concept of neighborhood container and our contrastive sampling strategy to incorporate multi-modal inputs at each node. We continue by describing our inter-neighborhood learning strategy for both directed and undirected edges. This section is concluded by a summary of the loss function used in M3G. Problem Statement Unlike most of the previous studies that focus on specific modality (e.g., image, text, etc.) and specific geographic unit (e.g. census tract, county, etc.), we restate the general problem of Urban Neighborhood Embedding agnostic to both as the following: Definition 3.1 (Urban Neighborhood Embedding Problem). Given a metropolitan area A that is composed of a set of disjointed neighborhood geometries U = {u 1 , u 2 , ..., u N }, s.t. A = N i u i , the goal of urban neighborhood embedding is to learn a vector representation z i ∈ R d for each u i which encodes the characteristics and mutual relations of u i . Notice u i can be a raster tile of certain size (commonly used in remote sensing), a census tract or a county. Under our abstraction we do not assume all u i are of the same geographic unit. Geo-tagged data (i.e. data with GPS coordinates) is used to generate such embedding. Instead of categorising data by the modality, we use a more general approach of categorization based on how data is associated with the location(s): Definition 3.2 (Geo-Tagged Point Data). Geo-tagged point data is the kind of data characterizing one geolocation l: D p m = {(x m , l) } is the set of geo-tagged point data with an input x m of modality m at each geolocation. Examples of geo-tagged point data includes street views, POI check-in data and satellite images. Definition 3.3 (Geo-Tagged Reciprocal Data). Geo-tagged reciprocal data is the kind of data characterizing the relation between two geolocations l 1 and l 2 , but it does not have a direction and the relation is reciprocal: D r m = {(x m , l 1 , l 2 )} {(x m , l 2 , l 1 )} is the set of geo-tagged reciprocal data with an input x m of modality m between two geolocations. Examples of geotagged reciprocal data include spatial distance, road connectivity, and transaction volume. Definition 3.4 (Geo-tagged Irreciprocal Data). Geo-tagged reciprocal data is the kind of data characterizing the relation between two geolocations l 1 and l 2 with a direction: D ir m = {(x m , l 1 , l 2 )} is the set of geo-tagged irreciprocal data with an input x m of modality m between two geolocations. Examples of geotagged irreciprocal data include human mobility, commute time, and goods imports/exports. The three categories of data are corresponding to the node, undirected, and directed edges in our M3G model and will be further explained in the next two sections. For now, let us assume D = m,t D t m and introduce the concept of multimodal multi-graph: Definition 3.5 (Multi-Modal Multi-graph (M3G)). The Multi-Modal Multi-graph G D (U, E) is a multi-graph for neighborhoods U and their edge set E, characterized by the multi-modal geo-tagged dataset D. The nodes U have attributes defined by all geo-tagged points data D p m , which are described with more details in Section 3.2. The edges E are defined by all geo-tagged reciprocal/irreciprocal data D r m and D ir m , which are described in Section 3.3. Intra-Neighborhood Modalities Despite their vast difference in data structure, both POI meta information and street view images depict the urban characteristics at specific location. In this section, we will use them as examples of Intra-Neighborhood Modalities and demonstrate how we incorporate their information into the neighborhood embedding. Neighborhoods as Containers Given a set of geo-tagged street view images D p S = {(x S , l)}, where s is an image and l is its geolocation, we can easily assign each data point to the urban neighborhood u i it is located in: S i = {x S |(x S , l) ∈ D p S , s.t. l ∈ u i } Each S i is a bag of street view images for neighborhood u i . Similarly, we can construct the feature container with the POIs D p P = {(x P , l)}, where p is a POI and l is its geolocation. To represent each POI p, we further disassemble the textual information of p, which are extracted from the POI category, price, and customer reviews, into a bag of words {t}. By pooling bags of words of all POIs inside a neighborhood, we obtain the bag of POI words for each neighborhood u i in M3G. P i = {t|(x P , l) ∈ D p P , s.t. t ∈ x P and l ∈ u i } t denotes a word. We can extend this approach to incorporate other textual data such as geo-tagged social media posts. Intra-Neighborhood Contrastive Learning Objective With the node feature containers S i and P i constructed, we here propose our intra-neighborhood contrastive-sampling strategy: For each pass, we sample one neighborhood u a uniformly at random from U, i.e. u a u ∼ U, as our anchor neighborhood. Then we sample one context street view image s c u ∼ S a and one negative street view image s n u ∼ S −a , with S −a = i =a S a . Our proposed triplet loss [Schroff et al., 2015] formulates as: L S (z a , s c , s n ) = [M + ||z a − f θ (s c )|| 2 − ||z a − f θ (s n )|| 2 ] +(1) , where [·] + is a rectifier and a positive constant M is used to prevent infinitely large difference between these two distances. z a is the embedding vector for neighborhood u a . f θ (·) is the learnable encoder for images, e.g. a convolutional neural network with parameters θ. Similarly, given a random sample u a from U, we can sample POI word t c u ∼ P a and t n u ∼ P −a = i =a P a and construct the triplet loss for POI data: L P (z a , t c , t n ) = [M + ||z a − g φ (t c )|| 2 − ||z a − g φ (t n )|| 2 ] +(2) The definitions of [·] + and M are the same as above. g φ (·) is the learnable encoder for word with parameters φ. Inter-Neighborhood Modalities Without data characterizing the relations between neighborhoods, the neighborhood embedding obtained by minimizing (1) and (2) can only incorporate information within neighborhoods . In this section, we will describe how D r j and D ir j characterizes the edges in graph G and introduce our learning strategy for inter-neighborhood modalities. We include both spatial distance D r D and human mobility D ir M as examples of inter-neighborhood modalities. Multi-Modal Multi-Edges Spatial distance can be measured between any pair of neighborhoods (u i , u j ). We can define the outgoing edge sets of u i induced from the spatial distance as: E D i = {(u i ,u j , x D )|(x D , l 1 , l 2 ) ∈ D r D s.t. l 1 ∈ u i and l 2 ∈ u j } Here x D = 1 dij , which is the reciprocal of geospatial distance between u i and u j . Notice that D r D already includes both directions of a same undirected edge according to Definition 3.4. Similarly we can define the outgoing edge sets of u i induced from the human mobility D ir M : E M i = {(u i ,u j , x M )|(x M , l 1 , l 2 ) ∈ D ir M s.t. l 1 ∈ u i and l 2 ∈ u j } Here x M is the total number of trips from a geolocation in u i to a geolocation in u j . Once we add both sets of edges to the graph G, it is likely there can be multiple edges between u i and u j from different modalities. Inter-Neighborhood Contrastive Learning Objectives Like Section 3.2, we first sample one neighborhood u a at random from U, i.e. u a u ∼ U. Instead of defining the context and negative set explicitly as in Section 3.2, we draw samples of context neighborhood by sampling each edge with the probability proportional to the weights associated with it. Specifically, edge (u, v, w) has weight of p m (w) being sampled, with p m (·) a designed thresholding function using the prior on modality m. For example, for the spatial distance, we can set p S (w) = 1, if w > 1 500 0, otherwise to sample a context neighborhood within a radius of 500 meters. Hence, for modality m ∈ {D, M}, the probability of u j being sampled as a context neighborhood u c is: P m a,j = (u,v,w)∈E m a p m (w)1 a (u)1 j (v) (u,v,w)∈E m a p m (v)1 a (u)(3) Here 1 x (·) is the indicator function with the value 0 everywhere except for x. The negative neighborhood u n is sampled uniformly at random from the set of rest of nodes {u j |P m a,j = 0}. Finally, we have the inter-neighborhood triplet loss for each modality m ∈ {D, M}: L m (z a , z c , z n ) = [M + ||z a − z c || 2 − ||z a − z n || 2 ] + (4) The definitions of [·] + and M are the same as above. By default, we sample balanced number of triplets for each modality. Together with Equation (1) and (2), we are able to train our neighborhood embedding with any modality of inter/intra-neighborhood data. Next section will demonstrate our framework with experiments on real-world datasets. Experiment To demonstrate the effectiveness of our framework, we conduct experiments on 1294 census tracts in Chicago and 1310 census tracts in New York City. We demonstrate our framework at census-tract level because the reference data for prediction (e.g., American Community Survey (ACS)) are readily available at this level. Our framework can be easily applied to other geographic divisions (e.g. block groups) or even customized units (e.g. raster tiles). Data Description The street view images and POI features we used are obtained from Google Street view API 2 and Yelp Fusion API 3 , respectively. We randomly sample 50 street views for each census tract. The human mobility data is provided by Safe-Graph 4 . Specifically, we use Core Places and Weekly Patterns datasets, which include, for each POI, the exact location, as well as the aggregated weekly estimates of the home CBGs of visitors. We preprocess the weekly patterns in Chicago and New York City from Jan 2018 to Dec 2019. Each visit is encoded as a directed edge between neighborhoods of POI and visitor's home; both are aggregated at the census tract level. Their statistics are summarized in Table 2. (1)). The encoder for POI words(i.e., g φ (·) in Equation (2)) is a look-up table with weights initialized by GloVe [Pennington et al., 2014]. During training, we minimize loss (1), (2), (4) sequentially in a three-stage process. When we sample inter-neighborhood triplet, for spatial distance, we sample u c uniformly at random from the 5 closest neighbors and sample u n uniformly at random from the rest. We obtain M3G neighborhood embeddings using three different configurations of edge modalities (1) Spatial distance only (M3G DIST); (2) Mobility only (M3G MOB); (3) Both spatial distance and mobility (M3G DIST+MOB). We compare the embedding with the one derived using Ur-ban2Vec method , which rely solely on intra-neighborhood modalities, and GAE [Kipf and Welling, 2016], which extract information from mobility graph using Graph Autoencoder. Results and Discussion Predicting Demographics and Economics In this task, we treat trained neighborhood embeddings as input features to predict ACS demographic and economic attributes for each census tract. We choose Median Age, Years of Education, and Percentage of White Population as demographic attributes, and Poverty Rate, Average Household In-come and Employment Rate as economic attributes. We apply PCA to reduce the embedding dimensions to 50 before running the regression model. In this work, we try both linear regression and random forest regression. Census tracts are split into training set (85%), and test set (15%). We use R 2 as the major metrics and randomly reshuffle train/test split for 20 rounds to estimate variance of the performance. As is shown in Figure 2, two models trained with single edge modality outperform one another on different attributes: For example, for Median Age and Years of Education, M3G DIST outperforms M3G MOB, while M3G MOB has a higher average R 2 for Percentage of White Population and Employment Rate. However, by combing both modalities, M3G DIST + MOB always outperform both of them and the baseline models Urban2Vec and GAE on all demographic and economic attributes, indicating the benefits of incorporating both intra-and inter-neighborhood modalities to capture mult-perspective urban characteristics. Linear regression results from Table 1 follow a similar pattern: M3G DIST+MOB outperforms all other models on all attributes except Percentage of White Population. Training with Multi-City Data Since we adopt a contrastive sampling approach to learn the graph structure, we can easily scale up experiments to multiple cities without facing any memory issue. In this experiment, we investigate the improvements from training with merged data of both Chicago and New York City. Table 3 shows the mean of R 2 for predicting all 6 demographic and economic attributes using linear regression. As is shown, using multi-city training set in Chicago yields better prediction performance but not for New York City. This may be explained by the relative sparse mobility data in New York City. Qualitative Analysis of the Embedding Space Clustering of Neighborhood Embeddings To interpret the neighborhood embeddings learned from our models, we apply k-means clustering on the generated embedding. Figure 3 shows the results for k = 6 in Chicago. As the plot shows, Downtown Chicago and South Chicago, which have a high number of crime reports 5 , are clustered into one group (red), while neighborhoods in the north like Evanston are clustered into other groups (yellow and orange). Correlation with Geospatial and Mobility Proximity In this analysis, we investigate the correlations between interneighborhood embedding distance and their real-world proximity in terms of geo-distance or mobility. In Figure 4, we sample 0.1% of the 1.6 M pairs of census tracts in Chicago and measure the L2 distances between their embedding vectors. With a larger number of aggregated visitors in between, neighborhoods tend to have representations closer in the embedding space; as spatial distance becomes larger, two neighborhoods tend to fall further apart in the embedding space. Such trends demonstrate that the embedding indeed captures both the geospatial and mobility relations through training. Neighborhood Embedding and Input Data Embedding We are also interested in whether the neighborhood embedding incorporates information from the geo-tagged point data. We apply PCA to extract the first two principal components of the embeddings of both neighborhoods and street views and plot their distribution in Figure 5. Large points with black borders denote neighborhoods; small points denote street view images, with the color indicating the neighborhood they belong to. Here, we randomly selected three census Conclusion In this work, we develop M3G, a framework to model urban neighborhoods as a multi-modal multi-graph and thus learn the neighborhood representation. To demonstrate our framework, we use street view images and POIs as two modalities of data inside the neighborhood and both geospatial proximity and mobility pattern as two modalities of "edges" between neighborhoods. We show the neighborhood embedding from our framework outperforms the ones from other multi-modal models in the downstream prediction tasks while preserving both proximity/mobility connections between neighborhoods, and relations between the neighborhood and street views. The method we propose here is a general framework to learn representation for a graph with multi-modal "node" and multi-modal "edge". Such a framework can further integrate other modalities like satellite imagery (as components of the "nodes") and inter-region transactions (as "edges"), and even be extended to learn the representation of other graph-structured data such as websites, which will be an important task in our future work. Figure 2 : 2Prediction R 2 on neighborhood attributes with random forest model in Chicago. Left: Demographic attributes. Right: Economic attributes. Figure 5 : 5Positions of embeddings in the plane of the first two PCA components, for both neighborhood and street view images. 1 . [Chu et al., 2019; Mac Aodha et al., 2019] generate geo-aware prior based on the geo-coding of coordinates. Tile2vec Table 1 : 1Prediction R 2 on demographic and economic attributes with linear regression model in Chicago.Area (km 2 ) # Edges Average in/out degree Chicago 606 143, 235 110 New York City 1212 120, 470 92 Table 2 : 2Safegraph mobility data statistics4.2 Training Details For all experiments we set embedding dimension d = 200 for images, POI words, and neighborhood. We use an Inception- v3 [Szegedy et al., 2016] architecture as the encoder for street view images (i.e., f θ (·) in Equation Table 3 : 3Average prediction R 2 , training on single-/multi-city data.Figure 3: Color-coded map based on Left: Total number of crimes Right: Embedding clusters derived by k-means (k=6) for Chicago. Figure 4 : 4Correlation between geospatial/mobility proximity of node pairs in the graph and the corresponding embedding distance in Chicago. Left: The horizontal axis is the total number of visitors (bidirectional) between each pair from January 2018 to December 2019. Right: The horizontal axis is the spatial distance measured in km.tracts for visualization. Census tracts in Orange, Blue, and Green have average household income of $34,407, $43,836, and $113,479, respectively. As the plot shows, street view embeddings scatter around their corresponding neighborhood embedding. Though all three sampled images contain large portion of vegetation, their visual difference (e.g. trimmed or not, road landscape) can be reflected by their proximity in embedding space. Median age Years of education Percentage of white population Poverty rate Average household income Employment rateModel Demographic characteristics Economic characteristics Urban2Vec [Wang et al., 2020] 4.081 0.724 0.186 0.076 20, 270 0.047 GAE [Kipf and Welling, 2016] 4.283 0.740 0.182 0.073 20, 531 0.046 M3G DIST 3.983 0.642 0.153 0.072 19, 295 0.043 M3G MOB 3.975 0.600 0.128 0.064 17, 794 0.039 M3G DIST+MOB 3.861 0.583 0.129 0.064 17, 509 0.038 Table 4 : 4Prediction MAE on demographic and economic attributes with linear regression model in Chicago Median age Years of education Percentage of white population Poverty rate Average household income Employment rateModel Demographic characteristics Economic characteristics Urban2Vec [Wang et al., 2020] 4.181 0.739 0.193 0.079 18, 728 0.048 GAE [Kipf and Welling, 2016] 4.104 0.716 0.140 0.070 18, 693 0.041 M3G DIST 3.747 0.608 0.140 0.064 16, 493 0.039 M3G MOB 4.014 0.690 0.114 0.064 17, 088 0.036 M3G DIST+MOB 3.716 0.587 0.064 0.064 15, 578 0.035 Table 5 : 5Prediction MAE on demographic and economic attributes with random forest model in Chicago Median age Years of education Percentage of white population Poverty rate Average household income Employment rate Urban2Vec Model Demographic characteristics Economic characteristics 0.430 0.634 0.496 0.494 0.533 0.508 GAE [Kipf and Welling, 2016] 0.419 0.648 0.512 0.510 0.557 0.529 M3G DIST 0.453 0.680 0.572 0.523 0.580 0.544 M3G MOB 0.450 0.702 0.614 0.568 0.617 0.579 M3G DIST+MOB 0.472 0.717 0.618 0.572 0.627 0.584 Table 6 : 6Prediction Kendall's τ on demographic and economic attributes with linear regression model in Chicago Median age Years of education Percentage of white population Poverty rate Average household income Employment rateUrban2Vec Model Demographic characteristics Economic characteristics 0.398 0.618 0.455 0.473 0.546 0.485 GAE [Kipf and Welling, 2016] 0.414 0.632 0.581 0.502 0.579 0.548 M3G DIST 0.487 0.694 0.603 0.569 0.619 0.582 M3G MOB 0.436 0.658 0.642 0.556 0.631 0.596 M3G DIST+MOB 0.493 0.711 0.673 0.567 0.648 0.624 Table 7 : 7Prediction Kendall's τ on demographic and economic attributes with random forest model in Chicago # Street views # POIs # Neighborhoods (census tract)Chicago 64, 739 38, 445 1, 294 New York City 67, 271 50, 697 1, 310 Table 8 : 8Street views and POI data statistics "Everything is related to everything else, but near things are more related than distant things." https://developers.google.com/maps/documentation/streetview 3 Available at https://www.yelp.com/fusion 4 See data catalog at https://docs.safegraph.com/docs/. Chicago crime data available at https://www.chicago.gov/city/ en/dataset/crime.html Geo-aware networks for finegrained recognition. [ References, Chu, Proceedings of the IEEE International Conference on Computer Vision Workshops. the IEEE International Conference on Computer Vision WorkshopsReferences [Chu et al., 2019] Grace Chu, Brian Potetz, Weijun Wang, Andrew Howard, Yang Song, Fernando Brucher, Thomas Leung, and Hartwig Adam. Geo-aware networks for fine- grained recognition. In Proceedings of the IEEE Interna- tional Conference on Computer Vision Workshops, pages 0-0, 2019. Efficient region embedding with multi-view spatial networks: A perspective of localityconstrained spatial autocorrelations. [ Fu, Proceedings of the AAAI Conference on Artificial Intelligence. the AAAI Conference on Artificial Intelligence33[Fu et al., 2019] Yanjie Fu, Pengyang Wang, Jiadi Du, Le Wu, and Xiaolin Li. 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The "cvpaper.challenge" is a group composed of members from AIST, Tokyo Denki Univ. (TDU), and Univ. of Tsukuba that aims to systematically summarize papers on computer vision, pattern recognition, and related fields. For this particular review, we focused on reading the ALL 602 conference papers presented at the CVPR2015, the premier annual computer vision event held in June 2015, in order to grasp the trends in the field. Further, we are proposing "DeepSurvey" as a mechanism embodying the entire process from the reading through all the papers, the generation of ideas, and to the writing of paper.

1605.08247

2f7e9b45255c9029d2ae97bbb004d6072e70fa79

A review of CVPR2015 and DeepSurvey 26 May 2016 cvpaper.challenge in 2015 Hirokatsu Kataoka · Yudai Miyashita · Tomoaki Yamabe · Soma Shirakabe · Shin'ichi Sato Hironori Hoshino Ryo Kato · Kaori Abe · Takaaki Imanari · Naomichi Kobayashi · Shinichiro Morita Akio Nakamura A review of CVPR2015 and DeepSurvey 26 May 2016 cvpaper.challenge in 2015Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor) The "cvpaper.challenge" is a group composed of members from AIST, Tokyo Denki Univ. (TDU), and Univ. of Tsukuba that aims to systematically summarize papers on computer vision, pattern recognition, and related fields. For this particular review, we focused on reading the ALL 602 conference papers presented at the CVPR2015, the premier annual computer vision event held in June 2015, in order to grasp the trends in the field. Further, we are proposing "DeepSurvey" as a mechanism embodying the entire process from the reading through all the papers, the generation of ideas, and to the writing of paper. Introduction cvpaper.challenge is a joint project aimed at reading papers mainly in the field of computer vision and pattern recognition 1 . Currently the project is run by around ten members representing different organizations; namely, AIST, TDU, and University of Tsukuba 2 . Reading international conference papers clearly provides various advantages other than gaining an understanding of the current standing of your own research, such as acquiring ideas and methods used by researchers around the world. In reality, however, although this input of knowledge is important, researchers and engineers are too busy to have time to do it, and the process takes Hirokatsu Kataoka Tsukuba, Ibaraki, Japan Tel.: +81-29-861-2267 E-mail: [email protected] 1 Further reading: Twitter @CVPaperChalleng (https://twitter.com/cvpaperchalleng), SlideShare @cvpaper.challenge (http://www.slideshare.net/cvpaperchallenge) 2 In 2016, we are now around 30 members including the University of Tokyo and Keio University. a great amount of time and effort for undergraduate and graduate students (particularly masters course students) who lack research experience and entails sacrificing their time for classes and research. Assigning this work, however, to non-experts who are not familiar with the field of computer vision, results in a great amount of time needed for interpreting the papers. As a way to address this problem, we believe that we can make it relatively easier to grasp advanced technologies if we share and systematize knowledge using the Japanese language. We therefore undertook to extensively read papers, summarize them, and share them with others working in the same field. The IEEE-sponsored Conference on Computer Vision and Pattern Recognition (CVPR) is known as the premier conference in the field of computer vision, pattern recognition, and related fields. CVPR, which is held annually in the U.S., has on average around 20% acceptance rate for submitted papers, making it a very difficult conference to hurdle, and pointing to the high quality of the accepted papers. Also, CVPR is also known to comprehensively cover papers in the different fields in computer vision and pattern recognition. A number of prominent international researchers and research groups choose their research themes after a comprehensive grasp of almost all papers presented in premier conferences and an understanding of research trends. We believe that the accuracy by which research themes are chosen can be improved by constantly being updated on cutting-edge technologies and discussing these new technology trends within the research groups as part of their regular activities. Further, a survey of papers presented in premier conferences is also an essential way to gather tools needed for research. We therefore believe that gaining an understanding of papers presented in premier conferences is the best method for authors to comprehend the lat-est trends in computer vision, pattern recognition, and related fields. As the first step of this endeavor, we undertook to read all the 602 papers accepted during the CVPR2015 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602]. This review summarizes all the papers of CVPR2015 we read as the first project of cvpaper.challenge. In this paper, we will describe the characteristics of CVPR2015 and discuss the trends and leading methods used in three areas; namely, recognition, 3D, and imaging/image processing. Further, we will enumerate the proposed datasets and new research problems presented at the conference and propose the concept of "DeepSurvey". Finally, we will give a summary and discuss future steps. We would like to stress, however, that this paper mainly focuses on a survey of the research trends, and does not cover the details of all the 602 papers, which are beyond the scope of this paper. Related initiatives and positioning of this project An example of a related initiative is the Japanese CHI Study Group that undertook to read all the papers presented at the ACM CHI Conference on Human Factors in Computing Systems, the top conference for user interfaces [603]. In 2015, the Study Group was held jointly in Tokyo and Hokkaido using a remote conference system to read within one day all the 485 papers presented at CHI2015. Speakers were assigned one session each and introduced one paper in approximately 30 seconds. The CHI Study Group started in 2006, and is on its 10th year in 2015. This initiative is a very effective way to grasp trends in the user interface domain, which is a very progressive field. It is noteworthy that several Japanese researchers have received the Best Paper and Honorable Mention awards at CHI2015. Considering the rapid progress in the field, the authors focused on "reading all the papers included in the international conference and summarizing trends through the project," as well as on listening to the introduction of the papers by project members. The CHI Study Group, therefore, serves its role in terms of covering all the papers. However, understanding the trends in a research field entails comprehensively reading all papers outside particular domains and holding discussions after reading the papers. We believe that trends can only be properly identified through discussions within the small group that undertook to read all the papers. Trends at CVPR2015 This paper is mainly divided into three main areas; namely, (i) Recognition, (ii) 3D, and (iii) Imaging/Image processing. Before we proceed with discussing the details of each area, we will first explain the features and perspectives gleaned from the titles of the papers and of the papers selected for oral presentation. 3.1 CVPR2015, as seen from the titles of the papers Figure 1 shows a visualization of the titles of papers at CVPR2015 using Wordle [608], a word-visualization service. We see that other than "image" as the most frequent word, which is expected, we also see that the words "deep," "learning," "object," and "recognition" are very prominent. As can be seen from this word visualization and goes without saying, deep learning (DL) is a widely used tool in image recognition in researches presented at CVPR2015. Results of a search among CVPR2015 papers showed that the term was found in 250 out of the 602 papers. Although deep learning was mainly used for object recognition, since R-CNN [609] was proposed, it also came to be used more for object detection. Also, with detection and recognition becoming more accurate, there are now more research initiatives to address semantic segmentation, which is an even more difficult problem. Meanwhile, the paper that received the Best Paper Award dealt with the method called DynamicFusion [38], which pertains to real-time construction of high-resolution 3D models. The research was recognized for being able to successfully reconstruct non-rigid models in real time, in addition to being able to reconstruct in higher resolution than KinectFusion [610]. As shown by the prominence of the words "reconstruction," "depth," and "shape," 3D research is also progressing at a steady pace. Also, deep learning is also being applied in stereo matching and 3D features, indicating the widening applications of deep learning. Even in imaging technologies, deblurring and sensing technologies are being updated, wherein examples of applications of feature extraction through deep learning were reported. CVPR2015 as seen from the oral presentations The 12 oral sessions are presented in CVPR2015 as below: -CNN Architectures -Depth and 3D Surfaces -Discovery and Dense Correspondences -3D Shape: Matching, Recognition, Reconstruction -Images and Language -Multiple View Geometry -Segmentation in Images and Video -3D Models and Images -Action and Event Recognition -Computational Photography -Learning and Matching Local Features -Image and Video Processing and Restoration Recognition With CNN as the most widely used approach in deep learning, the theme of the first oral presentation was on CNN Architecture. First we would like to mention about GoogLeNet [1], the winner of the ILSVRC2014 with a record 6.67% top-5 error rate. GoogLeNet is a 22-layers CNN architecture, where convolutional structures are recursively connected to make a deep structure. Some presentations dealt with addressing the conventional problems in image recognition through deep learning, such as methods to implement multi-layer or multi-instance learning in order to achieve flexibility through shape change [43], implementing optimization and repetition through Bayesian Optimization in the periphery of highly likely candidate regions in order to correct errors in object detection using R-CNN [28], proposal of a robust expression to withstand positional invariability and deformations [108] [372], inputting images results in an output wherein images segmented by pixel are outputted as fully connected layers. [168] reports on the use of deep learning for morphing. A corresponding model of the chair is generated when the type of chair, camera viewpoint, and conversion parameters are inputted. There was also a paper on the output of multiple attributes from deep learning for crowded human environment [504]. It includes 94 attributes, and tags explaining where, what kind of person, what is the person doing, etc. are returned. Also, there was a research on visualization of features of each layer of CNN [562], pointing to progress in the understanding of deep learning. A prominent session in the oral presentations for the area of recognition is Images and Language for image generation captions (image descriptions). In recent years, the level of research in this area has increased due to progress in research on recognition performance and natural language processing [339,342,285,296,324,254,347,161]. The Action and Event Recognition session formerly dealt with saliency and extension of human action recognition. Shu et al. reported on taking aerial videos using drones (unmanned aerial vehicles (UAV)) to extract human lines of movement and recognize their group actions [495]. Fernando et al. proposed Video-Darwin as a mechanism for capturing slowly changing actions in videos [583]. Ma et al. expressed the hierarchy of each part of action recognition through integration of tree structures [544]. Khatoonabadi et al. [596] presented a method on saliency and Park et al. presented a method on social saliency prediction [517]. In [596], they presented a method to achieve saliency and segmentation while reducing amount of information, based on image compression methods. Social saliency prediction [517] infers the area where attention is focused based on gaze directions of multiple persons. 3D With DynamicFusion [38] at the head of the list, new methods on 3D were proposed. DynamicFusion is a method for conducting more precise 3D shape restoration in real time by chronologically integrating depth images obtained through Kinect and other methods. SUN RGB-D [62] was proposed as a large-scale data set that captures indoor space in three-dimension. Their dataset contains a total of 10,335 RGB-D images, and they presented several important issues on the topic. Research on occlusion using 3D models has also progressed. Xiang et al. [207] used 3D Voxel Patterns (3DVP) to carry out 3D detection and enabled detection of missing parts of objects where occlusion or defects have occurred, using a model-based method. Reconstructing the World in Six Days is an example of research on large-scale space [356]. They carried out 3D reconstruction through world-scale SfM of 100 million images of worldwide landmarks found in flickr. Hengel et al. used silhouette obtained from multiple images to carry out meaningful resolution of each part of the 3D model [96]. The 3D structure was realized through Block World [611]. Richter et al. proposed a method for discriminatively resolving Shape-from-Shading [123]. Albl et al. came up with a mechanism for properly operating, in global shutters as well as in local shutters, the perspective-n-point problem (PnP), which is considered important in SfM, inferring camera position, and calculating odometry [249]. Due to the problem of having an arbitrary rotation matrix, in their proposed method, they proposed an R6P algorithm to make more dense calculation of rotation matrix. Song et al. proposed a method to infer the 3D location of vehicles using monocular cameras [404]. Kulkarni et al. proposed Picture (Probabilistic Programming Language), which is a stochastic expression of 3D modeling, to enable expressing a more complicated generation model [475]. Wang et al. conducted 3D scene recognition in outdoor environment using GPS positional information as preliminary data [429]. CRF was applied to assign segments and their significance to 3D positional information. Barron et al. devised an optimized method to enable generation of effective stereo images [483]. Use of Defocus and Fast Bilateral Filter eliminates the need for calculating all corresponding points. Wang et al. devised a method for searching the 3D model from the 2D sketch [204]. A sketch image as seen from multiple perspectives is generated from the 3D model of one sample, and a 3D model is searched through comparison with inputted sketch image and presented to the user. Brubaker et al. carried out 3D molecular model reconstruction of highresolution image from low-resolution image using electron cryomicroscopy [336]. Chin et al. realized improvement of robust matching such as RANSAC through optimization by A*search [262]. Image processing/imaging In regard to image processing and imaging, advances in research through new themes were seen. For example, Tanaka et al. presented their research resolution of paintings that are physically separated into multiple layers, such as pencil sketches or colored paintings [592], enabling the extraction of even deeper components. [554] presented the problem of finding an efficient border ownership (where the borderline is, whether an area is part of the background or foreground) in 2D images. The authors addressed the problem by using structural random forests (SRF) to differentiate borders. The problem regarding realizing photometric stereo under natural light rather than controlled light sources was also presented [489]. In order to apply photometric stereo in outdoor environment, the authors assumed a hemispherical experimental space and used GPS timestamp as preliminary information, and separately carried out light source estimation of sunlight. There were several proposals regarding the problem of inferring depth images from input images and videos, as well as a paper on simultaneous solution for image correction from fogged images and for inference of depth images [540]. Research on super-resolution was also included in the oral presentations [563]. The authors used self-similarity based super-resolution, and at the same time carried out inference of affine transformation parameters and localized shape variations. CVPR2015 as seen by area of study In the previous section we looked at CVPR2015 based on the titles and papers selected for oral presentation. In this section we will enumerate papers in more detail by area of study. Here we will comprehend the current trends in the field of computer vision by looking at all papers, regardless of whether they were presented orally or as posters. Recognition Deep learning architecture. We will cite papers that discuss the overall structure, as well as those that deal with problem-based structures, parameter adjustments, and architecture evaluation. Two examples of papers that discuss overall structure are on GoogLeNet [1] and DeepID-Net [261]. DeepID-Net uses Deformation Constrained (Def) pooling as alternative to max pooling and average pooling in order to improve expressiveness against changes in shape and position, as in DPM [616], contributing to improvement of accuracy in object detection. There were also many examples of attempts to carry out improvements under the framework of existing CNN methods [48, 405,93]. Wan et al. combined the advantages of DPM and CNN and, further, implemented Non-maximum Suppression (NMS) in order to correct effects of positional discrepancies [93]. DPM is a method for preserving parts and position in latent variables, while CNN has the advantage of being able to automatically learn features that are useful for object recognition. Other papers dealt with the characteristics of CNN [47,43,108], increasing speed of learning [88], initiatives to search for parameters [580], and visualization of features [562]. Lenc et al. carried out robust CNN feature expression to address image rotation by implementing a transformation layer for geometric transformation of convoluted features [108]. Liu et al. succeeded in reducing computational complexity and CNN calculation time by implementing sparse representation to address convolution [88]. They succeeded in significantly reducing calculation time by sparsing of kernels computed at every convolution, and improved calculation to enable operation even on a CPU. He et al. studied depth of structure, filter size, stride, and other trade-offs pertaining to CCN architectural parameters [580], and showed that depth is important. Other papers dealt with improvement of convolution layers [365], method to calculate similarity of patches [355,471], and research on morphing under the CNN framework [168]. Liang et al. claimed that better features can be obtained if CNN convolution frameworks are recursively convoluted [365]. This structure is called Recurrent Convolutional Layter (RCL). In MatchNet, architecture is configured for the purpose of measuring similarity between patches, and is partitioned to a network for generating features through pooling and convolution of four layers and a network for evaluating similarity through total combination of three layers [355]. Zagoruyko et al. also discussed a framework for calculating patch similarities in CNN [471]. They extracted the features based on convolutions of paired patches and calculated similarity in the later layers. Human recognition. We will introduce papers in Human Recognition by dividing them into Face Recog-nitionPedestrian DetectionHuman TrackingPose Esti-mationAction RecognitionEvent RecognitionCrowd Anal-ysisEgocentric Visionand Person Re-identification. First, in face recognition, FaceNet was presented as a system for handling high-precision recognition [89]. DeepFace, which has been recently proposed in 2014 [619], brought about significant improvements in accuracy, but FaceNet has achieved an even higher accuracy than DeepFace. Sun et al. improved their conventional face recognition, DeepNet [623], and applied features extracted from early convolution layers to improve face recognition accuracy particularly of face profiles and occlusions [314]. In pedestrian detection, Tian et al. were able to improve accuracy by combining CNN features and attributes for detection of pedestrians [550]. They accomplished this by including other attributes, such as positional relationships between pedestrians and environment, as well as learning of pedestrians and backgrounds. Honsang et al. implemented evaluation of features using CNN to carry out pedestrian detection [441]. In pose estimation, a research on marker-less motion capture using CNN features was presented [412]. For practical use, it is possible to significantly reduce installation costs if estimation can be implemented through maker-less MoCap using 23 cameras. In human tracking, there were reports featuring more advanced methods. Milan et al. were able to simultaneously carry out tasks of chronological area estimation and positioning by using Superfixel and CRF [585]. They established a method for combining low-and highlevel information and finely dividing background and foreground. A method for carrying out accurate tracking of multiple objects using Target Identity-aware Network Flow (TINF), which probabilistically resolves network nodes, was also presented [125]. The method constructs the optimum network using graph theory and carries out optimization through Lagrangian relaxation optimization. In action recognition, Gkioxari et al. used R-CNN [609] as basis for proposing a mechanism for recognizing actions, including position of the human subjects [83]. In order to extract the action area, candidate areas where extracted from an assembly of optical flows to extract CNN-based features. And in order to extract features from chronological actions, convolution was implemented for chronological images that stored optical flows and RGB visible images. To improve accuracy, researchers proposed a method based on Dense Trajectories (DT) [620,621] and on TDD, an action descriptor that combines CNN features [583]. In regard to the DT-based method, researchers adopted HOG, HOF, and MBH to accurately recognize actions, as well as applied CNN features to action recognition through normalization of the feature map. Lan et al. In event recognition, architecture specialized for event recognition called Deep Event Network (DevNet) was proposed [279]. The system enabled extracting not only pre-defined events, but also clues for important chronological events. Xiong et al. carried out recognition of complex events by combining multiple identification results and factors for still images and combined CNN features and results of object/human/face detection results to recognize events [175]. Shu et al. carried out event recognition from aerial images taken using unmanned aerial vehicles (UAV) [495]. They proposed a Space-time AND-OR Graph to analyze various clues from images from drones, such as positional adjustment of images containing egomotion, group action recognition, and human interaction. In crowd analysis, a mechanism that allows crossscene crowd counting was proposed [91]. They used a CNN model that allows switching the crowd density map and human count model. Although these two models are different, they are correlated and complement each others accuracy. Yi et al. analyzed crowd models from videos taken from surveillance cameras and measured routine pedestrian path directions [378]. They predicted crowd attributes and pedestrian destinations and enabled detection of abnormal actions as well as prediction of paths taken to reach destinations. A method for editing ones own videos taken using egocentric vision was also proposed [590]. Research to solve face recognition problems, such as recognition of severely occluded faces and small and far faces in images, has progressed. Huang et al. proposed a hand region segmentation method for egocentric vision to determine what tasks the person taking the video is performing [73]. Person re-identification deals with the problem of personal authentication between different cameras for surveillance and other cameras. Shi et al. inferred semantic attributes regarding humans and clothing at the patch level, and applied them in person re-identification [453]. They obtained clothing and other external appearance features and were able to improve expressivity by us-ing attributes. Chen et al. carried Multiple Similarity Function Learning using PCA compression color and texture features from images with segregated regions [171]. Zheng et al. evaluated effectiveness of features and enabled feature integration needed for Re-ID using Late Fusion [190]. Person re-identification using low-resolution images was also addressed [76]. Generally, images from surveillance cameras are of poor quality, and to address this, Jing et al. carried out superresolution to propose a mechanism for improving performance even for low-resolution images. Neural network architecture to improve robustness against feature variations between cameras was also proposed [423]. Given a pair of images as input, the authors used the difference of activation functions extracted from each patch after convolution and pooling as features for recognition. Object recognition and detection. The problem of recognizing objects appearing in images is currently an intensively studied area. This section also deals with object detection that includes recognition of position, scene recognition, search of hashed images, as well as fine-grained image recognition. Papers on object recognition have dramatically increased after AlexNet was proposed [614] at ILSVRC2012, and object recognition has also been applied to scene recognition and other problems. Research on object detection expanded after the proposal of R-CNN [609]. These trends are clearly evident in CVPR2015. A study was conducted to improve accuracy and streamline recognition by carrying out selection of CNN factors [106]. Association Rules [626] widely used in the data mining field were applied, and only features that are useful for identification were selected as a subset from among the CNN feature space. In object detection, there were many researches addressing the problem of inaccurate localization, which is one of the vulnerabilities of R-CNN. As previously mentioned, Zhang et al. proposed a method for optimization to correct inaccurate localization in R-CNN to address this vulnerability [28]. Tsai et al. considered the diversity of internal changes and variations of objects for detection, and compensated for inaccurate localization by improving feature pooling [80]. Oquab et al. used weakly supervised learning to investigate solutions for discrimination and localization of objects based only on labeling of image levels [75]. Fine-grained image discrimination is a problem that entails more detailed classification of objects, such as dog breeds or vehicle types. Due to high visual similarity of objects, such detailed classification is very difficult to carry out. It was found that adaptively extracting features useful for discrimination by dividing images into parts and extracting features only from particular regions is an effective method [630]. Using CNN architecture, Xiao et al. extracted candidate patches from major categories (e.g. dog, bird) and detailed categories (e.g. fine classification of dogs and birds) in a layered structure, and simultaneously implemented feature selection and discrimination [92]. Xie et al. carried out learning by applying multitask learning in multiple structured classes as well as in limited task data extensions [287]. They succeeded in simultaneously learning relationships through multitask learning of major and minor classifications. Lin et al. [182] proposed Deep Localization, Alignment and Classification (DeepLAC) as a mechanism to correct changes in regional position and angles, which is needed for finegrained image recognition, within the back-propagation algorithm framework. Segmentation. Segmentation requires implementing object recognition at the pixel level, making it a difficult procedure in terms of distinguishing borders between foreground and background. The number of papers dealing with semantic segmentation, which deals with assignment of meaning to segmentation areas, has increased. Hariharan et al. demonstrated the increase in accuracy of semantic segregation by using features extracted in the middle layers, not only from the fully connected layer, in regard to CNN architecture [49]. In particular, they used the 2nd pooling layer, the 4th convolution layer, and the 7th fully connected layer, and by combining these they were able to simultaneously implement low-, mid-, and high-level feature expression. In saliency-based segmentation, a method was proposed for extracting multi-scale CNN features [591]. Itti et al.s saliency model is well known [627], and, although they conducted multi-scale calculations, Li et al. extracted saliency and applied it segmentation by replacing CNN features. Although it overlaps with 3D reconstruction, we would like to mention here that Martinovic et al. proposed research for implementing semantic segmentation of 3D urban models [482]. Data generation. Data generation is an important issue in addressing recognition problems. In this section we will cite papers on data collection and selection. Hattori et al. generated learning images for pedestrian detection [413]. They conducted learning of 36 types of pedestrians, various kinds of walking, and occlusion patterns using CG. Russakovsky et al. cited an annotation method leveraging crowdsourcing, in order to efficiently and accurately detect objects [231]. The method deals with the usability and accuracy of labeling and is aimed at minimizing human annotation costs, wherein machines and humans interactively carry out annotation based on results from baseline recognition equip-ment. Xiao et al. discussed a framework for efficient labeling and learning, in an effort to reduce annotation operations for massive data [292]. 3-Dimension There were also many examples of applications of CNN even for 3D object recognition. Fang et al. proposed Deep Shape Descriptor (DeepSD) as a method for expressing 3D shapes [252]. They proposed a robust 3D feature that can handle structural variations in shape, noise, and shapes that include three-dimensional incompleteness. Xie et al. proposed DeepShape, a CNN feature to address problems in 3D object matching and retrieval [139]. They used a shape descriptor based on an auto-encoder to search 3D shapes. Abdelrahman et al. proposed a 3D non-rigid texture descriptor based on Weighted Heat Kernel Signature (W-HKS) [21]. There was also a proposal for a mechanism to extract information useful for recognition even from a limited learning sample using Deep Boltzmann Machine (DBM) and design of object recognition features through RGB-D [327]. They proposed an effective descriptor even for complex 3D objects by combining geometric shape information as well as color information. In RGB-D input, a problem was reported in giving tasks, such as 3D recognition and inferring positions that can be grasped by robots, in complex indoor environment [498]. Superfixel was applied as a preliminary processing step, and recognition of cuboid models and spatial smoothing through Conditional Random Fields (CRF) was carried out. Matsuo et al. also proposed a method for enhancing depth images (particularly planes) by combining low-resolution depth images and high-resolution RGB images [387]. They adjusted position and connection of tangent planes in 3D space and used JBU filter to reconstruct rough surfaces. Gupta et al. conducted research on extracting object position and 3D segmentation results from RGB-D image input [512]. They expressed object features through learning by CNN of surface normal line images. They then roughly estimated object pose based on 3-layered CNN and inferred detailed object pose and segment by comparison with the 3D model. Image processing/imaging CNN was also used for blur removal [84]. Non-uniform motion blurs arising from shaking of camera, etc. were corrected through learning of blurred/non-blurred patch pairs. There was also a research on fusion of multiple kernels [41]. The authors adopted a method using kernels for fusing multiple deblurring methods in order to develop a more advanced blur removal method. By using Gaussian Conditional Random Fields (GCRF), they were able to carry out kernel fusion based on learning. Eriksson et al. proposed a method for noise removal that takes sparsity into consideration [363]. To solve the k-support norm optimization and normalization problem, Eriksson et al. carried optimization by considering this problem as the minimum convex set that includes the set given as Convex Envelopes. Research on blur removal for videos was also reported [437]. There are two methods for blur removal for videos. One is by independently removing blur within the frames and splicing the frames together. The other is by inferring camera motion between frames. Zhang et al. combined these two methods. In regard to the problem of super-resolution, a method using Self-Similarity based Super-Resolution was reported [563]. The method simultaneously infers affine transformations and localized shape variations. Comparison with external/internal dictionaries enabled mapping to clear images. A method using a reference dictionary that accommodates shape variations was also reported to address the super-resolution problem for single images [587]. Gradient Ridge Image processing was performed as a preliminary processing step, and resolution was enhanced through matching with the dictionary. Schulter et al. solved the single-image superresolution problem as a linear regression problem using Random Forests [410]. A method for inferring shadow regions using CNN was reported for basic algorithms in image processing [225]. Shen et al. also proposed DeepContour, which is a CNN architecture for contour detection [431]. Deep-Contour involves learning contour/non-contour regions and is composed of a 6-layered architecture (four convolution layers and two fully connected layers). DeepEdge was also proposed as an application of CNN architecture for edge detection [474]. DeepEdge carries out more accurate edge detection by using higher-level features. Experimentally, they were able to show that unlike Can-nyEdge, where there was noise contamination, DeepEdge was able to better remove backgrounds as well as extract edges from objects. Teo et al. also proposed a method for effectively extracting borderlines in 2D images [554]. By using Structural Random Forests (SRF), they were able to rapidly determine where the borders are, and whether the area belongs to the background or the foreground. A research on the application of Linear Spectral Clustering (LSC) to Superpixel methods was also presented [148]. In device research, a hyperspectral camera that can acquire chronological images was proposed [535]. Sequences of multiple hyperspectral cameras were alter-nately complemented, and image reconstruction based on dictionary learning was conducted, in order to obtain clear images even at high-speed (100 fps) observation. Ti et al. developed a ToF sensor using a monocular camera and LED [469]. They developed the ToF sensor by attaching a total of four LEDs to the upper, lower, right, and left sides of the camera and capturing the reflection of LED light using the camera. To improve accuracy of ToF cameras, Naik et al. resolved the problem of Multipath Interference (MPI), where multiple optical reflections appear and are mixed up in the pixel [9]. MPI also occurs in natural scenes, such as in an environment where multiple reflected lights occur or reflected light is diffused. These reflections were divided into Phase and Amplitude, both directly and globally, in order to reduce depth image errors due to MPI. Ye et al. proposed an enhanced Kinect sensor by attaching Ultrasonic Sensor to Kinect [529]. They inferred the plane by applying Bayesian Network to the inference point obtained through the Ultrasonic Sensor. Datasets In this section, we will discuss new research problems as well as research on datasets. Datasets. An example of a dataset is the SUN RGB-D, an expansion of SUNdatabase (which is a problem in scene recognition mentioned in the previous section) to RGB-D [62]. It is an attempt to expand the data set into more advanced scene recognition, such as segmentation and detection of objects within scenes, other than merely for recognition. A similar research problem is on the dataset for estimating indoor layout proposed by Liu et al [370]. The dataset for indoor environment included information on the entire room, walls, doors, windows, and their positional information. A research for outputting detailed explanations of medical images was also reported [119]. This research problem pertained to outputting sentence descriptions from an input of medical images. Detailed explanations of symptoms are generated by learning in pairs the actual medical images and the corresponding medical examination results. There was also an attempt to increase recognition capability by creating a much larger-scale dataset in the field of fine-grained recognition [65]. NABirds is a dataset for fine-grained recognition of birds, the scope of which was expanded by increasing the number of classes. There was also a report on a dataset for categorizing cars [430]. The study provided data for fine-grained classification of cars, which previously were only categorized into the class called cars. There was also a study on creating data for detection of pedestrians through the use of images contain-ing a higher amount of information. Hwang et al. used a hyperspectral camera to acquire richer image information in order to improve detection of pedestrians at nighttime as well as daytime [113]. A dataset was also proposed for analyzing each person in a crowd by focusing on the spectators rather than on the sport itself [222]. They analyzed individual reactions of persons in a crowd, categorized crowds, and determined the type of spectators. In regard to pedestrian detection, a dataset was proposed for estimating gender, age, weight, clothing, etc., of pedestrians as well their location [594]. This dataset is intended for fine-grained recognition of persons. Thus, there was more focus on addressing fine-grained detection of pedestrians. There is more research being conducted on generation of image descriptions, with one oral session devoted to the topic. In particular, Rohrbach et al. proposed a dataset for movie description [347]. For action recognition datasets, Heilbron et al. published a dataset called ActivityNet [105], which is a large-scale dataset similar to ImageNet and includes a significantly greater amount of data and action variations. The dataset includes 203 trimmed data classes and 137 untrimmed classes, for a total of 849 video hours. Also in action recognition, Xu et al. proposed a dataset that maps attributes in advance to actors and actions [246]. New research problems. Here we introduce new research problems proposed at CVPR2015. Lin et al. proposed the research problem of identifying locations of aerial images using images taken on the streets as query [542]. Although ground images and aerial images are completely different in nature, the authors presented a possible approach to the problem by proposing Where-CNN. Akhter et al. conducted estimation of 3D human pose from 2D joint angles, and by adding a joint angle limit they were able to add a process for reducing poses with inscrutable motion [158]. Peng et al. proposed two new aspects on human emotions predicted from images [94]. There was a paper on detecting persons or animals in a best relationship, i.e., with a high co-occurrence relationship with another person or animal based on Best-Buddies Similarity [220]. The authors proposed a method based on template matching to visualize the co-occurrence relationship. There was also a paper that addressed the problem of identifying very important people (VIP) within a group [526]. The authors used im2text to solve the problem by classifying level of importance of images and texts. Traditional machine learning methods map input and output vectors as pairs, but Wang et al. assigned hidden information to images to further improve flexibility [538]. On the basis of this concept, they proposed that hidden information be handled as features or second objective functions. Zhang et al. proposed a method to address the problem of counting items in an image as well as finding saliency from images [438]. They claimed that it can be used for egocentric lifelogs and image thumbnails. Not only in sensing, but there will also be a need to carry out person recognition in next-generation camera images whose resolution has been lowered for security and privacy protection. This problem is addressed by Pittaluga et al. by carrying out face and pose recognition that can handle low-resolution images and resist changes in light source, proposing the method to be used for privacy protection [35]. There was also a proposal on object recognition that takes into consideration what kind of tasks are completed using particular tools [310]. The authors constructed 3D models of objects using 3D sensors and inferred the position by which the person carries the object based on joint angle, and measured how the task is being carried out. Measurement was made not only on 2D and 3D images, but they also calculated the impulse strength using voice data. Handling of the tool was inferred based on joint angle trajectory. There was also a proposal for inferring what a store is selling based on the storefront image [185]. Streetview images were used to extract characters through OCR, and ontology from those characters was used to classify stores according to business category. DeepSurvey We are proposing DeepSurvey (see Figure 2) as a mechanism for the systematization of knowledge, the generation of ideas, and as well as the writing of papers (specially for new research problems) based on an extensive reading of papers. DeepSurvey architecture is devised based on DeepLearning, which has flourished in recent years, and is composed of the following elements: -Input: Input the papers read (knowledge) -1st ideas: Individually generate ideas (from knowledge to ideas) -1st discussion: Group discussion (consolidation of ideas) -2nd ideas: Generate more ideas based on consolidated ideas -2nd discussion: Further refinement of ideas -1st implementation: Pick-up and hackathon -2nd implementation: Full-scale implementation and experiment -Output: Paper In comparison with general Convolutional Neural Networks (CNN) [631], ideas can be replaced with convolution layer, discussion with pooling, and implementation with fully connected layer to make it easier to understand. In pooling (discussion), multiple ideas are collected and good ideas are inputted as they are to the next layer, thus, it is closely similar to Lp pooling, which simultaneously possesses characteristics of max pooling and average pooling. The strategy is to repeat generation of ideas and discussion, and proceed to implementation once ideas have taken shape. The current counting of layers include convolutional layers and fully connected layers, thus, the architecture is a four-layer configuration. The most important feature of this architecture is the method for "becoming a part of the neuron." Under this framework, since the entire group works as one neural network architecture in real, rather than in virtual space, the group is able to write papers as the final output. (Thankfully, we got first output of DeepSurvey [634] which includes a conceptual subject integrating semantic segmentation into change detection.) It is also characterized by project members actually doing the thinking, reading, and writing of papers to enable them to grow, wherein the network itself grows and matures. For 2015, there was little time left for implementation and writing of papers, but we would like to write a more refined paper in the next year as well as be able to propose new research problems. Recently, since the structure of the architecture is also becoming deeper (VGGNet [632]: 16/19 layers; ResNet [633] 50/101/152 layers), going forward, we would like to generate more ideas, hold more discussions, and produce more refined ideas, research problems, and papers. Summary and future trends In this survey we comprehensively read papers presented at CVPR2015 to gain an understanding of the trends in computer vision. Further, we devised Deep-Survey as a mechanism to generate ideas from knowledge and eventually write a paper. We divided the papers into three areas; namely, recognition, 3D, and imaging/image processing, and sought to identify new research areas, as a means to expand the limits of the field. Here we are proposing DeepSurvey, and, going forward, we have started addressing some of its problems. The authors are sorting out the current issues and believe that conducting surveys that include a study of technologies is essential also for identifying the next research problems. Further, there is a need to gain the ability to view the field from a wider perspective aside from actually testing the survey results to better understand the issues. We hope that this initiative would serve as a useful step towards that end. Fig. 1 1Example of weighting and visualization based on titlesD proposed Multi-skip Feature Stacking (MIFS), a method for extract features by configuring multiple gradations to a chronological offset [23]. Fig. 2 2DeepSurvey architecture: ( ) shows the actual number of papers and the number of ideas and implementations. , etc. Meanwhile, Nguyen et al. automatically generated features that are mistakenly recognized by deep learning and showed that CNN features are not universal [47]. In Long et al.s segmentation method (FCN) 17 . 17Tianzhu Zhang, Si Liu, Changsheng Xu, Shuicheng Yan, Bernard Ghanem, Narendra Ahuja, Ming-Hsuan Yang, "Structural Sparse Tracking", in CVPR2015. 18. HyeokHyen Kwon, Yu-Wing Tai, Stephen Lin, "Data-Driven Depth Map Refinement via Multi-Scale Sparse Representation", in CVPR2015. 19. Feng Lu, Imari Sato, Yoichi Sato, "Uncalibrated Photometric Stereo Based on Elevation Angle Recovery From BRDF Symmetry of Isotropic Materials", in CVPR2015. 20. Ran Tao, Arnold W.M. Smeulders, Shih-Fu Chang, "Attributes and Categories for Generic Instance Search From One Example", in CVPR2015. 21. Mostafa Abdelrahman, Aly Farag, David Swanson, Moumen T. El-Melegy, "Heat Diffusion Over Weighted Manifolds: A New Descriptor for Textured 3D Non-Rigid Shapes", in CVPR2015. 22. Christopher Zach, Adrian Penate-Sanchez, Minh-Tri Pham, "A Dynamic Programming Approach for Fast and Robust Object Pose Recognition From Range Images", in CVPR2015. 23. Zhengzhong Lan, Ming Lin, Xuanchong Li, Alex G. Hauptmann, Bhiksha Raj, "Beyond Gaussian Pyramid: Multi-Skip Feature Stacking for Action Recognition", in CVPR2015. 24. Dongping Li, Kaiming He, Jian Sun, Kun Zhou, "A Geodesic-Preserving Method for Image Warping", in CVPR2015. 25. Shaoxin Li, Junliang Xing, Zhiheng Niu, Shiguang Shan, Shuicheng Yan, "Shape Driven Kernel Adaptation in Convolutional Neural Network for Robust Facial Traits Recognitio", in CVPR2015. 26. Marko Ristin, Juergen Gall, Matthieu Guillaumin, Luc Van Gool, "From Categories to Subcategories: Large-Scale Image Classification With Partial Class Label Refinement", in CVPR2015. 27. 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Christian Hane, ?ubor Ladicky, Marc Pollefeys, "Direction Matters: Depth Estimation With a Surface Normal Classifier", in CVPR2015. 43. George Papandreou, Iasonas Kokkinos, Pierre-Andre Savalle, "Untangling Local and Global Deformations in Deep Learning: Epitomic Convolution, Multiple Instance Learning, and Sliding Window Detection", in CVPR2015. 44. Yezhou Yang, Cornelia Fermuller, Yi Li, Yiannis Aloimonos, "Grasp Type Revisited: A Modern Perspective on a Classical Feature for Vision", in CVPR2015. 45. Sheng Huang, Mohamed Elhoseiny, Ahmed Elgammal, Dan Yang, "Learning Hypergraph-Regularized Attribute Predictors", in CVPR2015. 46. Roozbeh Mottaghi, Yu Xiang, Silvio Savarese, "A Coarse-to-Fine Model for 3D Pose Estimation and Sub-Category Recognition", in CVPR2015. 47. Anh Nguyen, Jason Yosinski, Jeff Clune, "Deep Neural Networks Are Easily Fooled: High Confidence Predictions for Unrecognizable Images", in CVPR2015. 48. 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Generic programming (GP) is an increasingly important trend in programming languages. Well-known GP mechanisms, such as type classes and the C++0x concepts proposal, usually combine two features: 1) a special type of interfaces; and 2) implicit instantiation of implementations of those interfaces.Scala implicits are a GP language mechanism, inspired by type classes, that break with the tradition of coupling implicit instantiation with a special type of interface. Instead, implicits provide only implicit instantiation, which is generalized to work for any types. This turns out to be quite powerful and useful to address many limitations that show up in other GP mechanisms.This paper synthesizes the key ideas of implicits formally in a minimal and general core calculus called the implicit calculus (λ⇒), and it shows how to build source languages supporting implicit instantiation on top of it. A novelty of the calculus is its support for partial resolution and higher-order rules (a feature that has been proposed before, but was never formalized or implemented). Ultimately, the implicit calculus provides a formal model of implicits, which can be used by language designers to study and inform implementations of similar mechanisms in their own languages. . According to that study many languages provide some support for GP. However, Haskell did particularly well, largely due to type classes. A direct consequence of that work was to bring the two main lines of work on GP closer together and promote crosspollination of ideas. Haskell adopted associated types[4,3], which was the only weak point found in the original comparison. For the C++ community, type classes presented an inspiration for developing language support for concepts[23,11,34].Several researchers started working on various approaches to concepts (see Siek's work [33] for a historical overview). Some researchers focused on integrating concepts into C++[7,11], while others focused on developing new languages with GP in mind. The work on System F G [34, 35] is an example of the latter approach: Building on the experience from the C++ generic programming community and some of the ideas of type classes, Siek and Lumsdaine developed a simple core calculus based on System F which integrates concepts and improves on type classes in several respects. In particular, System F G supports scoping of rules 1 .During the same period Scala emerged as new contender in the area of generic programming. Much like Haskell, Scala was not originally developed with generic programming in mind. However Scala included an alternative to type classes: implicits. Implicits were initially viewed as a poor man's type classes[26]. Yet, ultimately, they proved to be quite flexible and in some ways superior to type classes. In fact Scala turns out to have very good support for generic programming[28,29].A distinguishing feature of Scala implicits, and a reason for their power, is that resolution works for any type. This allows Scala to simply reuse standard OO interfaces/classes (which are regular types) to model concepts, and avoids introducing another type of interface in the language. In contrast, with type classes, or the various concept proposals, resolution is tightly coupled with the type class or concept-like interfaces.Limitations of Existing MechanismsTwenty years of programming experience with type classes gave the FP community insights about the limitations of type classes. Some of these limitations were addressed by concept proposals. Other limitations were solved by implicits. However, as far as we know, no existing language or language proposal overcomes all limitations. We discuss these limitations next.Global scoping: In Haskell, rules 2 are global and there can be only a single rule for any given type[18,2,6,8]. Locally scoped rules are not available. Several researchers have already proposed to fix this issue: with named rules[18]or locally scoped ones[2,6,8]. However none of those proposals have been adopted.Both proposals for concepts and Scala implicits offer scoping of rules and as such do not suffer from this limitation.Second class interfaces:Haskell type classes are second-class constructs compared to regular types: in Haskell, it is not possible to abstract over a type class[13]. Yet, the need for first-class type classes is real in practice. For example, Lämmel and Peyton Jones [21] desire the following type class for their GP approach:class (Typeable α, cxt α) ⇒ Data cxt α where gmapQ :: (∀β.Data cxt β ⇒ β → r ) → α → [r ]

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A New Foundation for Generic Programming 20 Mar 2012 Bruno C D S Oliveira Seoul National University Universiteit Gent Seoul National University Seoul National University Seoul National University Tom Schrijvers [email protected] Seoul National University Universiteit Gent Seoul National University Seoul National University Seoul National University Wontae Choi [email protected] Seoul National University Universiteit Gent Seoul National University Seoul National University Seoul National University Wonchan Lee [email protected] Seoul National University Universiteit Gent Seoul National University Seoul National University Seoul National University Kwangkeun Yi Seoul National University Universiteit Gent Seoul National University Seoul National University Seoul National University A New Foundation for Generic Programming 20 Mar 2012Extended Report: The Implicit CalculusD32 [Programming Lan- guages]: Language Classifications-Functional LanguagesObject- Oriented Languages; F33 [Logics and Meanings of Programs]: Studies of Program Constructs General Terms Languages Keywords Implicit parameterstype classesC++ conceptsgeneric programmingHaskellScala Generic programming (GP) is an increasingly important trend in programming languages. Well-known GP mechanisms, such as type classes and the C++0x concepts proposal, usually combine two features: 1) a special type of interfaces; and 2) implicit instantiation of implementations of those interfaces.Scala implicits are a GP language mechanism, inspired by type classes, that break with the tradition of coupling implicit instantiation with a special type of interface. Instead, implicits provide only implicit instantiation, which is generalized to work for any types. This turns out to be quite powerful and useful to address many limitations that show up in other GP mechanisms.This paper synthesizes the key ideas of implicits formally in a minimal and general core calculus called the implicit calculus (λ⇒), and it shows how to build source languages supporting implicit instantiation on top of it. A novelty of the calculus is its support for partial resolution and higher-order rules (a feature that has been proposed before, but was never formalized or implemented). Ultimately, the implicit calculus provides a formal model of implicits, which can be used by language designers to study and inform implementations of similar mechanisms in their own languages. . According to that study many languages provide some support for GP. However, Haskell did particularly well, largely due to type classes. A direct consequence of that work was to bring the two main lines of work on GP closer together and promote crosspollination of ideas. Haskell adopted associated types[4,3], which was the only weak point found in the original comparison. For the C++ community, type classes presented an inspiration for developing language support for concepts[23,11,34].Several researchers started working on various approaches to concepts (see Siek's work [33] for a historical overview). Some researchers focused on integrating concepts into C++[7,11], while others focused on developing new languages with GP in mind. The work on System F G [34, 35] is an example of the latter approach: Building on the experience from the C++ generic programming community and some of the ideas of type classes, Siek and Lumsdaine developed a simple core calculus based on System F which integrates concepts and improves on type classes in several respects. In particular, System F G supports scoping of rules 1 .During the same period Scala emerged as new contender in the area of generic programming. Much like Haskell, Scala was not originally developed with generic programming in mind. However Scala included an alternative to type classes: implicits. Implicits were initially viewed as a poor man's type classes[26]. Yet, ultimately, they proved to be quite flexible and in some ways superior to type classes. In fact Scala turns out to have very good support for generic programming[28,29].A distinguishing feature of Scala implicits, and a reason for their power, is that resolution works for any type. This allows Scala to simply reuse standard OO interfaces/classes (which are regular types) to model concepts, and avoids introducing another type of interface in the language. In contrast, with type classes, or the various concept proposals, resolution is tightly coupled with the type class or concept-like interfaces.Limitations of Existing MechanismsTwenty years of programming experience with type classes gave the FP community insights about the limitations of type classes. Some of these limitations were addressed by concept proposals. Other limitations were solved by implicits. However, as far as we know, no existing language or language proposal overcomes all limitations. We discuss these limitations next.Global scoping: In Haskell, rules 2 are global and there can be only a single rule for any given type[18,2,6,8]. Locally scoped rules are not available. Several researchers have already proposed to fix this issue: with named rules[18]or locally scoped ones[2,6,8]. However none of those proposals have been adopted.Both proposals for concepts and Scala implicits offer scoping of rules and as such do not suffer from this limitation.Second class interfaces:Haskell type classes are second-class constructs compared to regular types: in Haskell, it is not possible to abstract over a type class[13]. Yet, the need for first-class type classes is real in practice. For example, Lämmel and Peyton Jones [21] desire the following type class for their GP approach:class (Typeable α, cxt α) ⇒ Data cxt α where gmapQ :: (∀β.Data cxt β ⇒ β → r ) → α → [r ] Introduction Generic programming (GP) [23] is a programming style that decouples algorithms from the concrete types on which they operate. Decoupling is achieved through parametrization. Typical forms of parametrization include parametrization by type (for example: parametric polymorphism, generics or templates) or parametrization by algebraic structures (such as a monoid or a group). A central idea in generic programming is implicit instantiation of generic parameters. Implicit instantiation means that, when generic algorithms are called with concrete arguments, the generic arguments (concrete types, algebraic structures, or some other form of generic parameters) are automatically determined by the compiler. The benefit is that generic algorithms become as easy to use as specialized algorithms. To illustrate implicit instantiation and its benefits consider a polymorphic sorting function: sort [α] : (α → α → Bool ) → List α → List α with 3 parameters: the type of the elements in the list (α); the comparison operator; and the list to be compared. Instantiating all 3 parameters explicitly at every use of sort would be quite tedious. It is likely that, for a given type, the sorting function is called with the same, explicitly passed, comparison function over and over again. Moreover it is easy to infer the type parameter α. GP greatly simplifies such calls by making the type argument and the comparison operator implicit. isort : ∀α.(α → α → Bool ) ⇒ List α → List α The function isort declares that the comparison function is implicit by using ⇒ instead of →. It is used as: [2,1,3], isort [5,9,3]) implicit {cmpInt : Int → Int → Bool } in (isort The two calls of isort each take only one explicit argument: the list to be sorted. Both the concrete type of the elements (Int) and the comparison operator (cmpInt) are implicitly instantiated. The element type is automatically inferred from the type of the list. More interestingly, the implicit comparison operator is automatically determined in a process called resolution. Resolution is a type-directed process that uses a set of rules, the implicit environment, to find a value that matches the type required by the function call. The implicit construct extends the implicit environment with new rules. In other words, implicit is a scoping construct for rules similar to a conventional let-binding. Thus, in the subexpression (isort [2,1,3], isort [5,9,3]), cmpInt is in the local scope and available for resolution. Existing Approaches to Generic Programming The two main strongholds of GP are the C++ and the functional programming (FP) communities. Many of the pillars of GP are based on the ideas promoted by Musser and Stepanov [23]. These ideas were used in C++ libraries such as the Standard Template Library [24] and Boost [1]. In the FP community, Haskell type classes [42] have proven to be an excellent mechanism for GP, although their original design did not have that purpose. As years passed the FP community created its own forms of GP [14,10,21]. Garcia et al.'s [9] comparative study of programming language support for GP was an important milestone for both communi-In this type class, the intention is that the ctx variable abstracts over a concrete type class. Unfortunately, Haskell does not support type class abstraction. Proposals for concepts inherit this limitation from type classes. Concepts and type classes are usually interpreted as predicates on types rather than types, and cannot be abstracted 1 In the context of C++ rules correspond to models or concept maps. 2 In the context of Haskell rules correspond to type-class instances. over as regular types. In contrast, because in Scala concepts are modeled with types, it is possible to abstract over concepts. Oliveira and Gibbons [28] show how to encode this example in Scala. No higher-order rules: Finally type classes do not support higher-order rules. As noted by Hinze and Peyton Jones [12], nonregular Haskell datatypes like: data Perfect f α = Nil | Cons α (Perfect f (f α)) require type class instances such as: instance (∀β.Show β ⇒ Show (f β), Show α) ⇒ Show (Perfect f α) which Haskell does not support, as it restricts instances (or rules) to be first-order. This rule is higher-order because it assumes another rule, ∀β.Show β ⇒ Show (f β), that contains an assumption itself. Also note that this assumed rule is polymorphic in β. Both concept proposals and Scala implicits inherit the limitation of first-order rules. Contributions This paper presents λ⇒, a minimal and general core calculus for implicits and it shows how to build a source language supporting implicit instantiation on top of it. Perhaps surprisingly the core calculus itself does not provide implicit instantiation: instantiation of generic arguments is explicit. Instead λ⇒ provides two key mechanisms for generic programming: 1) a type-directed resolution mechanism and 2) scoping constructs for rules. Implicit instantiation is then built as a convenience mechanism on top of λ⇒ by combining type-directed resolution with conventional type-inference. We illustrate this on a simple, but quite expressive source language. The calculus is inspired by Scala implicits and it synthesizes core ideas of that mechanism formally. In particular, like Scala implicits, a key idea is that resolution and implicit instantiation work for any type. This allows those mechanisms to be more widely useful and applicable, since they can be used with other types in the language. The calculus is also closely related to System F G , and like System F G , rules available in the implicit environment are lexically scoped and scopes can be nested. A novelty of our calculus is its support for partial resolution and higher-order rules. Although Hinze and Peyton Jones [12] have discussed higher-order rules informally and several other researchers noted their usefulness [40,30,28], no existing language or calculus provides support for them. Higher-order rules are just the analogue of higher-order functions in the implicits world. They arise naturally once we take the view that resolution should work for any type. Partial resolution adds additional expressive power and it is especially useful in the presence of higher-order rules. From the GP perspective λ⇒ offers a new foundation for generic programming. The relation between the implicit calculus and Scala implicits is comparable to the relation between System F G and various concept proposals; or the relation between formal calculi of type classes and Haskell type classes: The implicit calculus is a minimal and general model of implicits useful for language designers wishing to study and inform implementations of similar GP mechanisms in their own languages. In summary, our contributions are as follows. • Our implicit calculus λ⇒ provides a simple, expressive and general formal model for implicits. Despite its expressiveness, the calculus is minimal and provides an ideal setting for the formal study of implicits and GP. • Of particular interest is our resolution mechanism, which is significantly more expressive than existing mechanisms in the literature. It is based on a simple (logic-programming style) query language, works for any type, and it supports partial resolution as well as higher-order rules. • The calculus has a polymorphic type system and an elaboration semantics to System F. This also provides an effective implementation of our calculus. The elaboration semantics is proved to be type-preserving, ensuring the soundness of the calculus. • We present a small, but realistic source language, built on top of λ⇒ via a type-directed encoding. This language features implicit instantiation and a simple type of interface, which can be used to model simple forms of concepts. This source language also supports higher-order rules. • Finally, both λ⇒ and the source language have been implemented and the source code for their implementation is available at http://ropas.snu.ac.kr/~bruno/implicit. Organization Section 2 presents an informal overview of our calculus. Section 3 shows a polymorphic type system that statically excludes ill-behaved programs. Section 4 shows the elaboration semantics of our calculus into System F and correctness results. Section 5 presents the source language and its encoding into λ⇒. Section 6 discusses comparisons and related work. Section 7 concludes. Overview of the Implicit Calculus λ ⇒ Our calculus λ⇒ combines standard scoping mechanisms (abstractions and applications) and typesà la System F, with a logicprogramming-style query language. At the heart of the language is a threefold interpretation of types: types ∼ = propositions ∼ = rules Firstly, types have their traditional meaning of classifying terms. Secondly, via the Curry-Howard isomorphism, types can also be interpreted as propositions -in the context of GP, the type proposition denotes the availability in the implicit environment of a value of the corresponding type. Thirdly, a type is interpreted as a logicprogramming style rule, i.e., a Prolog rule or Horn clause [19]. Resolution [20] connects rules and propositions: it is the means to show (the evidence) that a proposition is entailed by a set of rules. Next we present the key features of λ⇒ and how these features are used for GP. For readability purposes we sometimes omit redundant type annotations and slightly simplify the syntax. Fetching values by types: A central construct in λ⇒ is a query. Queries allow values to be fetched by type, not by name. For example, in the following function call foo ?Int the query ?Int looks up a value of type Int in the implicit environment, to serve as an actual argument. Constructing values with type-directed rules: λ⇒ constructs values, using programmer-defined, type-directed rules (similar to functions). A rule (or rule abstraction) defines how to compute, from implicit arguments, a value of a particular type. For example, here is a rule that computes an Int × Bool pair from implicit Int and Bool values: (|(?Int + 1, ¬ ?Bool ) : {Int , Bool } ⇒ Int ×Bool|) The rule abstraction syntax resembles a type-annotated expression: the expression (?Int +1, ¬ ?Bool ) to the left of the colon is the rule body, and to the right is the rule type {Int , Bool } ⇒ Int ×Bool. A rule abstraction abstracts over a set of implicit values (here {Int , Bool }), or, more generally, over rules to build values. Hence, when a value of type Int × Bool is needed (expressed by the query ?(Int ×Bool)), the above rule can be used, provided that an integer and a boolean value are available in the implicit environment. In such an environment, the rule returns a pair of the incremented Int value and negated Bool value. The implicit environment is extended through rule application (analogous to extending the environment with function applications). Rule application is expressed as, for example: (|(?Int + 1, ¬ ?Bool ) : {Int , Bool } ⇒ Int ×Bool |) with {1, True } With syntactic sugar similar to a let-expression, a rule abstractionapplication combination is denoted more compactly as: implicit {1, True } in (?Int + 1, ¬ ?Bool ) which returns (2, False). Higher-order rules: λ⇒ supports higher-order rules. For example, the rule (|?(Int ×Int) : {Int , {Int } ⇒ Int ×Int } ⇒ Int ×Int|), when applied, will compute an integer pair given an integer and a rule to compute an integer pair from an integer. Hence, the following rule application returns (3,4): implicit {3, (|(?Int , ?Int + 1) : {Int } ⇒ Int ×Int|)} in ?(Int ×Int) Recursive resolution: Note that resolving the query ?(Int ×Int) involves applying multiple rules. The current environment does not contain the required integer pair. It does however contain the integer 3 and a rule (|(?Int , ?Int + 1) : {Int } ⇒ Int ×Int|) to compute a pair from an integer. Hence, the query is resolved with (3,4), the result of applying the pair-producing rule to 3. Locally and lexically scoped rules: Rules can be nested and resolution respects the lexical scope of rules. Consider the following program: Polymorphic rules and queries implicit {1} in implicit {True, (| if ?Bool then 2 : {Bool } ⇒ Int |)} in ?Int The query ?Int is not resolved with the integer value 1. Instead the rule that returns an integer from a boolean is applied to the boolean True, because those two rules can provide an integer value and they are nearer to the query. So, the program returns 2 and not 1. Overlapping rules: Two rules overlap if their return types intersect, i.e., when they can both be used to resolve the same query. Overlapping rules are allowed in λ⇒ through nested scoping. The nearest matching rule takes priority over other matching rules. For example consider the following program: implicit {λx .x : ∀α.α → α} in implicit {λn.n + 1 : Int → Int } in ?(Int → Int ) 1 In this case λn.n + 1 : Int → Int is the lexically nearest match in the implicit environment and evaluating this program results in 2. However, if we have the following program instead: implicit {λn.n + 1 : Int → Int } in implicit {λx .x : ∀α.α → α} in ?(Int → Int ) 1 Then the lexically nearest match is λx .x :∀α.α → α and evaluating this program results in 1. The λ ⇒ Calculus This section formalizes the syntax and type system of λ⇒. Syntax This is the syntax of the calculus: (Simple) Types τ : := α | Int | τ1 → τ2 | ρ Rule Types ρ ::= ∀ α.ρ ⇒ τ Expressions e ::= n | x | λx : τ.e | e1 e2 | ?ρ | (|e : ρ|) | e[ τ ] | e with e : ρ Types τ are either type variables α, the integer type Int , function types τ1 → τ2 or rule types ρ. A rule type ρ = ∀ α.ρ ⇒ τ is a type scheme with universally quantified variables α and an (implicit) contextρ. This context summarizes the assumed implicit environment. Note that we use o to denote an ordered sequence o1, . . . , on of entities andō to denote a set {o1, . . . , on}. Such ordered sequences and sets can be empty, and we often omit empty universal quantifiers and empty contexts from a rule type. The base case of rule types is whenρ is the empty set (∀ α.{} ⇒ τ ). Expressions include integer constants n and the three basic typed λ-calculus expressions (variables, lambda binders and applications). A query ?ρ queries the implicit environment for a value of type ρ. A rule abstraction (|e : ∀ α.ρ ⇒ τ |) builds a rule whose type is ∀ α.ρ ⇒ τ and whose body is e. Without loss of generality we assume that all variables x and type variables α in binders are distinct. If not, they can be easily renamed apart to be so. Note that, unlike System F, our calculus does not have a separate Λ binder for type variables. Instead rule abstractions play a dual role in the binding structure: 1) the universal quantification of type variables (which binds types), and 2) the context (which binds a rule set). This design choice is due to our interpretation of rules as logic programming rules 3 . After all, in the matching process of resolution, a rule is applied as a unit. Hence, separating rules into more primitive binders (à la System F's type and value binders) would only complicate the definition of resolution unnecessarily. However, elimination can be modularized into two constructs: type application e[τ ] and rule application e with e : ρ. Using rule abstractions and applications we can build the implicit sugar that we have used in Sections 1 and 2. implicit e : ρ in e1 : τ def = (|e1 : ρ ⇒ τ |) with e : ρ For readability purposes, when we use implicit we omit the type annotation τ . As we shall see in Section 5 this annotation can be automatically inferred. For brevity and simplicity reasons, we have kept λ⇒ small. In examples we may use additional syntax such as built-in integer operators and boolean literals and types. Figure 1 presents the static type system of λ⇒. The typing judgment Γ | ∆ ⊢ e : τ means that expression e has type τ under type environment Γ and implicit environment ∆. The auxiliary resolution judgment ∆ ⊢r ρ expresses that type ρ is resolvable with respect to ∆. Here, Γ is the conventional type environment that captures type variables; ∆ is the implicit environment, defined as a stack of contexts. Figure 1 also presents lookup in the implicit environment (∆ τ ) and in contexts (ρ τ ). Type System We will not discuss the first four rules ((TyInt), (TyVar), (TyAbs) and (TyApp)) because they are entirely standard. For now we also ignore the gray-shaded conditions in the other rules; they are explained in Section 3.3. Rule (TyRule) checks a rule abstraction (|e : ∀ α.ρ ⇒ τ |) by checking whether the rule's body e actually has the type τ under the assumed implicit type contextρ. Rule (TyInst) instantiates a rule type's type variables α with the given types τ , and rule (TyRApp) instantiates the type contextρ with expressions of the required rule types e : ρ. Finally, rule (TyQuery) delegates queries directly to the resolution rule (TyRes). Resolution Principle The underlying principle of resolution in λ⇒ originates from resolution in logic. Following the Curry-Howard correspondence, we assign to each type a corresponding logical interpretation with the (·) † function: Definition 3.1 (Logical Interpretation). α † = α † Int † = Int † (τ1 → τ2) † = τ † 1 → † τ † 2 (∀ α.ρ ⇒ τ ) † = ∀ α † . ρ∈ρ ρ † ⇒ τ † Here, type variables α map to propositional variables α † and the primitive type Int maps to the propositional constant Int † . Unlike Curry-Howard, we do not map function types to logical implications; we deliberately restrict our implicational reasoning to rule types. So, instead we also map the function arrow to an uninterpreted higher-order predicate → † . Finally, as already indicated, we map rule types to logical implications. Resolution in λ⇒ then corresponds to checking entailment of the logical interpretation. We postulate this property as a theorem that constrains the design of resolution. Theorem 3.1 (Resolution Specification). If ∆ ⊢r ρ, then ∆ † |= ρ † . 3 In Prolog these are not separated either. Type Environments Γ ::= · | Γ; x : τ Implicit Environments ∆ ::= · | ∆;ρ Γ | ∆ ⊢ e : τ (TyInt) Γ | ∆ ⊢ n : Int (TyVar) (x : τ ) ∈ Γ Γ | ∆ ⊢ x : τ (TyAbs) Γ; x : τ 1 | ∆ ⊢ e : τ 2 Γ | ∆ ⊢ λx : τ 1 .e : τ 1 → τ 2 (TyApp) Γ | ∆ ⊢ e 1 : τ 2 → τ 1 Γ | ∆ ⊢ e 2 : τ 2 Γ | ∆ ⊢ e 1 e 2 : τ 1 (TyRule) ρ = ∀ α.ρ ⇒ τ unambiguous(ρ) Γ | ∆;ρ ⊢ e : τ α ∩ ftv (Γ, ∆) = ∅ Γ | ∆ ⊢ (|e : ρ|) : ρ (TyInst) Γ | ∆ ⊢ e : ∀ α.ρ ⇒ τ Γ | ∆ ⊢ e[ τ ] : [ α → τ ](ρ ⇒ τ ) (TyRApp) Γ | ∆ ⊢ e :ρ ⇒ τ Γ | ∆ ⊢ e i : ρ i (∀e i : ρ i ∈ e : ρ) Γ | ∆ ⊢ (e with e : ρ) : τ (TyQuery) ∆ ⊢r ρ unambiguous(ρ) Γ | ∆ ⊢?ρ : ρ ∆ ⊢r ρ (TyRes) ∆ τ =ρ ′ ⇒ τ ∆ ⊢r ρ i (∀ρ i ∈ρ ′ −ρ) ∆ ⊢r ∀ α.ρ ⇒ τ ∆ τ = ρρ τ = ρ no overlap(ρ, τ ) (∆;ρ) τ = ρ ρ τ = ⊥ ∆ τ = ρ (∆;ρ) τ = ρ ρ τ = ρ ρ ∈ρ ρ = ∀ α ′ .ρ ′ ⇒ τ ′ θτ ′ = τ ρ τ = θρ ′ ⇒ τ Figure 1. Type System Resolution for Simple Types The step from the logical interpretation to the (TyRes) rule in Figure 1 is non-trivial. So, let us first look at a simpler incarnation. What does resolution look like for simple types τ like Int ? (SimpleRes) ∆ τ =ρ ′ ⇒ τ ∆ ⊢r ρi (∀ρi ∈ρ ′ ) ∆ ⊢r τ First, it looks up a matching rule type in the implicit environment by means of the lookup function ∆ τ defined in Fig. 1. This partial function respects the nested scopes: it first looks in the topmost context of the implicit environment, and, only if it does not find a matching rule, does it descend. Within an environment context, the lookup function looks for a rule type whose right-hand side τ ′ can be instantiated to the queried τ using a matching unifier θ. This rule type is then returned in instantiated form. The matching expresses that the looked-up rule produces a value of the required type. To do so, the looked-up rule may itself require other implicit values. This requirement is captured in the context ρ ′ , which must be resolved recursively. Hence, the resolution rule is itself a recursive rule. When the contextρ ′ of the looked-up rule is empty, a base case of the recursion has been reached. Example Consider this query for a tuple of integers: Int ; ∀α.{α} ⇒ α × α ⊢r Int × Int Lookup yields the second rule, which produces a tuple, instantiated to {Int} ⇒ Int × Int with matching substitution θ = [α → Int]. In order to produce a tuple, the rule requires a value of the component type. Hence, resolution proceeds by recursively querying for Int. Now lookup yields the first rule, which produces an integer, with empty matching substitution and no further requirements. Resolution for Rule Types So far, so good. Apart from allowing any types, recursive querying for simple types is quite similar to recursive type class resolution, and λ⇒ carefully captures the expected behavior. However, what is distinctly novel in λ⇒, is that it also provides resolution of rule types, which requires a markedly different treatment. (RuleRes) ∆ τ =ρ ⇒ τ ∆ ⊢r ∀ α.ρ ⇒ τ Here we retrieve a whole rule from the environment, including its context. Resolution again performs a lookup based on a matching right-hand side τ , but subsequently also matches the context with the one that is queried. No recursive resolution takes place. Example Consider a variant of the above query: Int; ∀α.{α} ⇒ α × α ⊢r {Int} ⇒ Int × Int Again lookup yields the second rule, instantiated to {Int} ⇒ Int × Int. The context {Int } of this rule matches the context of the queried rule. Hence, the query is resolved without recursive resolution. Unified Resolution The feat that our actual resolution rule (TyRes) accomplishes is to unify these seemingly disparate forms of resolution into one single inference rule. In fact, both (SimpleRes) and (RuleRes) are special cases of (TyRes), which provides some additional expressiveness in the form of partial resolution (explained below). The first hurdle for (TyRes) is that types τ and rule types ρ are different syntactic categories. Judging from its definition, (TyRes) only covers rule types. How do we get it to treat simple types then? Just promote the simple type τ to its corresponding rule type ∀.{} ⇒ τ and (TyRes) will do what we expect for simple types, including recursive resolution. At the same time, it still matches proper rule types exactly, without recursion, when that is appropriate. Choosing the right treatment for the context is the second hurdle. This part is managed by recursively resolvingρ ′ −ρ. In the case of promoted simple types,ρ is empty, and the whole ofρ ′ is recursively solved; which is exactly what we want. In the caseρ ′ matchesρ, no recursive resolution takes place. Again this perfectly corresponds to what we have set out above for proper rule types. However, there is a third case, whereρ ′ −ρ is a non-empty proper subset ofρ ′ . We call this situation, where part of the retrieved rule's context is recursively resolved and part is not, partial resolution. Example Here is another query variant: Bool; ∀α.{Bool , α} ⇒ α × α ⊢r {Int} ⇒ Int × Int The first lookup yields the second rule, instantiated to {Bool , Int} ⇒ Int × Int, which almost matches the queried rule type. Only Bool in the context is unwelcome, so it is eliminated through a recursive resolution step. Fortunately, the first rule in the environment is available for that. Semantic Resolution Within the confines of the semantic constraint of Theorem 3.1 the rule (TyRes) implements a rather syntactic notion of resolution. In contrast, a fully semantic definition of resolution would coincide exactly with the semantic constraint and satisfy ∆ ⊢r ρ iff ∆ † |= ρ † For instance, it would allow to resolve Char ; Char ⇒ Int; Bool ⇒ Int ⊢r Int In this example, resolution gets stuck using the topmost rule in the environment. However, by using the next one down, the query can be resolved. The problem with supporting this semantic notion of resolution is that it requires backtracking. Because backtracking easily becomes a performance problem and because it is mentally hard to reason about for the programmer, we have decided against it. We have considered another definition of resolution, that avoids backtracking but is closer to the semantic notion: ∆ τ =ρ ′ ⇒ τ ∆,ρ ⊢r ρi (∀ρi ∈ρ ′ ) ∆ ⊢r ∀ α.ρ ⇒ τ This rule extends the environment ∆ with the queried rule type's contextρ for recursive resolution of the matching rule type's contextρ ′ . It resolves the following query that rule (TyRes) does not: Char ; Char ⇒ Int; Bool ⇒ Int ⊢r Char ⇒ Int However, we prefer our more syntactic definition of resolution, rule (TyRes), because it is much simpler: the environment does not grow recursively, but stays the same throughout the whole recursive resolution. We believe that this way it is more manageable for the programmer to perform resolution mentally. Moreover, the invariant environment in rule (TyRes) is much easier for deciding termination. Additional Type System Conditions The gray-shaded conditions in the type system are to check lookup errors (no overlap) and ambiguous instantiations (unambiguous). Avoiding Lookup Errors To prevent lookup failures, we have to check for two situations: • A lookup has no matching rule in the environment. • A lookup has multiple matching rules which have different rule types but can yield values of the same type (overlapping rules). The former condition is directly captured in the definition of lookup among a set of rule types. The latter condition is captured in the no overlap property, which is defined as: no overlap({ρ1, . . . , ρn}, τ ) def = ∀i, j. ρi = ∀ αi.ρi ⇒ τi ∧ ∃θi.θiτi = τ ∧ ρj = ∀ αj .ρj ⇒ τj ∧ ∃θj.θj τj = τ =⇒ i = j Avoiding Ambiguous Instantiations We avoid ambiguous instantiations in the same way as Haskell does: all quantified type variables ( α) in a rule type (∀ α.ρ ⇒ τ ) must occur in τ . We use the unambiguous condition to check in (TyRule) and (TyQuery): unambiguous(∀ α.ρ ⇒ τ ) = α ⊆ ftv (τ ) ∧ ∀ρi ∈ρ.unambiguous(ρi). If there is a quantified type variable not in type τ , the type may yield ambiguous instantiations (e.g. ∀α.{α} ⇒ Int). Type-Directed Translation to System F In this section we define the dynamic semantics of λ⇒ in terms of System F's dynamic semantics, by means of a type directed translation. This translation turns implicit contexts into explicit parameters and statically resolves all queries, much like Wadler and Blott's dictionary passing translation for type classes [42]. The advantage of this approach is that we simultaneously provide a meaning to well-typed λ⇒ programs and an effective implementation that resolves all queries statically. Figure 2 presents the translation rules that convert λ⇒ expressions into ones of System F extended with the integer and unit types. This figure essentially extends Figure 1 with the necessary information for the translation, but for readability we have omitted the earlier gray-shaded conditions. The syntax of System F is as follows: Type-Directed Translation Types T ::= α | T → T | ∀α.T | Int | () Expressions E ::= x | λ(x : T ).E | E E | Λα.E | E T | n | () The main translation judgment is Γ | ∆ ⊢ e : τ E, which states that the translation of λ⇒ expression e with type τ is System F expression E, with respect to type environment Γ and translation environment ∆. The translation environment ∆ relates each rule type in the earlier implicit environment to a System F variable x; this variable serves as value-level explicit evidence for the implicit rule. Lookup in the translation environment is defined similarly to lookup in the type environment, except that the lookup now returns a pair of a rule type and an evidence variable. Figure 2 also defines the type translation function | · | from λ⇒ types τ to System F types T. In order to obtain a unique translation of types, we assume that the types in a context are lexicographically ordered. Variables, lambda abstractions and applications are translated straightforwardly. Queries are translated by rule (TrQuery) using the auxiliary resolution judgment ⊢r, defined by rule (TrRes). Note that rule (TrRes) performs the same process that rule (TyRes) performs in the type system except that it additionally collects evidence variables. Rule (TrRule) translates rule abstractions to explicit type and value abstractions in System F, and rule (TrInst) translates instantiation to type application. Finally, rule (TrRApp) translates rule application to application in System F. Example We have that: · | · ⊢ (|(?α, ?α) : ∀α.{α} ⇒ α × α|) Λα.λ(x : α).(x, x) and also: (Int : x1), (∀α.{α} ⇒ α × α : x2) ⊢r Int × Int x2 Int x1 For brevity, Figure 2 omits the case where the context of a rule type is empty. To properly handle empty contexts, the translation of rule type should include |{} ⇒ τ | = () → |τ | and the translation rules (TrRule), (TrRApp) and (TrRes) should be extended in the obvious way. Theorem 4.1 (Type-preserving translation) . Let e be a λ⇒ expression, τ be a type and E be a System F expression. If · | · ⊢ e : τ E, then · ⊢ E : |τ |. ∆ ⊢r ρ E (TrRes) ∆(τ ) =ρ ′ ⇒ τ : Ex fresh ∀ρ i ∈ρ ′ : ∆ ⊢r ρ i E i , ρ i ∈ρ E i = x i , ρ i ∈ρ ∆ ⊢r ∀ α.ρ ⇒ τ Λ α.λ( x : | ρ|).(E E) ∆ τ = ρ : E ρ : x τ = ρ : E (∆; ρ : x) τ = ρ : E ρ : x τ = ⊥ ∆ τ = ρ (∆; ρ : x) τ = ρ ρ : x τ = ρ : E (ρ : x) ∈ ρ : x ρ = ∀ α ′ .ρ ′ ⇒ τ ′ θτ ′ = τ θ = [ α ′ → τ ] ρ : x τ = θρ ′ ⇒ τ : x | τ | |α| = α |Int| = Int |τ 1 → τ 2 | = |τ 1 | → |τ 2 | |∀ α.{ρ 1 , · · · , ρn} ⇒ τ | = ∀ α.|ρ 1 | → · · · → |ρn| → |τ | |Γ| = {(x : |τ |) | (x : τ ) ∈ Γ} |∆| = {(x : |ρ|) | (ρ : x) ∈ ∆}(p 1 ≡ p 2 , implicit {eqInt 2 } in p 1 ≡ p 2 ) Figure 3. Encoding the Equality Type Class Proof. (Sketch) We first prove 4 the more general lemma "if Γ | ∆ ⊢ e : τ E, then |Γ|, |∆| ⊢ E : |τ |" by induction on the derivation of translation. Then, the theorem is trivially proved by it. Dynamic Semantics Finally, we define the dynamic semantics of λ⇒ as the composition of the type-directed translation and System F's dynamic semantics. Following Siek's notation [34], this dynamic semantics is: eval (e) = V where · | · ⊢ e : τ E and E → * V with → * the reflexive, transitive closure of System F's standard single-step call-by-value reduction relation. Now we can state the conventional type safety theorem for λ⇒: Theorem 4.2 (Type Safety). If · | · ⊢ e : τ , then eval (e) = V for some System F value V . The proof follows trivially from Theorem 4.1. Source Languages and Implicit Instantiation Languages like Haskell and Scala provide a lot more programmer convenience than λ⇒ (which is a low level core language) because of higher-level GP constructs, interfaces and implicit instantiation. This section illustrates how to build a simple source language on top of λ⇒ to add the expected convenience. We should note that unlike Haskell this language supports local and nested scoping, and unlike both Haskell and Scala it supports higher-order rules. We present the type-directed translation from the source to λ⇒. Type-directed Translation to λ⇒ The full syntax is presented in Figure 4. Its use is illustrated in the program of Figure 3, which comprises an encoding of Haskell's equality type class Eq. The example shows that the source language features a simple type of interface I T (basically records), which are used to encode simple forms of type classes. Note that we follow Haskell's conventions for records: field names u are unique and they are modeled as regular functions taking a record as the first argument. So a field u with type T in an interface declaration I α actually has type ∀ᾱ.{} ⇒ I α → T . There are also other conventional programming constructs (such as let expressions, lambdas and primitive types). Unlike the core language, we strongly differentiate between simple types T and type schemes σ in order to facilitate type inference. Moreover, as the source language provides implicit rather than explicit type instantiation, the order of type variables in a quantifier is no longer relevant. Hence, they are represented by a set (∀ᾱ). We also distinguish simply typed variables x from let-bound variables u with polymorphic type σ. Figure 5 presents the type-directed translation G ⊢ E : T e of source language expressions E of type T to core expressions e, with respect to type environment G. The type environment collects both simply and polymorphic variable typings. The connection between source types T and σ on the one hand and core types τ and ρ on the other hand is captured in the auxiliary function · . Note that this function imposes a canonical ordering α on the set of quantifier variablesᾱ (based on their precedence in the left-to-right prefix traversal of the quantified type term). For the translation of records, we assume that λ⇒ is extended likewise with records. let and let-bound variables The rule (TyLet) in Figure 5 shows the type-directed translation for let expressions. This translation binds the variable u using a regular lambda abstraction in an expression e2, which is the result of the translation of the body of the let construct (E2). Then it applies that abstraction to a rule whose rule type is just the corresponding (translated) type of the definition (σ1), and whose body is the translation of the expression E1. The source language provides convenience to the user by inferring type arguments and implicit values automatically. This inference happens in rule (TyLVar), i.e., the use of let-bound variables. That rule recovers the type scheme of variable u from the environment G. Then it instantiates the type scheme and fires the necessary queries to resolve the context. Queries The source language also includes a query operator (?). Unlike λ⇒ this query operator does not explicitly state the type; that information is provided implicitly through type inference. For example, instead of using p1 ≡ p2 in Figure 5, we could have directly used the field eq as follows: eq ? p1 p2 When used in this way, the query acts like a Coq placeholder ( ), which similarly instructs Coq to automatically infer a value. The translation of source language queries, given by the rule (TyIVar), is fairly straightforward. To simplify type-inference, the query is limited to types, and does not support partial resolution (although other designs with more powerful queries are possible). In the translated code the query is combined with a rule instantiation and application in order to eliminate the empty rule set. Type Environments G ::= · | G, u : σ | G, x : T G ⊢ E : T e (TyIntL) G ⊢ n : Int n (TyVar) G(x) = T G ⊢ x : T x (TyAbs) G, x : T 1 ⊢ E e G ⊢ λx.E : T 1 → T 2 λx : T 1 .e (TyApp) G ⊢ E 1 : T 1 → T 2 e 1 G ⊢ E 2 : T 1 e 2 G ⊢ E 1 E 2 : T 2 e 1 e 2(TyLVar)G(u) = ∀α. σ ⇒ T ′ θ = [α → T ] T = θT ′ q i = (? θσ i ) : θσ i (∀σ i ∈ σ) G ⊢ u : T u[ T ] with q (TyLet) σ = ∀α.σ ⇒ T 1 G ⊢ E 1 : T 1 e 1 G, u : σ ⊢ E 2 : T 2 e 2 G ⊢ let u : σ = E 1 in E 2 : T 2 (λu : σ .e 2 ) (|e 1 : σ |) (TyImp) G ⊢ E : T e G(u i ) = σ i q i = u i : σ i (∀u i ∈ u) G ⊢ implicit u in E : T (|e : σ ⇒ T |) with q (TyIVar) G ⊢? : T ?({} ⇒ T ) with {} (TyRec) ∀i : G(u i ) = ∀ᾱ.{} ⇒ I α → T i G ⊢ E i : θT i e θ = [ α → T ] G ⊢ I u = E : I T I u = e α = α Int = Int T 1 → T 2 = T 1 → T 2 I T = I T ∀α.σ ⇒ T = ∀ α . σ ⇒ T Figure 5. Type-directed Encoding of Source Language in λ⇒ Implicit scoping The implicit construct, which has been already informally introduced in Section 1, is the core scoping construct of the source language. It is used in our example to first introduce definitions in the implicit environment (eqInt1 , eqBool and eqPair ) available at the expression (p1 ≡ p2, implicit {eqInt2 } in p1 ≡ p2) Within this expression there is a second occurrence of implicit, which introduces an overlapping rule (eqInt2 ) that takes priority over eqInt1 for the subexpression p1 ≡ p2. The translation rule (TyImp) of implicit into λ⇒ also exploits type-information to avoid redundant type annotations. For example, it is not necessary to annotate the let-bound variables used in the rule set u because that information can be recovered from the environment G. Higher-order rules and implicit instantiation for any type The following example illustrates higher-order rules and implicit instantiation working for any type in the source language. For brevity, we have omitted the implementations of showInt , comma and space; but showInt renders an Int as a String in the conventional way, while comma and space provide two ways for rendering lists. Evaluation of the expression yields ("1,2,3", "1 2 3"). Thanks to the implicit rule parameters, the contexts of the two calls to o control how the lists are rendered. This example differs from that in Figure 3 in that instead of using a nominal interface type like Eq, it uses standard functions to model a simple concept for pretty printing values. The use of functions as implicit values leads to a programming style akin to structural matching of concepts, since only the type of the function matters for resolution. Extensions The goal of our work is to present a minimal and general framework for implicits. As such we have avoided making assumptions about extensions that would be useful for some languages, but not others. In this section we briefly discuss some extensions that would be useful in the context of particular languages and the implications that they would have in our framework. Full-blown Concepts The most noticeable feature that was not discussed is a full-blown notion of concepts. One reason not to commit to a particular notion of concepts is that there is no general agreement on what the right notion of concepts is. For example, following Haskell type classes, the C++0x concept proposal [11] is based on a nominal approach with explicit concept refinement, while Stroustrup favors a structural approach with implicit concept refinement because that would be more familiar to C++ programmers [37]. Moreover, various other proposals for GP mechanisms have their own notion of interface: Scala uses standard OO hierarchies; Dreyer et al. use ML-modules [8]; and in dependently typed systems (dependent) record types are used [36,5]. An advantage of λ⇒ is that no particular notion of interface is imposed on source language designers. Instead, language designers are free to use the one they prefer. In our source language, for simplicity, we opted to add a very simple (and limited) type of interface. But existing language designs [29,8,36,5] offer evidence that more sophisticated types of interfaces, including some form of refinement or associated types, can be built on top of λ⇒. Type Constructor Polymorphism and Higher-order Rules Type constructor polymorphism is an advanced, but highly powerful GP feature available in Haskell and Scala, among others. It allows abstracting container types like List and Tree with a type variable f ; and applying the abstracted container type to different element types, e.g., f Int and f Bool . This type constructor polymorhism leads to a need for higherorder rules: rules for containers of elements that depend on rules for the elements. The instance for showing values of type Perfect f α in Section 1, is a typical example of this need. Extending λ⇒ with type constructor polymorphism is not hard. Basically, we need to add a kind system and move from a System F like language to System Fω like language. Subtyping Languages like Scala or C++ have subtyping. Subtyping would require significant adaptations to λ⇒. Essentially, instead of targetting System F, we would have to target a version of System F with subtyping. In addition, the notion of matching in the lookup function ∆ τ would have to be adjusted, as well as the no overlap condition. While subtyping is a useful feature, some language designs do not support it because it makes the system more complex and interferes with type-inference. Type-inference Languages without subtyping (like Haskell or ML) make it easier to support better type-inference. Since we do not use subtyping, it is possible to improve support for typeinference in our source language. In particular, we currently require a type annotation for let expressions, but it should be possible to make that annotation optional, by building on existing work for the GHC Haskell compiler [32,41]. Related Work Throughout the paper we have already discussed a lot of related work. In what follows, we offer a more detailed technical comparison of λ⇒ versus System F G and Scala implicits, which are the closest to our work. Then we discuss the relation with other work in the literature. System F G Generally speaking our calculus is more primitive and general than System F G . In contrast to λ⇒, System F G has both a notion of concepts and implicit instantiation of concepts 5 . This has the advantage that language designers can just reuse that infrastructure, instead of having to implement it. The language G [35] is based on System F G and it makes good use of these built-in mechanisms. However, System F G also imposes important design choices. Firstly it forces the language designer to use the notion of concepts that is built-in to System F G . In contrast λ⇒ offers a freedom of choice (see also the discussion in Section 5.2). Secondly, fixing implicit instantiation in the core prevents useful alternatives. For example, Scala and several other systems do provide implicit instantiation by default, but also offer the option of explicit instantiation, which is useful to resolve ambiguities [29,18,6,8]. This cannot be modeled on top of System F G , because explicit instantiation is not available. In contrast, by taking explicit instantiation (rule application) as a core feature, λ⇒ can serve as a target for languages that offer both styles of instantiation. There are also important differences in terms of scoping and resolution of rules. System F G only formalizes a very simple type of resolution, which does not support recursive resolution. Furthermore, scoping is less fine-grained than in λ⇒. For example, System F G requires a built-in construct for model expressions, but in λ⇒ implicit (which plays a similar role) is just syntactic sugar on top of more primitive constructs. Scala Implicits Scala implicits are integrated in a full-blown language, but they have only been informally described in the literature [29,27]. Our calculus aims at providing a formal model of implicits, but there are some noteworthy differences between λ⇒ and Scala implicits. In contrast to λ⇒, Scala has subtyping. As discussed in Section 5.2 subtyping would require some adaptations to our calculus. In Scala, nested scoping can only happen through subclassing and the rules for resolution in the presence of overlapping instances are quite ad-hoc. Furthermore, Scala has no (first-class) rule abstractions. Rather, implicit arguments can only be used in definitions. In contrast λ⇒ provides a more general and disciplined account of scoping for rules. Type Classes Obviously, the original work on type classes [42] and the framework of qualified types [15] around it has greatly influenced our own work, as well as that of System F G and Scala. There is a lot of work on Haskell type classes in the literature. Notably, there have been some proposals for addressing the limitations that arise from global scoping [18,6]. However in those designs, type classes are still second-class and resolution only works for type classes. The GHC Haskell compiler supports overlapping instances [17], that live in the same global scope. This allows some relief for the lack of local scoping. A lot of recent work on type classes is focused on increasingly more powerful "type class" interfaces. Functional dependencies [16], associated types [4,3] and type families [31] are all examples of this trend. This line of work is orthogonal to our work. Other Languages and Systems Modular type classes [8] are a language design that uses ML-modules to model type classes. The main novelty of this design is that, in addition to explicit instantiation of modules, implicit instantiation is also supported. In contrast to λ⇒, implicit instantiation is limited to modules and, although local scoping is allowed, it cannot be nested. Instance arguments [5] are an Agda extension that is closely related to implicits. However, unlike most GP mechanisms, implicit rules are not declared explicitly. Furthermore resolution is limited in its expressive power, to avoid introducing a different computational model in Agda. This design differs significantly from λ⇒, where resolution is very expressive and the scoping mechanisms allow explicit rule declarations. Implicit parameters [22] are a Haskell extension that allows named arguments to be passed implicitly. Implicit parameters are resolved by name, not by type and there is no recursive resolution. GP and Logic Programming The connection between Haskell type classes and Prolog is folklore. Neubauer et. al. [25] also explore the connection with Functional Logic Programming and consider different evaluation strategies to deal with overlapping rules. With Constraint Handling Rules, Stuckey and Sulzmann [38] use Constraint Logic Programming to implement type classes. Conclusion Our main contribution is the development of the implicit calculus λ⇒. This calculus isolates and formalizes the key ideas of Scala implicits and provides a simple model for language designers interested in developing similar mechanisms for their own languages. In addition, λ⇒ supports higher-order rules and partial resolution, which add considerable expressiveness to the calculus. Implicits provide an interesting alternative to conventional GP mechanisms like type classes or concepts. By decoupling resolution from a particular type of interfaces, implicits make resolution more powerful and general. Furthermore, this decoupling has other benefits too. For example, by modeling concept interfaces as conventional types, those interfaces can be abstracted as any other types, avoiding the issue of second class interfaces that arise with type classes or concepts. Ultimately, all the expressiveness offered by λ⇒ offers a widerange of possibilities for new generic programming applications. A. Termination of Resolution If we are not careful about which rules are made implicit, the recursive resolution process may not terminate. This section describes how to impose a set of modular syntactic restrictions that prevents non-termination. As an example of non-termination consider {{Char } ⇒ Int, {Int } ⇒ Char } ⊢r Int which loops, using alternatively the first and second rule in the implicit environment. The problem of non-termination has been widely studied in the context of Haskell's type classes, and a set of modular syntactic restrictions has been imposed on type class instances to avoid nontermination [39]. Adapting these restrictions to our setting, we obtain the following termination condition. Definition A.1 (Termination Condition). An implicit environment ∆ satisfies the condition, denoted term(∆), iff term(ρ) for every ρ = (∀ α.ρ ⇒ τ ) ∈ dom(∆), where: term(ρ) def ⇔ occα(τ ′ ) occα(τ ) (∀(∀ α ′ .ρ ′ ⇒ τ ′ ) ∈ρ, ∀α ∈ ftv (τ, τi)) \ α ′ ) ∧ |τi| < |τ | (∀τi ∈ρ) ∧ term(ρ ′ ) (∀ρ ′ ∈ρ) where occα(Int ) = 0 occα(α) = 1 occα(α ′ ) = 0 (α = α ′ ) occα(τ1 → τ2) = occα(τ1) + occα(τ2) occα(∀ α.ρ ⇒ τ ) = occα(τ ) + ρ∈ρ occα(ρ) |Int| = 1 |α| = 1 |τ1 → τ2| = 1 + |τ1| + |τ2| |∀ α.ρ ⇒ τ | = 1 + |τ | + ρ∈ρ |ρ|. B. Proofs Throughout the proofs we refer to the type system rules of System F listed in Figure 6. : λ⇒ allows polymorphic rules. For example, the rule (|(?α, ?α) : ∀α.{α} ⇒ α×α|) can be instantiated to multiple rules of monomorphic types {Int } ⇒ Int ×Int, {Bool } ⇒ Bool ×Bool, . . . Multiple monomorphic queries can be resolved by the same rule. The following expression returns ((3, 3), (True, True)): implicit {3, True, (|(?α, ?α) : ∀α.{α} ⇒ α×α|)} in (?(Int ×Int), ?(Bool ×Bool)) Polymorphic rules can also be used to resolve polymorphic queries: implicit {(|(?α, ?α) : ∀α.{α} ⇒ α×α|)} in ?(∀α.{α} ⇒ α×α) Combining higher-order and polymorphic rules: The rule (|(?((Int ×Int)×(Int ×Int ))) : {Int , ∀α.{α} ⇒ α×α} ⇒ (Int ×Int)×(Int ×Int)|) prescribes how to build a pair of integer pairs, inductively from an integer value, by consecutively applying the rule of type ∀α.{α} ⇒ α×α twice: first to an integer, and again to the result (an integer pair). For example, the following expression returns ((3, 3), (3, 3)): implicit {3, (|(?α, ?α) : ∀α.{α} ⇒ α×α|)} in ?((Int ×Int)×(Int ×Int)) Figure 2 . 2Type-directed Translation to System F interface Eq α = {eq : α → α → Bool } let (≡) : ∀α. {Eq α} ⇒ α → α → Bool = eq ? in let eqInt 1 : Eq Int = Eq {eq = primEqInt } in let eqInt 2 : Eq Int = Eq {eq = λx y.isEven x ∧ isEven y } in let eqBool : Eq Bool = Eq {eq = primEqBool } in let eqPair : ∀α β. {Eq α, Eq β } ⇒ Eq (α, β) =Eq {eq = λx y.fst x ≡ fst y ∧ snd x ≡ snd y } in let p 1 : (Int , Bool ) = (4, True) in let p 2 : (Int , Bool ) = (8, True) in implicit {eqInt 1 , eqBool , eqPair } in Figure 4 . 4Syntax of Source Language let show : ∀α. {α → String } ⇒ α → String = ? in let showInt : Int → String = . . . in let comma : ∀α. {α → String } ⇒ [α] → String = . . . in let space : ∀α. {α → String } ⇒ [α] → String = . . . in let o : {Int → String, {Int → String } ⇒ [Int ] → String } ⇒ String = show [1, 2, 3] in implicit showInt in (implicit comma in o, implicit space in o) Figure 6 . 6Lemma B.1. If Γ|∆ ⊢ e : τ E then |Γ|, |∆| ⊢ E : |τ | Proof. By structural induction on the expression and corresponding inference rule. (ÌÖÁÒØ) Γ|∆ ⊢ n : Int n It follows trivially from (F-Int) that |Γ|, |∆| ⊢ n : Int(ÌÖÎ Ö) Γ|∆ ⊢ x : x : T1 ⊢ E : T2 Γ ⊢ λx : T1.E : T1 → T2 (F-App) Γ ⊢ E1 : T2 → T1 Γ ⊢ E2 : T2 Γ ⊢ E1 E2 : T1 (F-TApp) Γ ⊢ E : ∀α.T2 Γ ⊢ E T1 : [α → T1]T2 (F-TAbs) Γ ⊢ E : T α ∈ ftv (Γ) Γ ⊢ Λα.E : ∀α.T System F Type SystemIt follows from (TrVar) that(x : τ ) ∈ ΓBased on the definition of | · | it follows ( x : x|τ |) ∈ |Γ| Thus we have by (F-Var) that |Γ|, |∆| ⊢ x : |τ | (ÌÖ ×) Γ|∆ ⊢ λx : τ1.e : τ1 → τ2 λx : |τ1|.E It follows from (TrAbs) that Γ; x : τ1|∆ ⊢ e : τ2 E and by the indution hypothesis that |Γ|, x : |τ1|, |∆| ⊢ E : |τ2| As all variables are renamed unique, it is easy to verify that this also holds: |Γ|, |∆|, x : |τ1| ⊢ E : |τ2| Hence, by (F-Abs) we have |Γ|, |∆| ⊢ λx : |τ1|.E : |τ1 → τ2| (ÌÖ ÔÔ) Γ|∆ ⊢ e1 e2 : τ1 E1 E2 By the induction hypothesis, we have: |Γ|, |∆| ⊢ E1 : |τ2 → τ1| and |Γ|, |∆| ⊢ E2 : |τ2| Then it follows by (F-App) that |Γ|, |∆| ⊢ E1 E2 : |τ1| (ÌÖÉÙ ÖÝ) Γ|∆ ⊢?ρ : ρ E From (TrQuery) we have ∆ ⊢ ρ E Based on Lemma B.2 we then know |∆| ⊢ E : |ρ| Hence, because all variables are unique |Γ|, |∆| ⊢ E : |ρ|(ÌÖÊÙÐ ) Γ|∆ ⊢ (|e : ρ|) : ρ Λ α.λ( x : | ρ|).EBased on (TrRule) and the induction hypothesis, we have|Γ|, |∆|, x : | ρ| ⊢ E : |τ | where ρ = ∀ᾱ.ρ ⇒ τ Thus, based on (F-Abs) we have |Γ|, |∆| ⊢ λ( x : | ρ|).E : |ρ1| → . . . → |ρn| → |τ | or, using the definition of | · | |Γ|, |∆| ⊢ λ( x : | ρ|).E : |ρ ⇒ τ | Moreover, because of (TrRule), we know α ∩ ftv (Γ, ∆) = ∅ and hence α ∩ ftv (|Γ|, |∆|) = ∅ So,finally, we may conclude from (F-TAbs) that |Γ|, |∆| ⊢ Λ α.λ( x : | ρ|).E : ∀ α.|ρ ⇒ τ | and again with | · | |Γ|, |∆| ⊢ Λ α.λ( x : | ρ|).E : |∀ α.ρ ⇒ τ | (ÌÖÁÒ×Ø) Γ|∆ ⊢ e[ τ ] : [ α → τ ](ρ ⇒ τ ) E | τ | By (TrInst) and the induction hypothesis, it follows that |Γ|, |∆| ⊢ E : |∀ α.ρ ⇒ τ | From which we have by definition of | · | |Γ|, |∆| ⊢ E : ∀ α.|ρ ⇒ τ | It follows from (F-TApp) that |Γ|, |∆| ⊢ E | τ | : [ α → | τ |]|ρ ⇒ τ | which is easily seen to be equal to |Γ|, |∆| ⊢ E | τ | : |[ α → τ ](ρ ⇒ τ )| (ÌÖÊ ÔÔ) Γ|∆ ⊢ e with e : ρ : τ E E From (TrApp) and the induction hypothesis we have: |Γ|, |∆| ⊢ E : |ρ ⇒ τ | and |Γ|, |∆| ⊢ Ei : |ρi|(∀i) Hence, base on the definition of | · | the first of these means |Γ|, |∆| ⊢ E : |ρ1| → . . . → |ρn| → |τ | Hence, based on (F-App) we know |Γ|, |∆| ⊢ E E : |τ | Proof. By induction on the derivation. From (TrRes) we have ∆ ⊢ ρ Λ α.λ( x : | ρ|).(E E) where ρ = ∀ α.ρ ⇒ τ Also from (TrRes) and the induction hypothesis, we have |∆| ⊢ Ei : |ρi| (ρi ∈ρ ′ −ρ) Also from (TrRes) and Lemma B.3, we have |∆| ⊢ E : | ρ ′ ⇒ τ | Assembling these parts using (F-App), (F-Abs) and (F-TAbs) we come to |∆| ⊢ Λ α.λ( x : | ρ|).(E E) Lemma B.3. If ∆ τ =ρ ⇒ τ : E then |∆| ⊢ E : |ρ ⇒ τ | Proof. This follows trivially from Lemma B.4. Lemma B.4. If ρ : x τ =ρ ⇒ τ : E then |ρ : x| ⊢ E : |ρ ⇒ τ | Proof. From the definintion of lookup we know that iff ρ : x τ = θρ ′ ⇒ τ : x | τ | then (ρ : x) ∈ ρ : x Hence, it trivially follows that (x : |ρ|) ∈ |ρ : x| Hence, from (F-Var) we have that |ρ : x| ⊢ x : |ρ| Following the definition of | · | we also know |ρ : x| ⊢ x : ∀ α.|ρ ′ ⇒ τ ′ | So |ρ : x| ⊢ x |ts| : |θ(ρ ′ ⇒ τ ′ )|Lemma B.2. If ∆ ⊢ ρ E then |∆| ⊢ E : |ρ| in the extra material of the submission Note that instantiation of type variables is still explicit. Interface Declarationsinterface I α = u : TInterface Implementation . The Boost C++ libraries. The Boost C++ libraries. http://www.boost.org/, 2010. Type inference for overloading without restrictions, declarations or annotations. C Camarão, L Figueiredo, FLOPS. C. Camarão and L. Figueiredo. Type inference for overloading without restrictions, declarations or annotations. In FLOPS, 1999. Associated type synonyms. M Chakravarty, G Keller, S L , Peyton Jones, ICFP. M. Chakravarty, G. Keller, and S. L. Peyton Jones. Associated type synonyms. In ICFP, 2005. Associated types with class. M Chakravarty, G Keller, S L , Peyton Jones, S Marlow, POPL. M. Chakravarty, G. Keller, S. L. Peyton Jones, and S. Marlow. Associated types with class. In POPL, 2005. On the bright side of type classes: Instance arguments in agda. D Devriese, F Piessens, ICFP. D. Devriese and F. Piessens. On the bright side of type classes: Instance arguments in agda. In ICFP, 2011. Making implicit parameters explicit. 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Concepts: linguistic support for generic programming in c++. D Gregor, J Järvi, J G Siek, B Stroustrup, G Dos Reis, A Lumsdaine, OOPSLA. D. Gregor, J. Järvi, J. G. Siek, B. Stroustrup, G. Dos Reis, and A. Lumsdaine. Concepts: linguistic support for generic programming in c++. In OOPSLA, 2006. Derivable type classes. Electronic Notes in Theoretical Computer Science. R Hinze, S L , Peyton Jones, 41R. Hinze and S. L. Peyton Jones. Derivable type classes. Electronic Notes in Theoretical Computer Science, 41(1):5 -35, 2001. Restricted data types in Haskell. J Hughes, HaskellJ. Hughes. Restricted data types in Haskell. In Haskell, 1999. Polytypic programming. P Jansson, J Jeuring, AFP. Springer-VerlagP. Jansson and J. Jeuring. Polytypic programming. In AFP. Springer- Verlag, 1996. Simplifying and improving qualified types. M P Jones, FPCA. M. P. Jones. Simplifying and improving qualified types. In FPCA, 1995. Type classes with functional dependencies. M P Jones, ESOP. M. P. Jones. Type classes with functional dependencies. In ESOP, 2000. Type classes: exploring the design space. S L Jones, M P Jones, E Meijer, Haskell Workshop. S. L. Peyton Jones, M. P. Jones, and E. Meijer. Type classes: exploring the design space. In Haskell Workshop, 1997. Named instances for Haskell type classes. W Kahl, J Scheffczyk, Haskell Workshop. W. Kahl and J. Scheffczyk. Named instances for Haskell type classes. In Haskell Workshop, 2001. Predicate logic as a programming language. R Kowalski, Proceedings of IFIP Congress. IFIP CongressR. Kowalski. Predicate logic as a programming language. In Proceedings of IFIP Congress, 1974. Linear resolution with selection function. R Kowalski, Donald , Kuehner , Artificial Intelligence. 2R. Kowalski, Donald, and Kuehner. Linear resolution with selection function. Artificial Intelligence, 2, 1971. Scrap your boilerplate with class: extensible generic functions. R Lämmel, S L , Peyton Jones, ICFP. R. Lämmel and S. L. Peyton Jones. Scrap your boilerplate with class: extensible generic functions. In ICFP, 2005. Implicit parameters: dynamic scoping with static types. J Lewis, J Launchbury, E Meijer, M Shields, POPL. J. Lewis, J. Launchbury, E. Meijer, and M. Shields. Implicit parameters: dynamic scoping with static types. In POPL, 2000. Generic programming. D Musser, A Stepanov, Symbolic and algebraic computation: ISSAC 88. SpringerD. Musser and A. Stepanov. Generic programming. In Symbolic and algebraic computation: ISSAC 88, pages 13-25. Springer, 1988. The STL Tutorial and Reference Guide: C++ Programming with the Standard Template Library. D R Musser, A Saini, Addison Wesley Longman Publishing Co., IncD. R. Musser and A. Saini. The STL Tutorial and Reference Guide: C++ Programming with the Standard Template Library. Addison Wesley Longman Publishing Co., Inc., 1995. Functional logic overloading. M Neubauer, P Thiemann, M Gasbichler, M Sperber, POPL. M. Neubauer, P. Thiemann, M. Gasbichler, and M. Sperber. Functional logic overloading. In POPL, 2002. Poor man's type classes. M Odersky, M. Odersky. Poor man's type classes. http://lamp.epfl.ch/~odersky/talks/wg2 July 2006. The Scala language specification, version 2.8. M Odersky, M. Odersky. The Scala language specification, version 2.8, 2010. Scala for generic programmers. B C D S Oliveira, J Gibbons, Journal of Functional Programming. 20B. C. d. S. Oliveira and J. Gibbons. Scala for generic programmers. Journal of Functional Programming, 20, 2010. Type classes as objects and implicits. B C D S Oliveira, A Moors, M Odersky, OOPSLA. B. C. d. S. Oliveira, A. Moors, and M. Odersky. Type classes as objects and implicits. In OOPSLA, 2010. Comparing libraries for generic programming in haskell. A Rodriguez, J Jeuring, P Jansson, A Gerdes, O Kiselyov, B C D S Oliveira, HaskellA. Rodriguez, J. Jeuring, P. Jansson, A. Gerdes, O. Kiselyov, and B. C. d. S. Oliveira. Comparing libraries for generic programming in haskell. In Haskell, 2008. 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A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations, that are fundamental to quantum mechanics, must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary representations of the central extension of the inhomogeneous symplectic group. This centrally extended group is the semidirect product of the cover of the symplectic group and the Weyl-Heisenberg group. Its unitary irreducible representations are computed explicitly using the Mackey representation theorems for semidirect product groups.

10.1063/1.4863896

1207.6787

77c3a8c4dfc69877ae742e4b5f2378d1201ef0d5

MAXIMAL QUANTUM MECHANICAL SYMMETRY: PROJECTIVE REPRESENTATIONS OF THE INHOMOGENOUS SYMPLECTIC GROUP Stephen G Low MAXIMAL QUANTUM MECHANICAL SYMMETRY: PROJECTIVE REPRESENTATIONS OF THE INHOMOGENOUS SYMPLECTIC GROUP A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations, that are fundamental to quantum mechanics, must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary representations of the central extension of the inhomogeneous symplectic group. This centrally extended group is the semidirect product of the cover of the symplectic group and the Weyl-Heisenberg group. Its unitary irreducible representations are computed explicitly using the Mackey representation theorems for semidirect product groups. Introduction The Heisenberg commutation relations are " p P i , p Q j ı " i δ i,j 1,(1) where i, j, ... " 1, ..., n. The hermitian operators p P i and p Q j represent quantum mechanical momentum and position observables acting on states |ψy that are elements of a Hilbert space H for which 1 is the unit operator. (We will use natural units in which " 1 throughout the paper.) These relations are fundamental to quantum mechanics in its original formulation. Weyl [1] established that these relations are the Hermitian representation of the algebra of a Lie group Hpnq that we now call the Weyl-Heisenberg group. The Weyl-Heisenberg Lie group is a semidirect product [2] of two abelian groups 1 Hpnq » Apnq b s Apn`1q,(2) where Apmq is the abelian Lie group isomorphic to the reals under addition, Apmq » pR m ,`q. Therefore, it has an underlying manifold diffeomorphic to R 2n`1 and is simply connected. In a global coordinate system p, q P R n , ι P R, the group product and inverse of the Weyl-Heisenberg group may be written Υpp 1 , q 1 , ι 1 qΥpp, q, ιq " Υpp 1`p , q 1`q , ι`ι 1`1 2`p 1¨q´q1¨p˘q , Υpp, q, ιq´1 " Υp´p,´q,´ιq. The identity element is e " Υp0, 0, 0q. Its Lie algebra is given by rP i , Q i s " δ i,j I, rP i , Is " 0, rQ i , Is " 0. The faithful unitary irreducible representations ξ of the Weyl-Heisenberg group may be written as ψ 1 pxq " pξpΥpp, q, ιqqψq pxq " e iλpι`x¨p´1 2 p¨qq ψpx´qq (6) where p, q, x P R n , ι P R. λ P Rzt0u label the irreducible representations and ψpxq " xx|ψy P H ξ » L 2 pR n , Cq. We label the Hilbert space with the unitary representation ξ as this Hilbert space, on which the unitary representation ξ acts, is determined by the unitary irreducible representation and is not given a priori. The Stone-von Neumann theorem [3], [4] establishes that (6) defines the complete set of faithful irreducible representations of the Weyl-Heisenberg group. This theorem is not constructive; it does not give a prescription to obtain these representations but only establishes that they are a complete set of faithful irreducible representations. However, as the Weyl-Heisenberg group has the form of the semidirect product given in (2), the unitary irreducible representations (6) can also be directly calculated using the Mackey theorems as these theorems are constructive. This is reviewed in Section 3.1. The position and momentum operators in (1) are given by the faithful 2 hermitian representation ξ 1 of the Weyl-Heisenberg algebra. (The prime designates the lift of the unitary representation ξ of the group to the algebra, ξ 1 " T e ξ.) p P i " ξ 1 pP i q, p P i " ξ 1 pP i q, p I " ξ 1 pIq. These operators also act on the Hilbert space H ξ » L 2 pR n , Cq. As the representation is a homomorphism, its lift preserves the Lie bracket, υ 1 prP i , Q i sq " " υ 1 pP i q, υ 1 pQ i q ‰˘" i δ i,j υ 1 pIq " i λδ i,j 1. The i appears simply because we are using hermitian rather than anti-hermitian operators 3 . Schur's lemma states that the representation of the central generators are a multiple of the identity for irreducible representations and so p I " λ1 where λ P Rzt0u. With λ " 1, these are the Heisenberg commutation relations given in (1). 2 There are also degenerate representations corresponding to the homomorphism π : Hpnq Ñ Ap2nq for which λ " 0 (See Appendix A, Theorem 4). These representations of the abelian group are not discussed further here. 3 In some neighborhood, the group element g is given in terms of an element X of the algebra by g " e X . Then for a unitary representation υ of the group, the representation υ 1 of the algebra is Hermitian (rather than non-Hermitian only if we insert an i, υpgq " e iυ 1 pXq . This follows as υpgq´1 " υpgq : implies´iυ 1 pXq " piυ 1 pXqq : and hence υ 1 pXq " υ 1 pXq : . Symmetry of Physical States. A basic assumption of quantum mechanics is that the Heisenberg commutation relations (1) are valid when acting on any physical state. Physically observable probabilities are given by the square of the modulus of the states. Therefore, physical states in quantum mechanics are rays Ψ that are equivalence classes of states |ψy in the Hilbert space that are equal up to a phase [5], [6], Ψ " r|ψys ,ˇˇr ψ E » |ψyˇˇr ψ E " e iθ |ψy .(9) The square of the modulus is the same for any representative state in the ray, P pα Ñ βq " | pΨ β , Ψ α q | 2 " | A r ψ β | r ψ α E | 2 " | xψ β |ψ α y | 2 .(10) Symmetry transformations between physical states (i.e. rays Ψ) are given by operators U that leave invariant the square of modulus, | pU Ψ β , U Ψ α q | 2 " | pΨ β , Ψ α q | 2 .(11) These transformations U are the representation of a group in the space U pHq of linear or anti-linear operators on H : G Ñ U pHq : g Þ Ñ U " pgq.(12) This operator also acts on any representative in the equivalence class of states that defines the ray, Ψ 1 " U Ψ,ˇˇψ 1 D " U |ψy .(13) Theorem 2 in Appendix A states that any representation of a Lie group [7], [8] that leaves invariant the square of the modulus is always equivalent to a linear unitary or anti-linear, anti-unitary operator mapping the Hilbert space H into itself. Furthermore, if the Lie group is connected 4 , it is always equivalent to a linear unitary operator. The representations are referred to as projective representations. If G is a connected Lie group, the fundamental Theorem 3 states that these projective representations are equivalent to the ordinary unitary representations υ of the central extension q G of G. We seek the maximal group with projective representations that preserve the Heisenberg commutation relations. As the Heisenberg commutation relations are a faithful unitary representation of the Lie algebra of the Weyl-Heisenberg group, the group we seek must be a subgroup of the automorphism group of the Weyl-Heisenberg algebra. As the Weyl-Heisenberg group is simply connected, the automorphism group of the algebra is equivalent to the automorphism group Aut Hpnq of the Weyl-Heisenberg group itself. Under the action of elements g P Aut Hpnq , the elements of the algebra transform to a new basis P 1 i " gP i g´1, Q 1 i " gQ i g´1, I 1 " gIg´1.(14) such that the form of the Lie algebra is preserved, " P 1 i , Q 1 i ‰ " δ i,j I 1 .(15) 4 In this paper, a connected group is abbreviation for a group for which every element is connected by a continuous path to the identity element. The element I 1 is central and as I spans the center of the algebra, we must have I 1 " dI with d P Rzt0u. Furthermore, the elements of the automorphism group that preserves the center of the algebra, I 1 " gIg´1 " I,(16) defines a subgroup. The group inner automorphisms of a group is isomorphic to the group itself. The full group of automorphisms always contains the group of inner automorphisms as a normal subgroup. For the case of the Weyl-Heisenberg group, this means that the Weyl-Heisenberg group is a normal subgroup of its automorphism group, Hpnq Ă Aut Hpnq . The projective representations of Aut Hpnq are equivalent to the unitary representations υ of its central extension } Aut Hpnq acting on a Hilbert space H υ . If we restrict υ to the normal subgroup Hpnq of inner automorphisms, these are the unitary representations of the Weyl-Heisenberg group, υ| Hpnq " ξ. Therefore, the Hilbert space H ξ is an invariant subspace of H υ . The generators of the Weyl-Heisenberg group transform under the action of elements U " υpgq, g P Aut Hpnq as p P 1 i " υ 1 pP 1 i q " υ 1 pgP i g´1q " υpgqξ 1 pP i qυpgq´1 " U p P i U´1, p Q 1 i " υ 1 pQ 1 i q " υ 1 pgQ i g´1q " υpgqξ 1 pQ i qυpgq´1 " U p Q i U´1, p I 1 " υ 1 pI 1 q " υ 1 pgIg´1q " υpgqξ 1 pIqυpgq´1 " U p IU´1,(17) For the faithful representation υ, the commutation relations for the transformed generators are, using (15), " p P 1 i , p Q 1 i ı " i δ i,j p I 1 " iλ 1 δ i,j 1(18) where p I 1 " d p I and so λ 1 " dλ. Now, as we have noted, the λ label the faithful irreducible representations of the Weyl-Heisenberg group. Furthermore, the physical cases corresponds to the choice λ " 1. This must also be true for the transformed operators and therefore λ 1 " 1 and so p I 1 " p I with d " 1. That is, the projective representation of the symmetry group of the Heisenberg commutation relations leaves the representation of the center of the Weyl-Heisenberg group invariant. As the representation is faithful, the symmetry group also must leave the central generator of the Weyl-Heisenberg algebra invariant, I 1 " I. Therefore, the maximal group of symmetries of the Heisenberg commutation relations are the projective representation of the subgroup of the automorphism group of the Weyl-Heisenberg group that leaves the central generator I invariant. The problem that this paper addresses is to determine the explicitly this symmetry group and its projective representations. We will show that the automorphism group of the Weyl-Heisenberg group is [2] Aut Hpnq » D b s HSpp2nq, where HSpp2nq » Spp2nq b s Hpnq, D » pRz t0u ,ˆq ,(20) where D is the reals excluding t0u viewed as a group under multiplication, D » pRzt0u,ˆq. We will show that the subgroup of the automorphism group that leaves the central generator I invariant is HSpp2nq. The group HSpp2nq is connected and is the central extension of the Inhomogeneous group, HSpp2nq » I | Spp2nq that is defined by the short exact sequence e Ñ Z b Ap1q Ñ HSpp2nq Ñ ISpp2nq Ñ e.(22) Z is the center of Spp2nq and Ap1q is the center of Hpnq. ISpp2nq is the inhomogeneous symplectic group familiar from classical Hamiltonian mechanics, ISpp2nq " Spp2nq b s Ap2nq.(23) To establish the above results we start by reviewing the Weyl-Heisenberg group. We then derive its automorphism group and the subgroup that leaves the center of the Weyl-Heisenberg group invariant. This is the maximal symmetry group. The projective representations of this symmetry group are equivalent to the unitary representations of its central extension. We use the Mackey theorems to compute the unitary irreducible representations of the symmetry group from first principles. (As the symmetry group contains the Weyl-Heisenberg group as normal subgroup, this first requires the computation of the faithful unitary irreducible representations of the Weyl-Heisenberg group itself using the Mackey theorems.) We will enumerate and comment on the degenerate cases. The symmetry group In this section, we review basic properties of the Weyl-Heisenberg group and determine its automorphism group. We then determine the subgroup leaving the center of the Weyl-Heisenberg group invariant and study certain of its properties. 2.1. The Weyl-Heisenberg group. The Weyl-Heisenberg Lie group is defined to be the semi-direct product of two abelian groups of the form given in (2). We first verify that these group product (3) and inverse (4) relations result in the semidirect product of this form. First, the group product and inverse (3-4) enable us to identify the abelian subgroups Υp0, q, ιq P Apn`1q, Υpp, 0, 0q P Apnq. where again p, q P R n and ι P R. These subgroups satisfy the group product and inverse relations Υp0, q 1 , ι 1 qΥp0, q, ιq " Υp0, q 1`q , ι`ι 1 q, Υp0, q, ιq´1 " Υp0,´q,´ιq (25) Υpp 1 , 0, 0qΥpp, 0, 0q " Υpp 1`p , 0, 0q, Υpp, 0, 0q´1 " Υp´p, 0, 0q.(26) Additional abelian subgroups are likewise given by Υpp, 0, ιq P Apn`1q, Υp0, q, 0q P Apn`1q(27) We calculate the inner automorphisms of the group using (3-4) to be 5 ς Υpp 1 ,q 1 ,ι 1 q Υpp, q, ιq " Υpp 1 , q 1 , ι 1 qΥpp, q, ιqΥpp 1 , q 1 , ι 1 q´1 " Υpp, q, ι`p 1 q´q 1¨p q. In particular, note that for each of the choices of the subgroups ς Υpp 1 ,q 1 ,ι 1 q Υp0, q, ιq " Υp0, q, ι`p 1 qq, ς Υpp 1 ,q 1 ,ι 1 q Υpp, 0, ιq " Υpp, 0, ι´q 1¨p q. This means that both of the Apn`1q subgroups given in (24), (27) are normal subgroups. Another special case of (3) is ς Υp0,0,ι 1 q Υpp, q, ιq " Υpp, q, ιq. and therefore the elements Υp0, 0, ι 1 q commute with all elements of the group. Furthermore, these are the only elements that commute with all other elements of the group. Therefore the Ap1q group that is defined by the elements Υp0, 0, ιq is the center of the group, Z » Ap1q. The final step to verify that the group relations defined by (3)(4) results in the Weyl-Heisenberg group having the structure of a semidirect product given in (2). We have already established that there are two choices for the Apnq subgroup and Apn`1q normal subgroup. It is clear in both cases that Apnq X Apn`1q " e,(32) as the identity Υp0, 0, 0q is the only element in both groups for both cases. It remains to show that Apn`1qApnq » Hpnq. Using the group product (3), for each of the cases (24), (27), this is Υp0, q, ιqΥpp, 0, 0q " Υpp, q, ι´1 2 q¨pq,(33) Υpp, 0, ιqΥp0, q, 0q " Υpp, q, ι`1 2 p¨qq. The map ϕ˘: Hpnq Ñ Hpnq : Υpp, q, ιq Þ Ñ Υ˘pp, q, ι˘q " Υpp, q, ι¯1 2 p¨qq is a homomorphism that is onto and the kernel is trivial. Therefore, the map ϕ˘is an isomorphism and the Weyl-Heisenberg group has the semidirect product structure given in (2) for either of the choices of abelian subgroup given by (24), (27). The Weyl-Heisenberg Lie group is a matrix group and may be realized by the 2n`2 dimensional square matrices Υpp, q, ιq "¨1 n 0 0 p 0 1 n 1 q q t´pt 1 2ι 0 0 0 1‹ ‹ ‚ .(36) 1 m denotes the unit matrix in m dimensions and the t superscript denotes the transpose. The group multiplication and inverse (3)(4) are realized by matrix multiplication and inverse. The Lie algebra of the Weyl-Heisenberg group may be computed from this matrix realization. The coordinates are nonsingular at the origin and therefore, choosing the unpolarized form, the generators are given by A general element of the algebra is then W " p i Q i`q i P i`ι I.(38) The nonzero commutation relations are, as expected, rP i , Q i s " δ i,j I(39) where I is a central generator. It is convenient to also introduce the notation that combines the p, q P R n into a single 2n tuple z " pp, qq, z P R 2n . Then the group product and inverse are Υpz 1 , ι 1 qΥpz, ιq " Υpz 1`z , ι`ι´1 2 z 1 t ζzq, Υpz, ιq´1 " Υp´z,´ιq(40) and the unpolarized matrix realization is Υpz, ιq "¨´1 2n 0 z z t ζ 1 2ι 0 0 1‚ , ζ "ˆ0 1 ń 1 n 0˙.(41) The Lie algebra has general element W pz, ιq " z α Z α`ι I,(42) α, β, ... " 1, ...2n where the matrix form of the algebra is W pz, ιq "¨´0 0 z z t ζ 0 2ι 0 0 0‚ ,(43) The generators satisfy the nonzero commutation relations rZ α , Z β s " ζ α,β I.(44) 2.2. The automorphism group of the Weyl-Heisenberg group. The automorphism group of a group G is the maximal group for which G is a normal subgroup. We have established in the previous section that the Weyl-Heisenberg group is a simply connected matrix group and this enables us to prove the following theorem. Theorem 1. The automorphism group of the Weyl-Heisenberg group Hpnq is Aut Hpnq » D b s Spp2nq b s Hpnq.(45) Hpnq is the Weyl-Heisenberg group, Spp2nq is the cover of the real symplectic group that leaves invariant a real skew symmetric form and D is the reals excluding zero viewed as a group under multiplication D » pRzt0u,ˆq. As the Weyl-Heisenberg group is simply connected, Theorem 7 states that the automorphism group of its algebra and group are equivalent. We can therefore establish the result by determining the maximal group for which its elements Ω satisfy ς Ω W " ΩW Ω´1 " W 1 .(46) W, W 1 are general elements of the algebra of the Weyl-Heisenberg group (42). The most general transformation between a primed and unprimed basis is Z 1 α " a β α Z β`xα I, I 1 " c α Z α`d I.(47) The commutator [Z 1 α , I 1 s " 0 requires c α " 0 so that I 1 " dI with d P Rzt0u. Next, ζ α,β I 1 " rZ 1 α , Z 1 β s " " a κ α Z κ`xα , a κ β Z κ`xβ ı " 1 d a δ α a γ β ζ δ,γ I 1 .(48) This has the solution a β α " δΣ β α and d " δ 2 . Therefore, for W pz, ιq " z κ Z κ`ι I we have W pz 1 , ι 1 q " ς Ω W pz, ιq " z 1 κ Z κ`ι 1 I (49) with[2] z 1 " δΣz, ι 1 " ιδ 2`z¨x .(50) To determine the group with elements Ω that satisfies (46, 50), we can use the matrix realization of the algebra given in (43). As Ω is nonsingular, (46) is equivalent to ΩW pz, ιq " W pz 1 , ι 1 qΩ.(51) where Ω is a 2n`2 dimensional square matrix. We can write Ω in terms of the submatrices Ω "¨a c z f d j g h ‚ (52) where j, d, r, h, e P R, c, w, f, g P R 2n and a is a 2n dimensional square submatrices and then solve (51) to obtain Ωpδ, Σ, z, ιq "¨δ Σ 0 ź δz t ζΣ δ 2 2ι 0 0 1‚(53) where z P R 2n , δ P D " Rzt0u, ι P R and Σ t ζΣ " ζ and so Σ P Spp2nq. Direct matrix multiplication shows that the elements Ωpδ, Σ, w, rq define a group that we call aut Hpnq with product and inverse Ωpδ 2 , Σ 2 , z 2 , ι 2 q " Ωpδ 1 , Σ 1 , z 1 , ι 1 qΩpδ, Σ, z, ιq " Ωpδ 1 δ, Σ 1 Σ, z 1`δ1 Σ 1 z, ι 1`δ1 2 ι´1 2 δ 1 z 1t ζΣ 1 zq,(54)Ωpδ, Σ, z, ιq´1 " Ωpδ´1, Σ´1,´δ´1Σ´1z,´δ´2ιq,(55) where the identity element is e " t1, 1 2n , 0, 0u. From these relations, we can explicitly compute that automorphisms of the algebra given in (46) W pz 1 , ι 1 q " ς Ωpδ,Σ,z 2 ,ι 2 q W pz, ιq(56) where z 1 " δΣz, ι 1 " ιδ 2´δ z 2 t ζΣ¨z.(57) Comparing with the general expression given in (50), they are equivalent where we identify x " δz 2 t ζΣ. As det Σ ‰ 0andδ ‰ 0, there is a bijection between values of x and z 2 . Using these relations, the next step is to show that the group aut Hpnq has the form of a semidirect product 6 aut Hpnq » pD b Spp2nqq b s Hpnq.(58) First, using the group product and inverse (54-55), we can establish that D, Spp2nq and Hpnq are subgroups of aut Hpnq with elements Ωpδ, 1 2n , 0, 0q P D, Ωp1, Σ, 0, 0q » Σ P Spp2nq Ωp1, 1 2n , z, ιq " Υpz, ιq P Hpnq (59) The direct product D b Spp2nq is immediately established from the special case of the group multiplication (54) Ωpδ, Σ, 0, 0q " Ωpδ, 1 2n , 0, 0qΩp1, Σ, 0, 0q " Ωp1, Σ, 0, 0qΩpδ, 1 2n , 0, 0q,(60) The semidirect product in (58) is established by first noting that pD b Spp2nqq X Hpnq » tΩpδ, Σ, 0, 0qu X tΩp1, 1 2n , z, ιqu » e,(61) Then, using the group product (54), Ωp1, 1 2n , z, ιqΩpδ, Σ, 0, 0q " Ωpδ, Σ, z, ιq.(62) Direct computation using (54-55) shows that the Weyl-Heisenberg subgroup Hpnq is a normal subgroup with the automorphisms given by ς Ωpδ 1 ,Σ 1 ,z 1 ,ι 1 q Υpz, ιq " Υpδ 1 Σ 1 z, δ 1 2 ι´δ 1 z 1 t ζΣ 1 zq.(63) This establishes that aut Hpnq has the semidirect product form given in (58). The right associative property of the semidirect product allows this to be written as aut Hpnq » pD b Spp2nqq b s Hpnq » D b s HSpp2nq(64) where HSpp2nq is a semidirect product of the form HSpp2nq » Spp2nq b s Hpnq(65) This the local characterization of the automorphism group. It remains to consider any global topological properties that could result in a larger group that behaves the same locally. The group D may be written as the direct product D » Z 2 b D`where D`» pR`,ˆq is the positive reals considered as a group under multiplication. Z 2 is the discrete group with two elements t˘1u. D`is simply connected but D has two components, D{D`» Z 2 . Therefore, the connected component of the group is aut c Hpnq » D`b s Spp2nq b s Hpnq.(66) Hpnq and D`are simply connected and Spp2nq is connected with fundamental group Z. Its simply connected universal cover is denoted Spp2nq with π : Spp2nq Ñ Spp2nq : Σ Ñ Σ " πpΣq, ker π » Z.(67) Therefore, by the universal covering theorem, 6 The definition of a semidirect product is reviewed in Definition 1 in Appendix A. Aut c Hpnq » aut c Hpnq » D`b s Spp2nq b s Hpnq,(68) is well defined and unique with the following group product and inverse Ωpδ 2 , Σ 2 , z 2 , ι 2 q " Ωpδ 1 , Σ 1 , z 1 , ι 1 qΩpδ, Σ, z, ιq " Ωpδ 1 δ, Σ 1 Σ, z 1`δ1 Σ 1 z, ι 1`δ1 2 ι´1 2 δ 1 z 1t ζΣ 1 zq,(69) Ωpδ, Σ, z, ιq´1 " Ωpδ´1, Σ´1,´δ´1Σ´1z,´δ´2ιq. where z P R 2n , δ P D`, ι P R and Σ P Spp2nq. Note that in these expressions Σ " πpΣq. The expression for automorphisms of the Weyl-Heisenberg subgroup remains the same as given in (63). The cover of a disconnected group may be defined to be the central extension of the group with a discrete central group. The problem is, that in general, this does not give a unique cover and so this must be checked on a case by case basis. This is discussed in Appendix C where we show that Aut Hpnq » aut Hpnq » D b s Spp2nq b s Hpnq(71) is unique and well defined. It has the group product and inverse given in (69-70) where now δ P D. Again, the automorphisms of the Weyl-Heisenberg subgroup remains the same as given in (63). The group Aut Hpnq is the largest group that the topological properties admit that is homomorphic to aut Hpnq and therefore we completed the proof of Theorem 1. 2.3. Subgroup of automorphism group with invariant center. The action of the automorphism group on the algebra is given in (57). Invariance of the central element requires δ " 1 which is the unit element for D`. Thus the maximal symmetry group that leaves the center of the Weyl-Heisenberg algebra invariant is HSpp2nq. As given in (22), the central extension of HSpp2nq is equivalent to the central extension of the inhomogeneous symplectic group familiar from classical mechanics. H | Spp2nq » HSpp2nq » I | Spp2nq (72) This is a very remarkable fact. The central extension of the Ap2nq is generally np2n´1q dimensional. However, because it is a subgroup of I | Spp2nq, the Lie algebra relations with the symplectic group constrain the central extension of the abelian normal subgroup to be precisely the one dimensional extension that is the Weyl-Heisenberg group. The group product and inverse are given by (69-70) with δ " 1. 2.3.1. Symplectic group factorization. The defining condition for the real symplectic group Spp2nq is Σ t ζΣ " ζ (73) where ζ is the symplectic matrix defined in (41). Matrix realizations of elements of the real symplectic group may be written as Σ "ˆΣ 1 Σ 2 Σ 3 Σ 4˙( 74) where Σ a , a " 1, .., 4 are nˆn submatrices. The symplectic condition (73) immediately results in the relations Σ t 1 Σ 4´Σ t 3 Σ 2 " 1 n , Σ t 1 Σ 3 " pΣ t 1 Σ 3 q t , Σ t 2 Σ 4 " pΣ t 2 Σ 4 q t .(75) A matrix realization of a Lie group is a coordinate system. As DetpΣq " 1, it follows that the determinate of at least one of the Σ a , a " 1, ..., 4, must be nonzero. These correspond to different coordinate patches for the manifold underlying the symplectic group. Assume DetpΣ 1 q ‰ 0. Then [9], Σpα, β, γq "ˆ1 n 0 γ 1 n˙ˆα´1 0 0 α t˙ˆ1 n β 0 1 n˙,(76) where we define α " pΣ 1 q´1, β " pΣ 1 q´1Σ 2 , γ " Σ 3 pΣ 1 q´1.(77) It follows from (75) that β " β t and γ " γ t . The matrix realizations of elements of the symplectic group factor as Σpα, β, γq " Σ´pγqΣ˝pαqΣ`pβq (78) where Σ˝pαq " Σpα, 1 n , 1 n q P Upnq, Σ`pβq " Σp1 n , β, 1 n q P Apmq, Σ´pγq " Σp1 n , 1 n , γq P Apmq.(79) and m " npn´1q 2 . Furthermore, note that ζΣ´pγqζ´1 " Σ`p´γq. A similar argument applies if we instead assume DetpΣ 4 q ‰ 0. Both of these coordinate patches contain the identity, 1 2n but neither contains the element ζ. These require us to consider the case with either Σ 2 or Σ 3 to be assumed to be nonsingular. In this case, define r Σ " Σζ´1 "ˆΣ 2´Σ1 Σ 4´Σ3˙.(81) The r Σ also satisfy the symplectic condition as ζ " Σ t ζΣ " ζ t r Σ t ζ r Σζ ñ r Σ t ζ r Σ " ζ.(82) This symplectic condition results in the identities Σ t 4 Σ 1´Σ t 2 Σ 3 " 1 n , Σ t 2 r Σ 4 " pΣ t 2 Σ 4 q t , Σ t 1 Σ 3 " pΣ t 1 Σ 3 q t .(83) We can now assume DetpΣ 2 q ‰ 0 and the analysis proceeds as before with α " pΣ 2 q´1, β "´pΣ 2 q´1Σ 1 , γ " Σ 4 pΣ 2 q´1, In this case the factorization must include the symplectic matrix from (82) Σpα, β, γq " Σ´pγqΣ˝pαqΣ`pβqζ. Finally a similar argument applies for the coordinate patch DetpΣ 3 q ‰ 0. Both of these coordinate patches contain the element ζ but do not contain the identity The expressions (78) and (85) can be combined into a single expression Σ pα, β, γq " Σ´pγqΣ˝pαqΣ`pβqζ . where P t0, 1u. 2.3.2. Lie Algebra. The Lie algebra of the symmetry group HSpp2nq is the same as the Lie algebra of HSpp2nq. It may be directly computed from its matrix realization. It is convenient to use a basis for the algebra of the symplectic group corresponding to the factorized form (78). Let the A i,j be the generators of the unitary subgroup with elements Σpαq P Upnq, and B i,j the generators of the abelian subgroup with elements Σpβq P Apmq and C i,j the generators of the abelian subgroup with elements Σpγq P Apmq. The abelian generators are symmetric, B i,j " B j,i and C i,j " C j,i . A general element is written as Z " α i,j A i,j`β i,j B i,j`γ i,j C i,j`p i Q i`q i P i`ι I.(87) Straightforward computation shows that these generators of Spp2nq satisfy the Lie algebra rA i,j , A k,l s " δ i,l A j,k´δj,k A i,l , rA i,j , B k,l s " δ j,k B i,l`δj,l B i,k , rA i,j , C k,l s "´δ i,k C j,l´δi,l C k,j , rB i,j , C k s " δ i,k A j,l`δi,l A j,k`δj,k A i,l`δj,l A i,k .(88) The nonzero commutators of the algebra of HSpp2nq are the above relations for the symplectic generators together with the Weyl-Heisenberg generators are rA i,j , Q k s " δ j,k Q i , rC i,j , Q k s " δ j,k P i`δi,k P j , rA i,j , P k s "´δ i,k P j , rB i,j , P k s " δ j,k Q i`δi,k Q j , rP i , Q j s " δ i,j I. The symplectic generators may be realized in the enveloping algebra up to a central element [10]. This will be important when we discuss the representations in Section 3.2. r A i,j " Q i P j , r B i,j " Q i Q j , r C i,j " P i P j .(90) Clearly B i,j " B j,i and C i,j " C j,i . Then, using the Weyl-Heisenberg commutation relations (5), this defines the commutation relations, up to the central element, I, " r A i,j , r A k,l ı " Ipδ i,l r A j,k´δj,k r A i,l q, " A i,j , r B k,l ı " Ipδ j,k r B i,l`δj,l r B i,k q, " A i,j , r C k,l ı "´Ipδ i,k r C j,l`δi,l r C k,j q, " r B i,j , r C k ı " Ipδ i,k r A j,l`δi,l r A j,k`δj,k r A i,l`δj,l r A i,k q.(91) Quantum symmetry: Projective representations The projective representations of the maximal symmetry group ISpp2nq are equivalent to the ordinary unitary representations of its central extension HSpp2nq. These unitary irreducible representations may be determined using the Mackey theorems for semidirect product groups. The first step in applying the Mackey theorem for semidirect products is to determine the unitary irreducible representations of the Weyl-Heisenberg normal subgroup. While these are well known, the method of constructing them using the Mackey theorems applied to the semidirect product of two abelian groups (2) does not appear to be as well known [11]. We review this briefly in order to introduce the Mackey theorems and also to show how the unitary irreducible representations of the symmetry group can be constructed completely from first principles. 3.1. Unitary irreducible representations of the Weyl-Heisenberg group. The Mackey theorem for semidirect products with an abelian normal subgroup are given in Theorem 10 in Appendix A [12]. We choose the normal subgroup (27) with elements Υpp, 0, ιq P Apn`1q. The unitary irreducible representations ξ of the abelian normal subgroup are the phases acting on the Hilbert space H ξ " C ξpΥpp, 0, ιqq|φy " e ipι p I`p i p Qiq |φy " e ipιλ`p¨αq |φy, |φy P C. The hermitian representation of the algebra has the eigenvalues that are given by p Q i |φy " ξ 1 pQ i q |φy " α i |φy, p I |φy " ξ 1 pIq |φy " λ |φy, where α P R n and λ P R. The characters ξ α,λ are parameterized by the eigenvalues α, λ and the equivalence classes that are elements of the unitary dual, rξ α,λ s P U Apn`1q » R n`1 . Each equivalence class has the single element rξ α,λ s " ξ α,λ . The action of the elements Υp0, q, 0q P Apnq of the homogeneous group on these representations is given by the dual automorphisms p ς Υp0,q,0q ξ α,λ˘p Υpp, 0, ιqq |φ D " ξ α,λ pς Υp0,q,0q Υpp, 0, ιqq|φ D " ξ α´λq,λ pΥpp, 0, ιqq |φy. In simplifying this expression, we have used (30) and (92). The little group is the set of Υp0, q, 0q P K˝that satisfy the fixed point equation (134), p ς Υp0,q,0q ξ α,λ " ξ α´λq,λ " ξ α,λ . The solution of the fixed point condition requires that α´λq " α. The λ " 0 solution for which the little group is Apnq is the degenerate case corresponding to the homomorphism Hpnq Ñ Ap2nq with kernel Ap1q. This is just the abelian group that is not considered further here. The faithful representation with λ ‰ 0 requires p " 0, and therefore has the trivial little group K˝» e » tΥp0, 0, 0qu. The stabilizer is G˝» Apn`1q. The orbits are O λ " p ς Υp0,q,0q rξ α,λ s|q P R n ( " tξ q,λ |q P R n u , λ P Rz t0u . All representations in the orbit are equivalent for the determination of the semidirect product unitary irreducible representations. A convenient representative of the equivalence class is ξ 0,λ . The unitary representations σ of the trivial little group are trivial and therefore the representations of the stabilizer are just ˝" ξ 0,λ . The Hilbert space H σ is also trivial and therefore the Hilbert space of the stabilizer is H ˝" H σ b H ξ » C.K " G{G˝" Hpnq{Apn`1q » Apnq » R n ,(97) with the natural projection π and a section Θ π : Hpnq Ñ K : Υpp, q, ιq Þ Ñ k q , Θ : K Ñ Hpnq : k q Þ Ñ Θpk q q " Υp0, q, 0q. These satisfy πpΘpa q qq " a q and so π˝Θ " Id K as required. Using (2), an element of the Weyl-Heisenberg group Hpnq can be written as, Υpp, q, ιq " Υp0, q, 0qΥpp, 0, ι`1 2 p¨qq.(99) The cosets are therefore defined by k q " Υp0, q, 0qΥpp, 0, ι`1 2 p¨qq|p P R n , ι P R ( " tΥp0, q, 0qApn`1qu(100) Note that Υpp, q, ιqk x " k x`q , x P R n .(101) The Mackey induced representation theorem can now be applied straightforwardly. First, the Hilbert space is H " L 2 pK, H ˝q » L 2 pR n , Cq.(102) Next the Mackey induction Theorem 8 yields ψ 1 pk x q " p pΥpp, q, ιqqψq´Υpp, q, ιq´1k x¯" ˝p Υpa˝, 0, ι˝qqψpk x´q q (103) Using the Weyl-Heisenberg group product (2), Υpp˝, q˝, ι˝q " Θpk x q´1Υpp, q, ιqΘpΥpp, q, ιq´1k x q " Υp0,´x, 0qΥpp, q, ιqΥp0, x´q, 0q " Υpp, 0, ι`p¨`x´1 2 q˘q. (104) We lighten notation using the isomorphism k x Þ Ñ x. The induced representation theorem then yields ψ 1 pxq " ξ 0,λ pΥpp, 0, ι`x¨p´1 2 p¨qqψpx´qq " e iλpι`x¨p´1 2 p¨qq ψpx´qq. Using Taylor expansion, we can write ψpx´qq " e´q i B Bx i ψpxq.(106) The Baker Campbell-Hausdorff formula [20] enables us to combine the exponentials ψ 1 pxq " e ipλι`λp i xi`q i i B Bx i q ψpxq " e ipι p I`p i p Qi`q i p Piq ψpxq.(107) The representation of the algebra is therefore p Iψpxq " λψpxq, p Q i ψpxq " λx i ψpxq, p P i ψpxq " i B Bx i ψpxq,(108) that satisfies the Heisenberg commutation relations (1). This analysis can also be carried out choosing Υp0, q, ιq P Apn`1q to be the elements of the normal subgroup and this yields the representation with p P i diagonal. Unitary irreducible representations of HSpp2nq. We consider next the unitary irreducible representations of the HSpp2nq group HSpp2nq » Spp2nq b s Hpnq. As HSpp2nq is the central extension of ISpp2nq, the projective representations of ISpp2nq are equivalent to the ordinary unitary representations of HSpp2nq. The unitary irreducible representations of HSpp2nq may be determined using Mackey Theorem 9 for the nonabelian normal subgroup case. The faithful unitary representations of the Weyl-Heisenberg group are given in the previous section (105). The next step in applying the Mackey's theorem is to determine the ρ representation of the stabilizer G˝Ă HSpp2nq. The unitary representation ρ acts on H ξ » L 2 pR n , Cq such that ρpΩ˝qξpΥpz, ιqqρpΩ˝q´1 " ξpς Ω˝Υ pz, ιqq, Ω˝P G˝. The representation ρ factors into ρpΩ˝pδ, Σ, w, rqq " ξpΥpw, rqqρpΣq, where again for notational brevity Σ " Ωp1, Σ, 0, 0q. We already have characterized the inner automorphisms. The automorphisms corresponding factor as ξpΥpw, rqqξpΥpz, ιqqξpΥpw, rqq´1 " ξpς Υpw,rq Υpz, ιqq, ρpΣqξpΥpz, ιqqρpΣq´1 " ξpς ΩpΣq Υpz, ιqq " ξpΥpπpΣqz, ιqq. (112) where Σ P Spp2nq and π : Spp2nq Ñ Spp2nq. The inner automorphisms are already characterized as we know the unitary irreducible representations ξ. Consider next the representation ρpΣq of the symplectic group Spp2nq. The hermitian representation of the symplectic generators is p A i,j " ρ 1 pA i,j q " λ p Q i p P j , p B i,j " ρ 1 pB i,j q " λ p Q i p Q j , p C i,j " ρ 1 pC i,j q " λ p P i p P j .(113) Clearly p B i,j " p B j,i and p C i,j " p C j,i . Then, using the Heisenberg commutation relations (1), this defines a hermitian realization of the Lie algebra of the automorphism group acting on the Hilbert space H ξ » L 2 pR n , Cq. " p A i,j , p A k,l ı " ipδ i,l p A j,k´δj,k p A i,l q, " p A i,j , p B k,l ı " ipδ j,k p B i,l`δj,l p B i,k q, " p A i,j , p C k,l ı "´ipδ i,k p C j,l`δi,l p C k,j q, " p B i,j , p C k ı " ipδ i,k p A j,l`δi,l p A j,k`δj,k p A i,l`δj,l p A i,k q,(114)" p A i,j , p Q k ı " iδ j,k p Q i , " p C i,j , p Q k ı " ipδ j,k p P i`δi,k p P j q, " p A i,j , p P k ı "´iδ i,k p P j , " p B i,j , p P k ı " ipδ j,k p Q i`δi,k p Q j q, " p P i , p Q j ı " iδ i,j p I.(115) Therefore, there exists a ρ 1 representation for the entire algebra of HSpp2nq and therefore the stabilizer is the group itself, G˝» HSpp2nq. This explicate construction of the algebra shows that the representation ρpΣq exists. Consequently, the Mackey induction theorem is not required. The ρpΣq representation is precisely (up to an overall phase) the metaplectic representation originally studied by Weil [13], [2]. We can construct this explicitly using the factorization of the symplectic group (86). We can consider each of the factors separately as ρpΣp , α, β, γq " ρpΣ´pγqqρpΣ˝pαqqρpΣ`pβqqρpζ q, and each of these factors can be applied separately to determine the ρ representation. The unitary representations of Σpβq P Apmq, m " npn`1q 2 in a basis with p Q i diagonal are ρpΣ`pβqq|ψ λ pxqy " e iα i,j p Bi,j |ψ λ pxqy " e i λ β i,j xixj |ψ λ pxqy . (117) The representations of the elements of the unitary group Σpαq P Upnq are ρpΣ˝pαqq|ψ λ pxqy " | det A|´1 2 |ψ λ pA´1xq D .(118) The symplectic matrix exchanges the p and q degrees of freedom, ς ζ Υpp, q, ιq " Υpq,´p, ιq. As is well known, the unitary representation of this is the Fourier transform, ρpζq " f where ρpΥpp, q, ιqqf |ψ λ pxqy " f ρpΥpq,´p, ιqq|ψ λ pxqy , where the Fourier transform is defined as usual by r ψpyq " f ψpxq " p2πiq´n 2 ż e´i x¨y ψpxqd n x,(120) and where p Q i |ψ λ pxqy " λx i |ψ λ pxqy, p P i | r ψ λ pyq E " y i | Ă ψ λ pyq E .(121) Finally, the ρpΣ`pβqq representation can be computed using (80) in a basis with p Q i diagonal giving ρpΣ´pγqq |ψ λ pxqy " f ρpΣ`p´γqqf´1|ψ λ pxq,(122) and the ρpΣ`p´γqq is given by (116). Putting all of these together gives the representation ρpΣq up to a phase. While one would expect the phase to be m P Z dependent, it actually only is two valued˘1 P Z 2 . The unitary representations of the double cover metaplectic group Mpp2nq are also a representation of Spp2nq due to the homomorphism (141). Of course, all of these calculations could also be done in a basis with p P i diagonal. As the stabilizer is the full group, Mackey induction is not required and the unitary irreducible representations υ of HSpp2nq are given by υpΩp1, Σ, z, ιqq|ψpxqy " σpΣq b ξpΥpz, ιqqρpΣq|ψpxqy where σ are ordinary unitary irreducible representations of Spp2nq, ρ are the metaplectic representation of Spp2nq given above and ξ are the unitary irreducible representations of Hpnq given in Section 3.1. The ordinary unitary representations of the symplectic group have been partially characterized [8][9]. A complete set of unitary irreducible representations of the covering group Spp2nq appears to be an open problem. Summary We have determined the projective representations of the inhomogeneous symplectic group. This is the maximal symmetry whose projective representations transform physical states such that the Heisenberg commutation relations are valid in all of the transformed states. The inhomogeneous symplectic symmetry is well known from classical mechanics. It acts on classical phase space with position and momentum degrees of freedom. The projective representations that define the quantum symmetry require its central extension which introduces the non-abelian structure of the Weyl-Heisenberg group, I | Spp2nq » Spp2nq b s Hpnq. The non-abelian structure is a direct result of the fact that transition probabilities are the square of the norm of physical states. Consequently, the physical states are defined up to a phase and the action of a symmetry group is given by the projective representations. This is the underlying reason for the non-abelian structure, or quantization. Any symmetry of quantum mechanics that preserves the position, momentum Heisenberg commutation relations must be a subgroup of this maximal symmetry. On the other hand, we now understand special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group [5], [6]. The central extension of this group does not admit an algebraic extension. For the connected component, the central extension is therefore the cover that we call the Poincaré group which for n " 3 is P " SLp2, Cq b s Ap4q. 7 Special relativistic quantum mechanics is formulated in terms of the unitary representations of the Poincaré group. There is however, no mention of the Weyl-Heisenberg group which plays a fundamental role in the original formulation of quantum mechanics. Symmetry is one of the most fundamental concepts of physics. We have the case where we have a quantum symmetry for the Weyl-Heisenberg of quantum 7 The full inhomogeneous group is given in terms of the orthogonal group Op1, nq that has 4 disconnected components. The discrete Z 2,2 symmetry is P, T and PT symmetry. Its central extension is not unique and it gives rise to the Pin group ambiguity. On the other hand the SOp1, nq group has 2 components but does have a unique central extension that is the Spin group. The discrete Z 2 symmetry is the PT symmetry. mechanics that is the projective representations of a classical symmetry on phase space. On the other hand, the quantum symmetry for the Minkowski metric of special relativity is given in terms of a classical symmetry on position-time space, that is, spacetime. Quantum mechanics and special relativity have at best, an uneasy marriage. Perhaps it is due to this underlying disparity in the most basic symmetries of these theories. The standard approach is to ignore the quantum symmetry described in this paper and formulate special relativistic quantum mechanics as the projective representations of the inhomogeneous group. If we truly are to bring together quantum mechanics and special relativity, we must first reconcile these basic symmetries and find a symmetry that encompasses both. This can be done in a remarkably straightforward manner and results in a theory that, in a physical limit, results in the usual formulation of special relativistic quantum mechanics. But, before the limit is taken, it points to a theory incorporating both symmetries that may give further understanding of the unification of quantum mechanics and relativity [14], [15], [16]. In this theory, physics takes place in extended phase space and there is no invariant global projection that gives physics in position-time space (i.e. space-time). Generally, local observers with general non-inertial trajectories construct different space-times as subspaces of extended phase space. The usual Lorentz symmetry continues to hold exactly for inertial trajectories but is generalized in a remarkable manner for non-inertial trajectories. Appendix A: Key Theorems In this appendix we review a set of definitions and theorems that are fundamental for the application of symmetry groups in quantum mechanics. We state the theorems only and refer the reader to the cited literature for full proofs. Definition 1. A group G is a semidirect product if it has a subgroup K (referred to as the homogeneous subgroup) and a normal subgroup N such that K X N " e and G » N K. Our notation for a semidirect product is G » K b s N 8 [17]. It follows directly that a semidirect product is right associative in the sense that D » pA b s Bq b s C implies that D » A b s pB b s Cq and so brackets can be removed. However D » A b s pB b s Cq does not necessarily imply D » pA b s Bq b s C as B is not necessarily a normal subgroup of A. Definition 2. An algebraic central extension of a Lie algebra g is the Lie algebra q g that satisfies the following short exact sequence where z is the maximal abelian algebra that is central in q g, 0 Ñ z Ñ q g Ñ g Ñ 0.(124) where 0 is the trivial algebra. Suppose tX a u is a basis of the Lie algebra g with commutation relations rX a , X b s " c c a,b X c , a, b " 1, ...r. Then an algebraic central extension is a maximal set of central abelian generators tA α u, where α, β, ... " 1, ..m, such that rA α , A β s " 0, rX a , A α s " 0, rX a , X b s " c c a,b X c`c α a,b A α .(125) The basis tX a , A α u of the centrally extended Lie algebra must also satisfy the Jacobi identities. The Jacobi identities constrain the admissible central extensions of the algebra. The choice X a Þ Ñ X a`Aa will always satisfy these relations and this trivial case is excluded. The algebra q g constructed in this manner is equivalent to the central extension of g given in Definition 2. Definition 3. The central extension of a connected Lie group G is the Lie group q G that satisfies the following short exact sequence where Z is a maximal abelian group that is central in q G e Ñ Z Ñ q G π Ñ G Ñ e.(126) The abelian group Z may always be written as the direct product Z » ApmqbA of a connected continuous abelian Lie group Apmq » pR m ,`q and a discrete abelian group A that may have a finite or countable dimension [10]. The exact sequence may be decomposed into an exact sequence for the topological central extension and the algebraic central extension, e Ñ A Ñ G πÑ G Ñ e, e Ñ Apmq Ñ q G r π Ñ G Ñ e.(127) where π " π˝˝r π. The first exact sequence defines the universal cover where A » ker π˝is the fundamental homotopy group. All of the groups is in the second sequence are simply connected and therefore may be defined by the exponential map of the central extension of the algebra given by Definition 2. In other words, the full central extension may be computed by determining the universal covering group of the algebraic central extension. Definition 4. A ray Ψ is the equivalence class of states |ψ γ y that are elements of a Hilbert space H up to a phase, Ψ " e iω |ψ D |ω P R ( , |ψ y P H.(128) Note that the physical probabilities that are the square of the modulus depend only on the ray | pΨ β , Ψ α q | 2 " |xψ β |ψ α y| 2 for all |ψ γ y P Ψ. For this reason, physical states in quantum mechanics are defined to be rays rather than states in the Hilbert space Definition 5. A projective representation of a symmetry group G is the maximal representation such that for | r ψ γ y " pgq|ψ γ y, the modulus is invariant |x r ψ β | r ψ α y| 2 " |xψ β |ψ α y| 2 for all |ψ γ y, | r ψ γ y P Ψ. Theorem 2. (Wigner, Weinberg): Any projective representation of a Lie symmetry group G on a separable Hilbert space is equivalent to a representation that is either linear and unitary or anti-linear and anti-unitary. Furthermore, if G is connected, the projective representations are equivalent to a representation that is linear and unitary [1], [11]. This is the generalization of the well known theorem that the ordinary representation of any compact group is equivalent to a representation that is unitary. For a projective representation, the phase degrees of freedom of the central extension enables the equivalent linear unitary or anti-linear anti-unitary representation to be constructed for this much more general class of Lie groups that admit representations on separable Hilbert spaces. (A proof of the theorem is given in Appendix A of Chapter 2 of [6]). The set of groups that this theorem applies to include all the groups that are studied in this paper. Noting that a representation is a homomorphism, This theorem follows straightforwardly from the properties of homomorphisms. As a consequence, the set of degenerate representations of a group is characterized by its set of normal subgroups. A faithful representation is the case that the representation is an isomorphism. Theorem 5. (Levi) Any simply connected Lie group is equivalent to the semidirect product of a semisimple group and a maximal solvable normal subgroup [18] As the central extension of any connected group is simply connected, the problem of computing the projective representations of a group always can be reduced to computing the unitary irreducible representations of a semidirect product group with a semisimple homogeneous group and a solvable normal subgroup. The unitary irreducible representations of the semisimple groups are known and the solvable groups that we are interested in turn out to be the semidirect product of abelian groups. Theorem 6. Any semidirect product group G » K b s N is a subgroup of a group homomorphic to the group of automorphisms of N [13]. The proof follows directly from the definition of the semidirect product and an automorphism group. Theorem 7. The automorphism group of a simply connected group is isomorphic to the automorphism group of its Lie algebra. [18] 5.1. Mackey theorems for the representations of semidirect product groups. The Mackey theorems are valid for a general class of topological groups but we will only require the more restricted case G » K b s N where the group G and subgroups K, N are smooth Lie groups. The central extension of any connected Lie group is simply connected and therefore generally has the form of a semidirect product due to Theorem 5 (Levi). Theorem 6 further constrains the possible homogeneous groups K of the semidirect product given the normal subgroup N . The first Mackey theorem is the induced representation theorem that gives a method of constructing a unitary representation of a group (that is not necessarily a semidirect product group) from a unitary representation of a closed subgroup. The second theorem gives a construction of certain representations of a certain subgroup of a semidirect product group from which the complete set of unitary irreducible representations of the group can be induced. This theorem is valid for the general case where the normal subgroup N is a nonabelian group. In the special case where the normal subgroup N is abelian, the theorem may be stated in a simpler form. Theorem 8. (Mackey). Induced representation theorem. Suppose that G is a Lie group and H is a Lie subgroup, H Ă G such that K » G{H is a homogeneous space with a natural projection π : G Ñ K, an invariant measure and a canonical section Θ : K Ñ G : k Þ Ñ g such that π˝Θ"Id K where Id K is the identity map on K. Let ρ be a unitary representation of H on the Hilbert space H ρ : ρphq : H ρ Ñ H ρ : |ϕ y Þ Ñ | r ϕ y " ρphq |ϕ y , h P H. Then a unitary representation of a Lie group G on the Hilbert space H , pgq : H Ñ H : |ψ y Þ Ñˇˇr ψ E " pgq |ψ y , g P G, may be induced from the representation ρ of H by defining r ψpkq " p pgqψq pkq " ρpg˝qψpg´1kq, g˝" Θpkq´1gΘpg´1kq, where the Hilbert space on which the induced representation acts is given by H » L 2 pK, H ρ q [14], [13]. The proof is straightforward given that the section Θ exists by showing first that g˝P kerpπq » H and therefore ρpg˝q is well defined. Definition 6. (Little groups): Let G " K b s N be a semidirect product. Let rξs P U N where U N denotes the unitary dual whose elements are equivalence classes of unitary representations of N on a Hilbert space H ξ . Let ρ be a unitary representation of a subgroup G˝" K˝b s N on the Hilbert space H ξ such that ρ |N " ξ. The little groups are the set of maximal subgroups K˝such that ρ exists on the corresponding stabilizer G˝» K˝b s N and satisfies the fixed point equation p ς ρpkq rξs " rξs , k P K˝.(130) In this definition the dual automorphism is defined bỳ p ς ρpgq ξ˘phq " ρpgqρphqρpgq´1 " ρpghg´1q " ξpς g hq(131) for all g P G˝and h P N . The equivalence classes of the unitary representations of N are defined by rξs " p ς ξphq ξ|h P N ( .(132) A group G may have multiple little groups K˝α whose intersection is the identity element only. We will generally leave the label α implicit. This major result and its proof are due to Mackey [14]. Our focus in this paper is on applying this theorem. 5.1.1. Abelian normal subgroup. The theorem simplifies for special cases where the normal subgroup N is an abelian group, N » Apnq. An abelian group has the property that its unitary irreducible representations ξ are the characters acting on the Hilbert space H ξ » C, ξpaq |φy " e ia¨ν |φy , ν P R n (133) The unitary irreducible representations are labeled by the ν i that are the eigenvalues of the hermitian representation of the basis tA i u of the abelian Lie algebra, p A i |φy " ξ 1 pA i q |φy " ν i |φy . The equivalence classes rξs P U Apnq each have a single element rξs » ξ as, for the abelian group, the expression (131) is trivial. The representations ρ act on H ξ » C and are one dimensional and therefore must commute with the ξ. Therefore, in equation (130), ρpgqξphqρpgq´1 " ξphq and (129) simplifies to ξpaq " ξpς k aq " ξpkak´1q, a P Apmq, k P K˝. H σ b C. The theorem then proceeds as in the case of the general Theorem 9. Appendix B: Polarized Realization of the Weyl-Heisenberg group The maps ϕ˘defined in (35) is an isomorphism. Therefore the Υ˘pp, q, ι˘q are elements of the Weyl-Heisenberg group realized in another coordinate system of matrices. These realizations are referred to as the polarized realizations [2]. The group products in these coordinates are computed directly from (3)(4) to be Υ`pp 1 , q 1 , ι 1 qΥ`pp, q, ιq " Υ`pp 1`p , q 1`q , ι`ι`p 1¨q q, Υ´pp 1 , q 1 , ι 1 qΥ´pp, q, ιq " Υ´pp 1`p , q 1`q , ι`ι´q 1¨p q, Υ˘pp, q, ιq´1 " Υ˘p´p,´q,´ι˘p¨qq. (136) Note that the polarized realizations factor directly Υ`p0, q, ιqΥ`pp, 0, 0q " Υ`pp, q, ιq, Υ´pp, 0, ιqΥ´p0, q, 0q " Υ´pp, q, ιq. The existence of the isomorphisms ϕ˘and these two different normal subgroups Apn`1q with elements Υpp, 0, ιq and Υp0, q, ιq whose intersection is the center Z » Ap1q is responsible for many of the remarkable properties of the Weyl-Heisenberg group. In fact, we shall see shortly that the choice of the normal subgroup in determining the unitary representations when applying the Mackey theorems results in unitary representations with either p or q diagonal. The matrix realization corresponds to a coordinate system of the Lie group and is therefore not unique. The polarized matrix realizations are given by the n`2 dimensional square matrices Υ`pp, q, ιq "¨1 q t ι 0 1 n p 0 0 1‚ , Υ´pp, q, ιq "¨1 p t ι 0 1 n q 0 0 1‚ . Appendix C: Extended Central Extension The central extension for a group that is not connected group is not necessarily unique. The central extension for a group that is not connected may be defined by requiring exact sequences both for the cover of the group and the homomorphisms onto the discrete group for the components. For the Z 2 b s HSpp2nq, these sequences are [19] e e e Ó Ó Ó e Ñ Z b Ap1q Ñ HSpp2nq Ñ ISpp2nq Ñ e Ó Ó Ó e Ñ D Ñ Z 2 b s HSpp2nq Ñ Z 2 b s ISpp2nq Ñ e Ó Ó Ó Z 2 Z 2 Z 2 Ó Ó Ó e e e(140) The solution is D » Z 2 bZbAp1q. Therefore the central extension of Z 2 b s ISpp2nq is unique and is given by Z 2 b s HSpp2nq. Q i " B Bp i Υpp, q, ιq| e , P i " BBq i Υpp, q, ιq| e , I "B BιΥpp, q, ιq| e . Mackey induction. The final step is to apply the Mackey induction theorem to determine the faithful unitary irreducible representations of the full Hpnq group. The induction requires the definition of the symmetric space 3. 2 . 1 . 21Stabilizer and ρ representation. The representation ρ of the stabilizer Ga cts on the Hilbert space H ξ and therefore the hermitian representations ρ 1 of the algebra of the stabilizer must be realized in the enveloping algebra of the Weyl-Heisenberg group. The ρ representation restricted to the Weyl-Heisenberg group are given by ρ| Hpnq " ξ where ξ are the unitary irreducible representations of the Weyl Heisenberg group. The faithful representations ξ are given in (105). Theorem 3 . 3(Bargmann, Mackey) The projective representations of a connected Lie group G are equivalent to the ordinary unitary representations of its central extension q G[7],[8].Theorem 2 states that are all projective representations of a connected Lie group are equivalent to a projective representation that is unitary. A phase is the unitary representation of a central abelian subgroup. Therefore, the maximal representation is given in terms of the central extension of the group. Theorem 4 . 4Let G,H be Lie groups and π : G Ñ H be a homomorphism. Then, for every unitary representation r of H there exists a degenerate unitary representation of G defined by " r ˝π. Conversely, for every degenerate unitary representation of a Lie group G there exists a Lie subgroup H and a homomorphism π : G Ñ H where kerpπq ‰ e such that " r ˝π where r is a unitary representation of H. Theorem 9 . 9(Mackey). Unitary irreducible representations of semidirect products. Suppose that we have a semidirect product Lie group G » K b s N , where K, N are Lie subgroups. Let ξ be the unitary irreducible representation of N on the Hilbert space H ξ . Let G˝» K˝b s N be a maximal stabilizer on which there exists a representation ρ on H ξ such that ρ| N " ξ. Let σ be a unitary irreducible representation of K˝on the Hilbert space H σ . Define the representation ˝" σ b ρ that acts on the Hilbert space H˝» H σ b H ξ . Determine the complete set of stabilizers and representations ρ and little groups that satisfy these properties, that we label by α,tpG˝, ˝, H ˝q α u. If for some member of this set G˝» G then for this case the representations are pG, , H q»pG˝, ˝, H ˝q . For the cases where the stabilizer G˝is a proper subgroup of G then the unitary irreducible representations pG, , H q are the representations induced (using Theorem 8) by the representations pG˝, ˝, H ˝q of the stabilizer subgroup. The complete set of unitary irreducible representations is the union of the representations Y α tpG, , H q α u over the set of all the stabilizers and corresponding little groups. ( 135 ) 135Theorem 10.(Mackey). Unitary irreducible representations of a semidirect product with an abelian normal subgroup. Suppose that we have a semidirect product group G » K b s A where A is abelian. Let ξ be the unitary irreducible representation (that are the characters) of A on H ξ » C. Let K˝Ď K be a Little group defined by (134) with the corresponding stabilizers G˝» K˝b s A. Let σ be the unitary irreducible representations of K˝on the Hilbert space H σ . Define the representation ˝" σ b ξ of the stabilizer that acts on the Hilbert space H ˝» We always use ς to define the similarity map ςgh " ghg´1 in what follows. Our notation follows[17]. Another notation commonly used is N¸K. It is just notation; the definition remains the same for both notations. Appendix D: HomomorphismsRepresentations are homomorphisms of a group G. If the homomorphism is an isomorphism, then the representation is said to be faithful and otherwise it is degenerate. Theorem 4 establishes that degenerate representations are faithful representations of groups homomorphic to G. The homomorphisms can be characterized by the normal subgroups that are the kernel of the homomorphism.First we consider the subgroup HSpp2nq that we have noted in(22)is the central extension of ISpp2nq with centerwhere Z is the center of Spp2nq and Ap1q is the center of Hpnq (31). The double cover of Spp2nq is the metaplectic group Mpp2nq. 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In this article we initiate research on locally compact C * -simple groups. We first show that every C * -simple group must be totally disconnected. Then we study C * -algebras and von Neumann algebras associated with certain groups acting on trees. After formulating a locally compact analogue of Powers' property, we prove that the reduced group C * -algebra of such groups is simple. This is the first simplicity result for C * -algebras of non-discrete groups and answers a question of de la Harpe. We also consider group von Neumann algebras of certain non-discrete groups acting on trees. We prove factoriality, determine their type and show nonamenability. We end the article by giving natural examples of groups satisfying the hypotheses of our work.

10.1515/crelle-2016-0026

1505.07793

b360fa7bca49eb83d228cf66a53dcbe0af44a4c0

C * -simplicity of locally compact Powers groups Sven Raum C * -simplicity of locally compact Powers groups In this article we initiate research on locally compact C * -simple groups. We first show that every C * -simple group must be totally disconnected. Then we study C * -algebras and von Neumann algebras associated with certain groups acting on trees. After formulating a locally compact analogue of Powers' property, we prove that the reduced group C * -algebra of such groups is simple. This is the first simplicity result for C * -algebras of non-discrete groups and answers a question of de la Harpe. We also consider group von Neumann algebras of certain non-discrete groups acting on trees. We prove factoriality, determine their type and show nonamenability. We end the article by giving natural examples of groups satisfying the hypotheses of our work. Introduction Group C * -algebras and group von Neumann algebras enjoy a long history. Group von Neumann algebras of discrete groups have been used by McDuff [McD69] to provide examples of a continuum of pairwise non-isomorphic II 1 -factors. Connes' conjecture about W * -superrigidity of group von Neumann algebras of discrete groups with property (T) [Con82] drew analogues with lattices in their ambient Lie groups. Only very recently in a breakthrough result Ioana-Popa-Vaes [IPV10] could provide the first examples of so called W * -superrigid groups at all, leaving Connes' conjecture untouched. Group von Neumann algebras equally well found applications in topology, where L 2 -Betti numbers can be defined thanks to a continuous notion of dimension provided by the operator algebraic viewpoint [Lüc01]. However, until now group von Neumann algebras associated with non-discrete groups drew only minor attention. Group C * -algebras of discrete groups have been investigated with a focus on the Baum-Connes conjecture and on simplicity results, after Kadison asked whether the reduced group C * -algebra C * red (F 2 ) is simple without non-trivial projections [Pow75]. Powers in [Pow75] showed with combinatorial methods that C * red (F 2 ) is simple. He used an averaging argument that formed the basis of most follow up results on C * -simplicity. His argument was put in an abstract context by de la Harpe and lead for example to simplicity results for free products, hyperbolic groups and Baumslag-Solitar groups [dlH07,dlHP09]. Very recently, the astonishing work of Kalantar-Kennedy and Breuillard-Kalantar-Kennedy-Ozawa [BKKO14] basically pushed the question of C * -simplicity to the point, where the only serious open problem had to be solved by group theorists. Shortly afterwards, the conjecture that a discrete group is C * -simple if and only if it has a trivial amenable radical could be proven wrong by Le Boudec in [LB15]. Further, in the same year Kennedy and Haagerup gave a characterisation of C * -simple groups. While Kennedy provides a group theoretic characterisation in terms of recurrent amenable subgroups [Ken15], Haagerup proves the equivalence of C * -simplicity and the Powers averaging property in [Haa15]. Also in representation theory group C * -algebras had a serious impact (see e.g. [Ros94]). The probably easiest approach to classical Peter-Weyl theory studies the group C * -algebra C * (K) of a compact group K . Further the notion of a type I group stems from C * -algebras [Kap49,Gli61]. Their representation theory is completely understood by its irreducible representations [Dix77]. Examples of type I groups include connected semisimple Lie groups [HC65], reductive algebraic groups over non-Archimedean fields [Ber74] and the full group of automorphisms of a tree [FTN91]. Also C * -simplicity has a representation theoretic interpretation, invoking the Fell topology on unitary representations [Fel60]. In fact, a group is C * -simple if and only if the support of its left regular representation is a closed point in the group's unitary dual. In this article, we initiate the study of non-discrete C * -simple groups. Our first result shows that every C * -simple group is totally disconnected, extending a result of Bekka-Cowling-de la Harpe. [BCdlH94,Proposition 4] Theorem A (Theorem 6.1). Let G be a locally compact C * -simple group. Then G is totally disconnected. We then provide first examples of non-discrete C * -simplicity groups, answering a question of de la Harpe [dlH07, Question 5]. Question (de la Harpe). Does there exist a non-discrete second countable locally compact group which is C * -simple? We adapt the combinatorial method of Powers averaging from a discrete setting to the setting of general totally disconnected groups. Then, inspired by work of de la Harpe-Préaux, we study the action of a closed subgroup of Aut(T ) on the boundary ∂T of the tree, in order to obtain by geometric means the necessary input to apply Powers averaging to group C * -algebras of some natural class of non-discrete locally compact groups. The main achievement of this article is hence twofold. On the one hand, we are able to answer de la Harpe's question, giving examples of C * -simple non-discrete second countable locally compact groups. On the other hand, we show how averaging techniques, well known for discrete groups, can be exploited in operator algebras associated with locally compact totally disconnected groups. The class of groups treated in this article, has a geometric and an algebraic formulation. We introduce the following notation, where N G (K) denotes the normaliser of a subgroup K ≤ G and G 0 denotes the kernel of the modular function of a locally compact group. Recall that a tree is called thick if each of its vertices has valency at least 3. Notation (See Theorem 3.6). We say that a locally compact group G satisfies condition ( * ) if one of the following equivalent conditions holds. • There is a locally finite tree T such that G ≤ Aut(T ) as a closed subgroup. Further, G is non-amenable without any non-trivial compact normal subgroup and there is a compact open subgroup K ≤ G such that N G (K) K contains an element of infinite order. • There is a locally finite thick tree T such that G ≤ Aut(T ) as a closed subgroup. Further, G acts minimally on ∂T and there is some point x ∈ ∂T such that G x ≤ G is open and G x ∩ G 0 contains a hyperbolic element. We can now formulate the main theorem of this article. Theorem B (Theorem 6.2). Let G be a group satisfying condition ( * ). Then C * red (G) is simple. We give two applications of our main result. Since group C * -algebras of totally disconnected groups and Hecke C * -algebras are related, our result also implies simplicity of certain reduced Hecke C *algebras (Section 2.5.1), which is in sharp contrast to results previously obtained. Theorem C (Corollary 6.3). Let T be thick tree and Γ ≤ Aut(T ) some not necessarily closed group acting without proper invariant subtree. Let Λ be some vertex stabiliser in Γ and assume that there is a finite index subgroup Λ 0 ≤ Λ such that N Γ (Λ 0 ) Λ 0 contains an element of infinite order. Then the reduced Hecke C * -algebra C * red (Γ, Λ) is simple. As a further corollary of Theorem B, we obtain a result about type I groups acting on trees. A locally compact group G is called type I group if each of its unitary representations generates a type I von Neumann algebra. Theorem D (Corollary 6.4). Let T be a thick tree and G ≤ Aut(T ) be a closed subgroup acting minimally on ∂T . Assume that there is x ∈ ∂T such that • Kx is finite for some compact open subgroup K ≤ G, and • there is some hyperbolic element in G 0 ∩ G x . Then G is not a type I group. Note that our result is in contrast to and motivated by the following conjecture on type I groups. Conjecture. Let G ≤ Aut(T ) be a closed subgroup. Assume that there is a compact open subgroup of G acting transitively on ∂T . Then G is a type I group. Since to a certain extend we are able to control weights on a group C * -algebra, our methods are also able to deal with von Neumann algebraic results. We obtain the following factoriality results for group von Neumann algebras of non-discrete groups and we are able to determine their type in terms of the modular function. We refer the reader to Section 2.4.1 for some facts about Connes' S-invariant S(M ) for a factor M . Theorem E (Theorem 7.2). Let G be a group satisfying condition ( * ). Further assume that some compact open subgroup of G is topologically finitely generated. Then the group von Neumann algebra L(G) is a factor and S(L(G)) = ∆(G). • If G is discrete, then L(G) is a type II 1 factor. • If G is unimodular but not discrete, then L(G) is a type II ∞ factor. • If ∆(G) = λ Z for some λ ∈ (0, 1), then L(G) is a type III λ factor. • If ∆(G) is not singly generated, then L(G) is a type III 1 factor. We also prove non-amenability of the group von Neumann algebra associated with certain groups acting on trees. Theorem F (Theorem 8.2). Let G be a group satisfying condition (*). Further assume that some compact open subgroup of G is topologically finitely generated. Then L(G) is not amenable. Finally, we give concrete examples of groups satisfying the hypotheses of our work. They arise as Schlichting completions of Baumslag-Solitar groups. Theorem G (Theorem 9.2). Let 2 ≤ m ≤ n and consider the relative profinite completion G(m, n) of the Baumslag-Solitar group BS(m, n). Then the following statements are true. • L(G(m, n)) is a non-amenable factor. • If m = n, then G(m.n) is discrete and L(G(m, n)) is of type II 1 . • If m ≠ n, then L(G(m, n)) is of type III m n . • C * red (G(m, n)) is simple. The fact that L(G(m, n)) is a factor and the calculation of its type was obtained with different methods in unpublished work of the author and C.Ciobotaru. Preliminaries Totally disconnected groups For a locally compact group G, we denote by G 0 the kernel of its modular function ∆ ∶ G → R >0 determined by µ(Ag) = ∆(g)µ(A)∆(g) = µ(g −1 (K ∩ gKg −1 )g) µ(K ∩ gKg −1 ) = µ(g −1 (K ∩ gKg −1 )g) µ(K) ⋅ µ(K) µ(K ∩ gKg −1 ) = [K ∶ K ∩ gKg −1 ] [K ∶ K ∩ g −1 Kg] ∈ Q , for all compact open subgroups K ≤ G. In particular, the modular function of a totally disconnected locally compact group takes only values in Q. We need the following two observations, about topologically finitely generated profinite groups. Lemma 2.1. Let G be a topologically finitely generated group. Then for every r ∈ N there are only finitely many closed subgroups of index r. Proof. Let G be a topological group. If H ≤ G is a closed subgroup of index n < ∞, then we can identify G H ≅ {1, . . . , n}. We obtain a continuous homomorphism π ∶ G → S n such that H = π −1 ((S n ) i ) for some i ∈ {1, . . . , n}, where (S n ) i is the stabiliser group of i. Now assume that G is topologically finitely generated. Then there are only finitely many continuous homomorphism G → S n for each n, since the image of G is determined by the image of its generators. Consequently, G has only finitely many closed subgroups of finite index. subgroup L ≤ G satisfies [K ∶ K ∩ L], [L ∶ K ∩ L] < ∞. So the proposition follows from the fact that topological finite generation passes between finite index inclusions. Schlichting completions Let Λ ≤ Γ be an inclusion of discrete groups. It is a discrete Hecke pair if [Λ ∶ Λ ∩ gΛg −1 ] < ∞ for all g ∈ Γ. We define the map ι ∶ Γ → Sym(Γ Λ) by left multiplication ι(g)hΛ = ghΛ. Equipping Sym(Γ Λ) with the topology of pointwise convergence, we put Γ Λ ∶= ι(Γ). This is the Schlichting completion of the Hecke pair Λ ≤ Γ. It is a totally disconnected group equipped with the natural compact open subgroup ι(Λ). Groups acting on trees Let T be a locally finite tree. Then the group of automorphisms Aut(T ) equipped with the topology of pointwise convergence is a totally disconnected locally compact group. Every vertex stabiliser is a compact open subgroup of Aut(T ) and every compact subgroup of Aut(T ) stabilises some vertex or some edge of T . The set of one-sided infinite geodesic rays in T modulo equality at some point is called the boundary ∂T of T . Formally, we have ∂T = {ξ ∶ N → T ∀n ∈ N ∶ d(ξ(0), ξ(n)) = n} ξ ∼ ξ ′ if ∃n 0 ∈ N, m ∈ Z ∀n ≥ n 0 ∶ ξ(n + m) = ξ ′ (n) . For ρ, η ∈ T we denote by [ρ, η], [ρ, η), (ρ, η] and (ρ, η) the set of vertices on the geodesic between ρ and η, excluding or not their starting and end points. Similarly, for x ∈ ∂T , we denote by [ρ, x) and (ρ, x) the set of vertices on the unique geodesic which represents x and starts at ρ. The boundary of a tree carries a natural topology called the shadow topology. Definition 2.3. For two vertices ρ ≠ η of T consider U ρ,η ∶= {x ∈ ∂T η ∈ [ρ, x)} . Then the topology generated by all U ρ,η , where (ρ, η) runs through all pairs of distinct vertices of T , is called the shadow topology on ∂T . The action of Aut(T ) on T induces a continuous action by homeomorphisms on ∂T . We remark that every compact subgroup K ≤ Aut(T ) which fixes a point in the boundary, automatically fixes a vertex of T . There are 3 types of elements in Aut(T ). Elliptic elements, inversions and hyperbolic elements. An element g ∈ Aut(T ) is called elliptic if it fixes a vertex of T . It is an inversion, if it does not fix a vertex of T , but an edge. In all other cases, g is called hyperbolic. Any hyperbolic element fixes exactly two points in x, y ∈ ∂T and acts by translation along the axis (x, y). If g translates in the direction of x, then x is called the attracting fixed point of g. Two hyperbolic elements are called transverse if they have no common fixed point. Proposition 2.4. Let G ≤ Aut(T ) be a closed non-amenable subgroup. Then there is a minimal G-invariant subtree T ′ ≤ T . If G does not contain any compact normal subgroup, then G ↷ T ′ is faithful. Proof. Since G is non-amenable, it contains some hyperbolic element. The smallest tree T ′ containing all axes of hyperbolic elements in G, is G-invariant. Further every G-invariant subtree contains T ′ . Next observe that the kernel of the restriction map G ↦ Aut(T ′ ) is G T ′ , which is a compact subgroup of G. It follows that G T ′ is trivial and hence G ↷ T ′ is faithful if G does not contain any compact normal subgroup. Now assume that G is not compact and there is a G-invariant subtree T ′ ≤ T . Since G is not compact, T ′ is infinite. So it contains at least one infinite geodesic ray. Let ρ ∈ T ∖ T ′ be some vertex. Since T ′ is convex, there is some neighbouring vertex ρ ∼ η ∈ T ∖ T ′ . So the open set O = U ρ,η ⊂ ∂T is not empty. If x ∈ ∂T is the endpoint of some geodesic ray in T ′ , then Gx ∩ O = ∅. So the orbit of x is not dense. We want to have some control over the action on the boundary ∂T of vertex stabilisers in Aut(T ). To this end, we make the following definition of the meet of two boundary points with respect to a fixed vertex in T . Notation 2.6. Fix a vertex ρ ∈ T . Then for x, y ∈ ∂T we set m ρ (x, y) ∶= max{d(ρ, η) η ∈ [ρ, x) ∩ [ρ, y)} ∈ N ∪ {+∞} . Remark 2.7. • Fix x ∈ ∂T and ρ ∈ T . Then a basis of compact open neighbourhoods of x in the shadow topology is given by the sets U x (n) ∶= {y ∈ ∂T m ρ (x, y) ≥ n}, n ∈ N. • The vertex stabiliser Aut(T ) ρ leaves m ρ invariant, i.e. for all x, y ∈ ∂T and all k ∈ Aut(T ) ρ we have m ρ (kx, ky) = m ρ (x, y). We can now quantify the dynamics of a hyperbolic element close to its fixed points. Proposition 2.8. Let g ∈ Aut(T ) be a hyperbolic element fixing a point x ∈ ∂T . Then for all ρ ∈ T there is d ∈ N such that for all y ∈ ∂T ∖ {x} with m ρ (x, y) ≥ d we have m ρ (x, y) ≠ m ρ (x, gy) . Proof. Replacing g by its inverse, we may assume that x is its attracting fixed point. Denote by x ′ the repelling fixed point of g and by η the vertex of T in which the geodesics [ρ, x) and [ρ, x ′ ) split. d(ρ, gξ) = d(ρ, η) + d(η, gξ) > d(ρ, η) + d(η, ξ) = d(ρ, ξ) . Since the geodesics [ρ, gy) and [ρ, x) split in gξ, we obtain m ρ (x, gy) = d(ρ, gξ) > d(ρ, ξ) = m ρ (x, y). This finishes the proof of the proposition. The next lemma gives us many invariant neighbourhoods of points in ∂T . Lemma 2.9. Let K ≤ Aut(T ) be a compact subgroup fixing x ∈ ∂T . Then there is a basis of K-invariant neighbourhoods of x. Proof. Since K is compact there is some ρ ∈ T fixed by K. Then K fixes the geodesic [ρ, x) pointwise. So K fixes elements of the neighbourhood basis of x U x (n) = {y ∈ ∂T m ρ (x, y) ≥ n} ,n ∈ N . C * -algebras and von Neumann algebras A C * -algebra is a Banach algebra A such that x * x = x 2 for all x ∈ A. It is called simple if every closed *-ideal in A is either trivial or equals A. Denote by B(H) the *-algebra of bounded operators on a Hilbert space H. The topology of pointwise convergence on B(H) is called strong topology. A von Neumann algebra is a strongly closed unital *-subalgebra of B(H). Note that every von Neumann algebra is also a C * -algebra. Throughout the text we assume all von Neumann algebras to act on a separable Hilbert space. A von Neumann algebra M is simple if every strongly closed *-ideal of M is either trivial or equals M . Simple von Neumann algebras are called factors. Let A be a C * -algebra. A projection in A is an element p ∈ A satisfying p = p 2 = p * . If p, q are projections, then we write p ≤ q in case pq = p. A state on A is a unital functional ϕ ∶ A → C such that ϕ(x * x) ≥ 0 for all x ∈ A. A state is tracial if ϕ(xy) = ϕ(yx) for all x, y ∈ A. If A ⊂ B(H) is a C * -algebra, then its multiplier algebra M(A) ∶= {x ∈ B(H) ∀y ∈ A ∶ yx, xy ∈ A} is a C * -algebra, independent of the representation of A in B(H). It carries the strict topology defined on nets by x λ → x if and only if x λ y − xy , yx λ − yx → 0 for all y ∈ A. Since every C * -algebra contains an approximate unit, it is a strictly dense two-sided ideal in its multiplier algebra. The type of a von Neumann algebra Before defining the type of a von Neumann algebra, let us remark that every von Neumann algebra is the norm closure of the linear span of its projections. So von Neumann algebras contain an abundance of projections. The type of a von Neumann algebra depends on how its projections behave. Definition 2.10. A factor M is called finite if it admits a a tracial state; a projection p ∈ M is finite if pM p is finite. • M is of type I if it contains a minimal projection. • M is of type II if it contain a finite by not a minimal projection. • In all other cases M is of type III. In We need the following theorem to calculate the type of group von Neumann algebras appearing in this article. We refer to Section 2.6 for a brief introduction to weights on von Neumann algebras. If ϕ is a normal semi-finite faithful weight ϕ on a von Neumann algebra M , we denote by ∆ ϕ its modular operator, by (σ ϕ t ) t = (Ad ∆ it ϕ ) t the modular flow on M and by M ϕ = {x ∈ M ∀t ∶ σ ϕ t (x) = x} the fixed point algebra of the modular flow. The spectrum of ∆ ϕ is denoted by σ(∆ ϕ ). Theorem 2.12 ([Con73, Corollary 3.2.7]). Let M be a factor with a normal semi-finite faithful weight ϕ. If M ϕ is a factor, then S(M ) = σ(∆ ϕ ). Group C * -algebras and group von Neumann algebras Let G be a locally compact group and denote by λ G ∶ G → U(L 2 (G)) the left regular representation of G. Then the group von Neumann algebra of G is L(G) = λ G (G) ′′ . The canonical unitaries λ G (g), g ∈ G are denoted u g ∈ L(G). Fixing a left Haar measure µ of G, we define λ G ∶ C c (G) → B(L 2 (G)) by λ G (f )ξ(g) = G f (h)ξ(h −1 g)dµ(h) for all f, g ∈ C c (G). The reduced group C * -algebra of G is C * red (G) = λ G (C c (G)) ⋅ . The group C * -algebra of G does not contain the unitaries u g unless G is discrete. But u g ∈ M(C * red (G)) and even u g C c (G) = C c (G) = C c (G)u g . As a matter of fact, we have strongly dense inclusions C * red (G) ⊂ L(G) and M(C * red (G)) ⊂ L(G). If K ≤ G is a compact open subgroup, we obtain an averaging projection p K ∈ C c (G) ⊂ C * red (G) described by p K ξ(g) = 1 µ(K) K ξ(kg)dµ(k) for a square integrable function ξ on G representing an element of L 2 (G). These averaging projections play an important role in the present paper. They form a natural approximate unit in C * red (G) and L(G). This is the content of the next proposition. Proposition 2.13. Let G be a totally disconnected group. Then the net (p K ), K ≤ G compact open subgroup, strictly converges to 1 in C * red (G). Further, it strongly converges to 1 in L(G). Proof. Since strict convergence implies strong convergence for bounded nets, it suffices to show that (p K ) K strictly converges to 1 in C * red (G). This in turn follows from a standard estimate for (p K x − x) 1 for x in the dense subalgebra C c (G) ⊂ C * red (G) and K some compact open subgroup of G. For later use we want to note how averaging projections interact with each other and with the canonical unitaries u g . We start by describing relations between the averaging projections p K for different compact open subgroups K ≤ G. For the next statement, recall that u g C c (G) = C c (G). Also note that u g p L = p L for all g ∈ L, so that the right hand side of the following equation is well-defined. Proposition 2.14. Let G be a locally compact group with compact open subgroups L ≤ K ≤ G. In C c (G) we have p K = 1 [K ∶ L] gL∈K L u g p L . Proof. Take L ≤ K ≤ G as in the statement and let f ∈ C c (G) be arbitrary. Let µ be a left Haar measure for G. Then for all h ∈ G 1 [K ∶ L] gL∈K L u g p L f (h) = 1 [K ∶ L] gL∈K L p L f (g −1 h) = 1 [K ∶ L] gL∈K L 1 µ(L) L f (l −1 g −1 h)dµ(l) = 1 µ(K) gL∈K L gL f (l −1 h)dµ(l) = (p K f )(h) . Since C c (G) is dense in L 2 (G), we find that p K = 1 [K∶L] ∑ gL∈K L u g p L . The next lemma shows that averaging projections behave well with respect to conjugation by canonical unitaries. Lemma 2.15. Let G be a locally compact group and K ≤ G be a compact open subgroup. Then in C c (G) for all g ∈ G we have u g p K u * g = p gKg −1 . Proof. Take K ≤ G as in the statement of the lemma and let µ be a left Haar measure for G. For f ∈ C c (G) and g, h ∈ G we obtain u g p K u * g f (h) = 1 µ(K) K f (gk −1 g −1 h)dµ(k) = 1 µ(K) ∆(g) gKg −1 f (k −1 h)dµ(k) = 1 µ(gKg −1 ) gKg −1 f (k −1 h)dµ(k) = p gKg −1 f (h) . This shows that u g p K u * g = p gKg −1 . We can next describe products of the form p K u g p K in C c (G). The second part of the following proposition has a reformulation in terms of Hecke algebras (Section 2.5.1). Proposition 2.16. Let K be a compact open subgroup and g ∈ G. Put L = K ∩ gKg −1 Then p K u g p K = 1 [K ∶ L] kL∈K L u kg p K . In particular for g, h ∈ G we have p K u h p K u g p K = 1 [K ∶ L] kL∈K L p K u hkg p K . Proof. By Proposition 2.14, we have p K = 1 [K ∶ L] kL∈K L u k p L . Using Lemma 2.15, this implies p K u g p K = 1 [K ∶ L] kL∈K L u k p L u g p K = 1 [K ∶ L] kL∈K L u kg p g −1 Lg p K = 1 [K ∶ L] kL∈K L u kg p K , which proves the first part of the proposition. The second part of the proposition follows directly from the first one. Hecke C * -algebras Given a discrete Hecke pair Λ ≤ Γ there is a natural convolution product on double cosets. Let C(Γ, Λ) be the vector space whose basis consists of v g , ΛgΛ ∈ Λ Γ Λ. We set R(g) ∶= [Λ ∶ Λ ∩ gΛg −1 ] and L(g) ∶= [Λ ∶ Λ ∩ g −1 Λg]. We define a multiplication on C(Γ, Λ) by v h v g = g ′ Λ⊂ΛgΛ R(h) R(hg ′ ) v hg ′ and an involution by v * g = R(g) L(g) v g −1 . There is a *-representation of C(Γ, Λ) on 2 (Λ Γ) via v h δ Λg = Λh ′ ⊂ΛhΛ δ Λh ′ g . The norm closure of C(Γ, Λ) in this representation is the reduced Hecke-C * -algebra of Λ ≤ Γ, denoted by C * red (Γ, Λ). In a completely analogous fashion, one associates with an inclusion K ≤ G of a compact open group into a locally compact group a reduced Hecke-C * -algebra C * red (G, K). Note that in this case the equality ∆(g) = [g −1 Kg ∶ K ∩ g −1 Kg] [K ∶ K ∩ g −1 Kg] = R(g) L(g) holds. There is a *-isomorphism p K C * red (G)p K ≅ C * red (G, K) given by p K u g p K ↦ R(g) 1 2 L(g) 1 2 v g . Moreover, if K ≤ G is the Schlichting completion of a discrete Hecke pair Λ ≤ Γ, then C * red (G, K) ≅ C * red (Γ, Λ) as Tzanev showed in [Tza03, Theorem 4.2]. Weight theory In this section we briefly recall weight theory and the Plancherel weight on the group C * -algebra and the group von Neumann algebra of a locally compact group. We refer the reader to [Tak03, Chapter VII] for more details about weight theory on von Neumann algebras and C * -algebras. For a short summary of weights on C * -algebras, we recommend [KV99, Section 1]. Weights should be thought of as integration against a possibly infinite measure on a noncommutative space. Definition 2.17. Let A be a C * -algebra. A function ϕ ∶ A + → [0, ∞] is called a weight if • ϕ(x + y) = ϕ(x) + ϕ(y) for all x, y ∈ A + , and • ϕ(rx) = rϕ(x) for all x ∈ A + , r ∈ R ≥0 . If ϕ is a weight on A then we denote by n ϕ ∶= {x ∈ A ϕ(x * x) < ∞} the space of square integrable elements and by m ϕ = n * ϕ n ϕ the space of integrable elements. Every element x ∈ m + ϕ satisfies ϕ(x) < ∞. There is a unique linear functional on m ϕ which extends ϕ m + ϕ . We denote it also by ϕ. ϕ is called densely defined, if m ϕ ⊂ A is dense. We say that ϕ is proper if it is non-zero densely defined and lower semi-continuous in the norm topology. A weight on a von Neumann algebra M is called semi-finite if m ϕ ⊂ M is strong-* dense. It is called normal if it is lower semi-continuous in the strong-* topology. A normal semi-finite faithful weight is called an nsff weight. The GNS-construction Definition 2.18. Let ϕ be a weight on a C * -algebra A. A GNS-construction for ϕ is a triple (H, Λ, π) where • H is a Hilbert space, • Λ ∶ n ϕ → H is a linear map with dense image such that ⟨Λ(x), Λ(y)⟩ = ϕ(y * x) for all x, y ∈ n ϕ . • π ∶ A → B(H) is a *-representation satisfying π(x)Λ(y) = Λ(xy) for all a ∈ A and all y ∈ n ϕ . Every weight has a GNS-construction, which is unique up to unitary equivalence. The Plancherel weight and its modular automorphism group Let G be a locally compact group and µ a left Haar measure on G. Then the Plancherel weight ϕ on L(G) satisfies ϕ(f ) = f (e) for all f ∈ C c (G) ⊂ L(G). It is an nsff weight. Its restriction to C * red (G) is a proper weight. Note that a Plancherel weight depends on the choice of µ via the embedding C c (G) ⊂ L(G). If K is a compact open subgroup, we associated the averaging projection p K = 1 µ(K) ∫ K λ k dµ(k) with it. The Plancherel weight ϕ satisfies ϕ(u g p K ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 µ(K) , g ∈ K 0 , g ∈ G ∖ K . We will see in Lemma 2.23 that this property almost characterises Plancherel weights. Plancherel weights are described in [Tak03, Chapter VII, §3]. The so called modular operator of a Plancherel weight is the maximal self-adjoint positive multiplication operator associated with the modular function ∆ of G. For a Plancherel weight ϕ, the modular operator is denoted by ∆ ϕ . Its spectrum is σ(∆ ϕ ) = ∆(G). Denote by σ ϕ t = Ad ∆ it the so called modular automorphism group of ϕ (see [Tak03,Chapter VIII,§1]. It satisfies σ ϕ t (u g ) = ∆(g) it u g for all g ∈ G. The set of elements x ∈ L(G) such that the map t ↦ σ ϕ t (x) can be extended to an entire function on C is called the set of analytic elements of ϕ. All elements in C c (G) are analytic for any Plancherel weight on L(G). Let us collect some remarks about the Plancherel weight and its modular automorphism group. Remark 2.19. Fix a locally compact group G with modular function ∆ and a left Haar measure µ. Let ϕ be the Plancherel weight of L(G) associated with µ. • Since σ ϕ t preserves C c (G) ⊂ L(G), it restricts to a one-parameter group of *-automorphisms on C * red (G). We refer to it as the natural one-parameter group of *-automorphism of the reduced group C * -algebra. • For a compact open subgroup K ≤ G we have ∆ K ≡ 1. For any z ∈ C, we hence obtain σ ϕ z (p K ) = σ ϕ z 1 µ(K) K u g dµ(g) = 1 µ(K) K σ ϕ z (u g )dµ(g) = p K . KMS-weights If ϕ is a normal semi-finite faithful weight on a von Neumann algebra M , then it satisfies ϕ(xy) = ϕ(yσ ϕ −i (x)) for all analytic square integrable elements x, y ∈ M . (Compare [Tak03, Chapter VIII, § 1, Definition 1.1]). A similar result for proper weights on C * -algebras does not hold in general. However, we are next going to define a class of weights on C * -algebras which admit such control. If (σ t ) t is a norm continuous one-parameter group of *-automorphisms, on a C * -algebra A, then the set of analytic elements for (σ t ) t is dense in A according to [Kus97, Section 1]. So the following definition makes sense. Definition 2.20. Let A be a C * -algebra and (σ t ) t a one-parameter group of *-automorphisms. A proper weight ϕ on A is called KMS-weight with respect to (σ t ) t if • ϕ ○ σ t = ϕ for all t ∈ R and • ϕ(x * x) = ϕ(σ i 2 (x)σ i 2 (x) * ) for all x ∈ D(σ i 2 ), where D(σ i 2 ) denotes the domain of σ i 2 . Since a Plancherel weight on C * red (G) is a restriction of a Plancherel weight on L(G), it is a KMSweight. For illustration, we explicitly work out the example of Plancherel weights of totally disconnected groups. Example 2.21. Let G be a totally disconnected locally compact group with left Haar measure µ and ϕ be the Plancherel weight associated with µ. Then ϕ is a KMS-weight with respect to the natural one-parameter group of *-automorphisms of C * red (G) from Remark 2.19. Indeed, let g, h ∈ G and K ≤ G be a compact open subgroup. Then by Proposition 2.16 ϕ(p K u g p K u * h p K ) = δ g,h 1 µ(K)[K ∶ K ∩ g −1 Kg] . Also, using ∆(g) = R(g) L(g) , we have ϕ(σ i 2 (p K u h p K )σ i 2 (p K u g p K ) * ) = ∆(g) −1 2 ∆(h) −1 2 ϕ(p K u h p K u * g p K ) = δ g,h ∆(g −1 ) 1 µ(K)[K ∶ K ∩ g −1 Kg] = δ g,h 1 µ(K)[K ∶ K ∩ gKg −1 ] = ϕ(p K u * g p K u h p K ) . Moreover, ϕ(σ t (p K u g p K )) = ∆(g) it ϕ(p K u g p K ) = ∆(g) it 1 µ(K) , if g ∈ K 0 , if g ∈ G ∖ K = ϕ(p K u g p K ) , by the fact that ∆ K ≡ 1. Since ϕ is proper, [Kus97,p K C c (G)p K ⊂ C * red (G) show that ϕ is a KMS-weight with respect to (σ t ) t . One can show that the modular automorphism group of a KMS-weight, is implemented by a modular operator, which is described in the following proposition. Proposition 2.22. Let ψ be a KMS-weight with respect to a one-parameter group of *-automorphisms (σ t ) t on a C * -algebra A. Let (H, Λ, π) be a GNS-construction for ψ. There is a unique positive self-adjoint operator ∆ ψ on H such that ∆ it ψ Λ(x) = Λ(σ t (x)) for all x ∈ n ψ . We refer to [Tak03, Chapter VIII, §1, proof of Theorem 1.2] and [KV99, Section 2.2] for more details. The notion of KMS-weights allows us to characterise the Plancherel weight on C * red (G) similar to the canonical trace on group C * -algebras of discrete groups. Recall that the natural one-parameter group of *-automorphisms of C * red (G) is the restriction of the modular flow of a Plancherel weight as described in Section 2.6.2. Lemma 2.23. Let ψ be a KMS-weight for the natural one-parameter group of *-automorphisms on C * red (G). If there is a left Haar measure µ on G such that for every g ∈ G and K ≤ G compact open we have ψ(u g p K ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 µ(K) , g ∈ K 0 , g ∉ K , then ψ is the Plancherel weight associated with µ. We refer to the poofs of Proposition 1.14 and Corollary 1.15 in [KV99]. Denoting by ϕ the Plancherel weight normalised to ϕ(p K ) = ψ(p K ), then the proofs given by Kustermans-Vaes can be taken over, if we observe that that p K , K ≤ G compact open, is a net converging to 1 strictly in C * red (G) by Proposition 2.13 and ψ and ϕ agree on elements of the form ap K . The following proposition can be found for example in [KV99, Proposition 1.13]. It says that we have similar control over KMS-weights on a C * -algebra as we have over normal semi-finite faithful weights on a von Neumann algebra. Proposition 2.24. Let ψ be a KMS-weight to with respect to (σ t ) t on a C * -algebra A. Denote by A the analytic subalgebra of (σ t ) t . If x, y ∈ A are square integrable with respect to ψ, then ψ(xy) = ψ(yσ −i (x)). The S-invariant of a group von Neumann algebra We explain how to determine Connes' S-invariant (Section 2.4.1) for group von Neumann algebras of totally disconnected groups. For the purpose of this paper, it suffices to analyse the modular operator of a Plancherel weight. We start by identifying the centraliser (see [Tak03, Chapter VIII, §2]) of a Plancherel weight. If ϕ is an nsff weight on a von Neumann algebra M , and (σ ϕ t ) t is the modular automorphism group of ϕ, then the fixed point algebra of (σ ϕ t ) t is denoted by M ϕ . It is called the centraliser of ϕ. For the next proposition recall from Section 2.1 that G 0 denotes the kernel of the modular function of a locally compact group G. Proposition 2.25. Let G be a totally disconnected group and ϕ a Plancherel weight on L(G). Then L(G) ϕ = L(G 0 ).. Proof. Since σ ϕ t (u g ) = ∆(g) it u g for all g ∈ G, it follows that L(G 0 ) ⊂ L(G) ϕ . We prove the converse inclusion. For x ∈ L(G) ϕ and a compact open subgroup K ≤ G, we have xp K ∈ L(G) ϕ . By Proposition 2.13 it suffices to prove that L(G) ϕ p K ⊂ L(G 0 ). Let x ∈ L(G) ϕ p K . Since p K ∈ n ϕ , we can consider Λ ϕ (x) = ∑ gK∈G K x g 1 gK for unique scalars x g ∈ C. By Proposition 2.22, we have Λ ϕ (x) = Λ ϕ (σ ϕ t (x)) = ∆ it ϕ Λ ϕ (x) = gK∈G K x gK ∆(g) it 1 gK . By uniqueness of the coefficients x g , gK ∈ G K, we see that Λ ϕ (x) ∈ L 2 (G 0 ). This shows that x ∈ L(G 0 ), which finishes the proof. Proposition 2.26. Let G be a locally compact group and ϕ be a Plancherel weight on L(G). Then σ(∆ ϕ ) = ∆(G). Proof. We saw in Section 2.6.2 that ∆ ϕ is the multiplication operator associated with g ↦ ∆(g). So the proposition follows right away. Theorem 2.27. Let G be a totally disconnected group such that L(G 0 ) is a factor. Then S(L(G)) = ∆(G). Proof. By Proposition 2.25 we may apply Theorem 2.12 to a Plancherel weight ϕ of L(G). Then Proposition 2.26 shows that S(L(G)) = σ(∆ ϕ ) = ∆(G). Groups acting on trees with open stabilisers of boundary points In this section we describe different aspects of groups G acting on a tree T such that G x ≤ G is open for some point x ∈ ∂T . In Theorem 3.6 we describe condition ( * ) from the introduction, which applies to all groups treated in the rest of this article. Proposition 3.1. Let G ≤ Aut(T ) be a closed subgroup and let x ∈ ∂T . Then the following statements are equivalent. • G x ≤ G is open. • For every compact open subgroup K ≤ G the orbit Kx is finite. • There is a compact open subgroup K ≤ G fixing x. Proposition 3.2. Let G ≤ Aut(T ) be a closed subgroup. Let g ∈ G be hyperbolic and denote by x the attracting fixed point of g. Then the following statements are equivalent. Proof. If G x is open, then Kx = [K ∶ K ∩ G x ] is • G x ≤ G is open. • There is a compact open subgroup K ≤ G such that g n Kg −n ≥ K for all n ∈ N. • There is a compact open subgroup K ≤ G such that [K ∶ K ∩ g n Kg −n ], n ∈ N, is bounded. • For all compact open subgroups K ≤ G the sequence [K ∶ K ∩ g n Kg −n ], n ∈ N, is bounded. Proof. Assume that G x ≤ G is open. Let ρ be a vertex of T on the axis of g. Then K ∶= G x ∩ G ρ = G [ρ,x) is a compact open subgroup of G. We have g n Kg −n = G [g n ρ,x) ≥ G [ρ,x) = K for all n ∈ N. Assume that there is K ≤ G such that g n Kg −n ≥ K for all n ∈ N. Then K ∩ g n Kg −n = K and hence [K ∶ K ∩ g n Kg −n ] = 1 is bounded in n. Assume that there is a compact open subgroup L ≤ G such that [L ∶ L ∩ g n Lg −n ] is bounded in n. Let K ≤ G be a compact open subgroup. Then [K ∶ K ∩ g n Kg −n ] ≤ [K ∶ K ∩ g n (K ∩ L)g −n ] ≤ [K ∶ K ∩ g n Lg −n ][L ∶ K ∩ L] ≤ [(K ∩ L) ∶ (K ∩ L) ∩ g n Lg −n ][K ∶ K ∩ L][L ∶ K ∩ L] ≤ [L ∶ L ∩ g n Lg −n ][K ∶ K ∩ L][L ∶ K ∩ L] . It follows that [K ∶ K ∩ g n Kg −n ] is bounded in n. Assume that for all compact open subgroups K ≤ G the sequence [K ∶ K ∩ g n Kg −n ], n ∈ N is bounded. Let ρ be a vertex on the axis of g. Then K ∶= G ρ is a compact open subgroup and K ∩g n Kg −n = G [ρ,g n ρ] is a descending sequence of compact open subgroups. Since [K ∶ K ∩g n Kg −n ] is bounded in n, the sequence K ∩ g n Kg −n becomes stationary. So ⋂ n≥0 K ∩ g n Kg −n is an open subgroup. Then also G x ≥ G [ρ,x) = ⋂ n≥0 K ∩ g n Kg −n is open. Before we describe hyperbolic elements both of whose fixed points have an open stabiliser, we need to note the following well-known lemma. Proof. Let g, h ∈ G be elliptic. There are ρ, η ∈ T fixed by g and h respectively. Let ξ be a point in [ρ, x) ∩ [η, x). Then g and h fix ξ, so gh fixes ξ and it is hence elliptic. Since H is the union of vertex stabilisers, it is open in G. Since G fixes x ∈ ∂T , it consists only of elliptic and hyperbolic elements. Hence G ∖ H consists entirely of hyperbolic elements. Now let g ∈ G ∖ H be a hyperbolic whose attracting fixed point is X. Let ρ be a vertex on the axis of g and K = G [ρ,x) . Then gKg −1 = G [gρ,x) ≥ K. If k ∈ H, then there is ρ ′ ∈ T fixed by k. Since kx = x, k fixes all points on the geodesic [ρ ′ , x). So k ∈ G [ρ ′ ,x)∩[ρ,x) ⊂ ⋃ n∈Z g n Kg −n . Proposition 3.4. Let G ≤ Aut(T ) be a closed subgroup and g ∈ G hyperbolic such that one of its fixed points in ∂T has an open stabiliser in G. Then the following statements are equivalent. • Both fixed points of g have an open stabiliser in G. • g lies in the kernel G 0 of the modular function of G. • g normalises a compact open subgroup of G. In particular, if g ∈ G 0 is hyperbolic, then either both or none of its fixed points have an open stabiliser. Proof. Throughout the proof, g ∈ G denotes a hyperbolic element such that G x is open for one of its fixed points x ∈ ∂T . Replacing g by its inverse if necessary, we may assume that its attracting fixed point has an open stabiliser in G. By Proposition 3.2, there is a compact open subgroup K ≤ G such that g n Kg −n ≥ K for all n ∈ N. Then [K ∶ K ∩ g −n Kg n ] = [g n Kg −n ∶ K] = ∆(g −n ) = ∆(g) −n . If the repelling fixed point of g has an open stabiliser, Proposition 3.2 and the previous equation show that ∆ G (g) −1 is a positive integer, whose powers are bounded. Hence ∆ G (g) = 1, which proves that g ∈ G 0 . If g ∈ G 0 , let K ≤ G be a compact open subgroup such that gKg −1 ≥ K, which is provided by Let us now describe condition ( * ) as it is mentioned in the introduction. To this end, we need a characterisation of amenable subgroups of Aut(T ). Proposition 3.5 (Adams-Ballmann [AB98]). Let G ≤ Aut(T ) be a closed subgroup. • If G is amenable, then it fixes some point in V (T ) ∪ E(T ) ∪ ∂T . In case G fixes an edge of T , then it contains an index 2 subgroup fixing a vertex of T . • If G fixes a point in V (T ) ∪ E(T ) ∪ ∂T , then it is amenable. Theorem 3.6. Let G be a topological group. Then the following two conditions on G are equivalent. • There is a locally finite tree T such that G ≤ Aut(T ) as a closed subgroup. Further, G is non-amenable without any non-trivial compact normal subgroup and there is a compact open subgroup K ≤ G such that N G (K) K contains an element of infinite order. • There is a locally finite thick tree T such that G ≤ Aut(T ) as a closed subgroup. Further, G acts minimally on ∂T and there is some point x ∈ ∂T such that G x ≤ G is open and G x ∩ G 0 contains a hyperbolic element. Proof. Assume that G satisfies our first condition and take G ≤ Aut(T ) as in the statement. Since G is non-amenable there is a minimal G-invariant subtree T ′ ≤ T by Proposition 2.4. Note that T ′ must be thick. Since G does not contain any compact normal subgroups, we can consider G ≤ Aut(T ′ ) by restriction. Since G ↷ T ′ admits no proper G-invariant subtree, it follows that G ↷ ∂T ′ is minimal by Proposition 2.5. Let K ≤ G be a compact open subgroup and g ∈ N G (K) an element whose image in N G (K) K has infinite order. Then H ∶= ⟨K, g⟩ ≤ G is a non-compact amenable open subgroup. So Proposition 3.5 says that there is x ∈ ∂T fixed by H. In particular, H ≤ G x ≤ G is open. Moreover, g ∈ G x is not contained in any compact subgroup, so g is hyperbolic. Now G ≤ Aut(T ′ ) satisfies all conditions of the second statement. Now assume that G satisfies the second condition of the theorem and take G ≤ Aut(T ) as in the statement. Note that G is not compact, since it contains a hyperbolic element. Since T is thick and G ↷ ∂T is minimal, it follows that G is non-amenable. Moreover, G ↷ T is minimal by Proposition 2.5. So any compact normal subgroup of G fixes T pointwise, showing that it is trivial. Take x ∈ ∂T such that G x is open and G x ∩G 0 contains a hyperbolic element g. By Propositions 3.2 and 3.4 there is a compact open subgroup K ≤ G such that g ∈ N G (K). Since g ∈ G 0 it follows that g ∈ N G (K). Since g is hyperbolic, it is not contained in any compact subgroup of G and hence its image in N G (K) K has infinite order. So G satisfies all conditions of the first statement. This finishes the proof of the theorem. Locally compact Powers groups In this section we introduce the notion of a locally compact Powers group. This generalises an idea of Powers [Pow75] and de la Harpe [dlH85] to prove C * -simplicity for discrete groups. We prove an analogue of Powers averaging. Further we adapt an idea of de la Harpe and Préaux [dlHP09] used to show C * -simplicity of HNN-extensions in order to prove that a natural class of groups acting on trees are Powers groups. This justifies our definition of Powers groups in the context this article. We however insist on the ad hoc character of the next definition. Definition 4.1 (Locally compact Powers group). Let G be a locally compact group. Let K ≤ G be a compact open subgroup, F ⊂ G ∖ K be a compact set and r ∈ N. We say that G satisfies Powers property with control r with respect to K and F if the following condition holds. For all n ∈ N × there are elements g 1 , . . . , g n ∈ G 0 and a decomposition of G into left K-invariant sets G = C ⊔ D such that • for all f ∈ F we have f C ∩ C = ∅, • the sets g 1 D, . . . , g n D are pairwise disjoint, and • [K ∶ K ∩ g i Kg −1 i ] ≤ r for all i ∈ {1, . . . , n}. If we do not need to specify control, we speak of Powers property with respect to K and F only. Remark 4.2. • A discrete group G is a Powers group in the sense of [dlH85] if and only if it has the Powers property with respect to {e} and F for all finite sets F ⊂ G ∖ {e}. • For the results of this paper it is important to introduce explicit control over the subgroup K and the constant r in our formulation of Powers property. This becomes clear from Proposition 4.5 and the proof of Theorem 6.2. • We may replace the control condition " [K ∶ K ∩ g i Kg −1 i ] ≤ r for all i ∈ {1, . . . , n}" by "[K ∶ K ∩ g −1 i Kg i ] ≤ r for all i ∈ {1, . . . , n}", since g 1 , . . . , g n ∈ G 0 . Indeed, [K ∶ K ∩ gKg −1 ] = ∆(g)[K ∶ K ∩ g −1 Kg] for all g ∈ G. The next proposition generalises Powers averaging to locally compact Powers groups. We adapt the proof given in [dlH07]. Proposition 4.3. Let G be a locally compact group. Assume that G has the Powers property with control r ∈ N with respect to the compact open subgroup K ≤ G and the compact set F ⊂ G∖K . Then for all x ∈ C c (G) whose support lies in F and all ε > 0 there is n ∈ N and elements g 1 , . . . , g n ∈ G 0 satisfying 1 n n i=1 u g i (xp K )u * g i < ε , and [K ∶ K ∩ g i Kg −1 i ] ≤ r for all i ∈ {1, . . . , n}. Proof. Take K ≤ G, F ⊂ G ∖ K and r ∈ N as in the statement of the proposition. Checking the definition of Powers property, we see that we may assume F = F K. Let x ∈ C c (G) have support in F and let ε > 0. For all n ∈ N × there are elements g 1 , . . . , g n ∈ G 0 and a decomposition of G into K-invariant sets G = C ⊔ D such that • for all f ∈ F ∖ K we have f C ∩ C = ∅, • the sets g 1 D, . . . , g n D are pairwise disjoint, and • [K ∶ K ∩ g i Kg −1 i ] ≤ r for all i ∈ {1, . . . , n}. LetF be a set of representatives for F K and write xp K = f ∈F x f u f p K for some scalars x f ∈ C, f ∈F . Since C is K-invariant, it is open. For i ∈ {1, . . . , n} we may consider the orthogonal projections q i ∶ L 2 (G) → L 2 (g i C) . Then for all f ∈ F u f p K u * g i q i L 2 (G) = L 2 (f C) and u * g i q i L 2 (G) = L 2 (C) . So ⟨q i u g i u f p K u * g i q i L 2 (G), L 2 (G)⟩ = ⟨u f p K u * g i q i L 2 (G), u * g i q i L 2 (G)⟩ = ⟨L 2 (f C), L 2 (C)⟩ = {0} . This shows q i u g i u f p K u * g i q i = 0. Moreover, the images of (1−q i )u g i u f p K u * g i , i ∈ {1, . . . , n} are pairwise orthogonal. Also, the supports of q i u g i u f p K u * g i (1 − q i ), i ∈ {1, . . . , n} are pairwise orthogonal. We can proceed to the following estimate. 1 n n i=1 u g i (xp K )u * g i ≤ 1 n n i=1 (1 − q i )u g i xp K u * g i + 1 n n i=1 q i u g i xp K u * g i (1 − q i ) ≤ √ n n sup (1 − q i )u g i xp K u * g i i ∈ {1, . . . , n} + sup q i u g i xp K u * g i (1 − q i ) i ∈ {1, . . . , n} ≤ 2 √ n n x . Taking n big enough so that 2 √ n n x < ε finishes the proof. In the rest of the section we are going to prove that groups satisfying condition ( * ) have the Powers property with respect to specific compact open subgroups. To this end, we need an abundance of pairwise transverse hyperbolic elements with uniform control over how well they commensurate compact open subgroups. Proof. Let O ⊂ ∂T be open. Since G gx = gG x g −1 for all x ∈ ∂T and all g ∈ G, the set of points x ∈ ∂T such that G x is open and G x ∩ G 0 contains a hyperbolic element is dense by minimality of G ↷ ∂T . So we may take h ∈ G 0 hyperbolic with attracting fixed point in O. Since G is not amenable, we may then find a conjugate g of h which is transverse to h. Replacing h by some positive power of itself, we may assume that h maps the fixed points of g into O. Then the elements g i ∶= h i gh −i , i ∈ N × are hyperbolic and their fixed points lie in O. Take a compact open subgroup K ≤ G. For all i ≥ 1 and l ∈ Z we have [K ∶ K ∩ g l i Kg −l i ] = [K ∶ K ∩ h i g l h −i K(h i g l h −i ) −1 ] = [h −i Kh i ∶ h −i Kh i ∩ g l h −i Kh i g −l ] ≤ [h −i Kh i ∶ h −i Kh i ∩ g l (K ∩ h −i Kh i )g −l ] ≤ [h −i Kh i ∶ h −i Kh i ∩ g l Kg −l ][K ∶ K ∩ h −i Kh i ] ≤ [(K ∩ h −i Kh i ) ∶ (K ∩ h −i Kh i ) ∩ g l Kg −l ][K ∶ K ∩ h −i Kh i ][h −i Kh i ∶ K ∩ h −i Kh i ] ≤ [K ∶ K ∩ g l Kg −l ][K ∶ K ∩ h −i Kh i ][K ∶ K ∩ h i Kh −i ] Since by Proposition 3.4 the fixed points of g, h ∈ G 0 have open stabilisers, Proposition 3.2 implies that [K ∶ K ∩ g l i Kg −l i ] is uniformly bounded for i ≥ 1 and l ∈ Z. Possibly passing to a subsequence of (g i ) i≥1 , we may assume that these elements are pairwise transverse, which finishes the proof. We proceed to show Powers property for groups satisfying condition ( * ). The proof is inspired by Proposition 8 of [dlHP09]. Proposition 4.5. Let G be a group satisfying condition ( * ). Then the following statements hold true. (i) There is r ∈ N and a compact open subgroup K ≤ G such that for all compact sets F ⊂ G ∖ K, G has the Powers property with control r with respect to (K, F ). (ii) For a compact open subgroup L ≤ G and x ∈ ∂G such that G x is open, set K ∶= L ∩ G x . Then G has the Powers property with respect to (K, L ∖ K). (iii) For all neighbourhoods of the identity N ⊂ G there is a compact open subgroup K ≤ G contained in N such that G has the Powers property with respect to (K, gK) for all g ∈ G ∖ K. Proof. We start to consider case (i). Fix x ∈ ∂T such that G x ≤ G is open and G x ∩ G 0 contains a hyperbolic element. By Lemma 3.3 and Proposition 3.4 we find a maximal compact open subgroup K ≤ G x such that G x ∖ K consists of hyperbolic elements only. Let F ⊂ G ∖ K be a compact set. We construct an open K-invariant subset O ⊂ ∂T such that f O ∩ O = ∅ for all f ∈ F . Consider F ∖ G x . It is covered by finitely many right K-cosets. Further f x ≠ x for all f ∈ F ∖ G x . By Lemma 2.9, we find an open Let g ∈ G 0 be some hyperbolic element whose attracting fixed point lies in O and whose repelling fixed point we denote by y ∈ ∂T . Let U ∋ y be a clopen neighbourhood. By Lemma 4.4 there is a sequence of pairwise transverse hyperbolic elements (h i ) i≥1 in G 0 whose fixed points lie in ∂T ∖ ({x} ∪ U). Also, we find r 0 ∈ N such that [K ∶ K ∩ h l i Kh −l i ] ≤ r 0 for all i ≥ 1 and all l ∈ Z. Note that r is independent of F . Denote the attracting fixed point of h i by α i and its repelling fixed point by ω i . Since α i , ω i ∈ ∂T ∖ ({x} ∪ U), i ≥ 1, there is some m ∈ N such that g m α i , g m ω i ∈ O for all i ≥ 1. Let g i ∶= g m h i g −m . Now calculation of Lemma 4.4 shows that Let us now consider case (iii). Take a neighbourhood of the identity N ⊂ G. Note that for any set Σ ⊂ ∂T containing at least three different points, G Σ fixes a point in T and it is hence compact. K-invariant set O 0 ∋ x such that (F ∖ G x )O 0 ∩ O 0 = ∅. Since F ⊂ G ∖ K,[K ∶ K ∩ g l i Kg −l i ] ≤ [K ∶ K ∩ g m Kg − Since G ↷ ∂T is faithful, in any dense set Σ 0 ⊂ ∂T we find a finite subset Σ ⊂ Σ 0 such that G Σ ⊂ N . The set Σ 0 ∶= {x ∈ ∂T G x is open and G x ∩ G 0 contains a hyperbolic element} is dense, because G ↷ ∂T is minimal. It follows that there is a finite subset Σ ⊂ Σ 0 such that G Σ ⊂ N . Put K ∶= G Σ . If g ∈ G ∖ K, then there is x ∈ Σ such that gx ≠ x. Let O be an open neighbourhood of x such that gO ∩ O = ∅. Since K fixes x, Proposition 2.9 says that we may make O smaller so as to assume that it is K-invariant. For the last part of the proof we set F ∶= gK. In all cases (i),(ii) and (iii), we have a compact open subgroup K ≤ G and a compact right K- invariant set F ⊂ G ∖ K together with an open K-invariant set O ⊂ ∂T satisfying f O ∩ O = ∅ for all f ∈ F . In case (i), K is independent of the choice of F . Moreover, in case (i) we already constructed a sequence (g i ) i≥1 of pairwise transverse hyperbolic elements whose fixed points lie in O. There is some r ∈ N independent of F such that [K ∶ K ∩ g l i Kg −l i ] ≤ r for all i ≥ 1 and all l ∈ Z. In cases (ii) and (iii), we may choose such a sequence (g i ) i≥1 according to Lemma 4.4. But in the latter case, r possibly depends on F . For the next paragraph, denote by ω i the attracting fixed point of g i , i ≥ 1. Now pick x ∈ ∂T and set C ∶= {g ∈ G gx ∈ O} , D ∶= {g ∈ G gx ∉ O} . Then C and D are left K-invariant, since O is K-invariant. Moreover, f O∩O = ∅ implies f C ∩C = ∅ for all f ∈ F . Fix n ∈ N. We can find pairwise disjoint open neighbourhoods of W i of ω i , i ∈ {1, . . . , n} and exponents l i ∈ N × such that g l i i (∂T ∖ O) ⊂ W i . Replacing each g i by g l i i , we may assume that g i (∂T ∖ O) ⊂ W i . Then g 1 D, . . . , g n D are pairwise disjoint sets. This finishes the proof of the proposition. Fullness of averaging projections In this section we take a first step to prove our C * -simplicity result. If a group G is C * -simple, then in particular the projections p K averaging over a compact open subgroup of G are full in C * red (G). We are not aware of any simple criterion ensuring fullness of p K . So the aim of this section is to prove that averaging projections in reduced group C * -algebras of groups satisfying condition ( * ) are full. We start with a lemma ensuring invertibility of certain averages in C * red (G) in the proof of Proposition 5.2. Lemma 5.1. Let G be a locally compact group and K ≤ G a compact open subgroup. Then for all g ∈ G we have p K u * g p K u g p K ≥ [K ∶ K ∩ gKg −1 ] −2 p K . Proof. Take a compact open subgroup K ≤ G and g ∈ G. It suffices to prove that for all ξ ∈ p K L 2 (G), we have ⟨p K u * g p K u g p k ξ, ξ⟩ ≥ [K ∶ K ∩ gKg −1 ] −2 ξ 2 . Let µ be the left Haar measure for G satisfying µ(K) = 1. For ξ = ∑ h ξ h 1 Kh ∈ p K L 2 (G), we have ⟨p K u * g p K u g p k ξ, ξ⟩ = p K u g ξ 2 = h G ξ h 2 (p K 1 gKh )(l) 2 dµ(l) = h G ξ h 2 ( K 1 gKh )(k −1 l)dµ(k) 2 dµ(l) . Since k −1 l ∈ gKh if and only if k ∈ lh −1 Kg −1 , we obtain for l ∈ gKh and k ∈ gKg −1 ∩ K that 1 gKh (k −1 l) = 1. We can hence continue the previous equation and obtain ⟨p K u * g p K u g p k ξ, ξ⟩ ≥ h ξ h 2 µ(gKh)µ(gKg −1 ∩ K) 2 = µ(gKg −1 ∩ K) 2 h ξ h 2 µ(Kh) = [K ∶ K ∩ gKg −1 ] −2 ξ 2 . This finishes the proof of the lemma. Proposition 5.2. Assume that G is a group satisfying condition ( * ). Then p K is a full projection in C * red (G), i.e. the closed two-sided ideal generated by p K equals C * red (G). Proof. We have to show that for all compact open subgroups K ≤ G the closed two-sided ideal generated by p K equals C * red (G). We start by proving the following claim. Claim. Let I ⊴ C * red (G) be a closed two-sided ideal such that p K ∈ I for some compact open subgroup K ≤ G. Then p K∩Gx ∈ I for all x ∈ ∂T that have an open stabiliser G x ≤ G. Fix I ⊴ C * red (G), K ≤ G and x ∈ ∂T as in the claim. By Proposition 4.5 the group G has the Powers property with respect to L ∶= K ∩ G x and F ∶= K ∖ L. We can write [K ∶ L]p K = ∑ gL∈K L u g p L , as shown by Proposition 2.14. So Proposition 4.3 says that there is r ∈ N such that for all ε > 0 there are g 1 , . . . , g n ∈ G 0 satisfying [L ∶ L ∩ g i Lg −1 i ] ≤ r and 1 n n i=1 u g i ([K ∶ L]p K − p L )u * g i < ε . Note that [L ∶ L ∩ g −1 i Lg i ] = [L ∶ L ∩ g i Lg −1 i ]∆(g −1 i ) ≤ r for all i ∈ {1, . . . , n}. By Lemma 5.1 we have p L u g i p L u * g i p L ≥ [L ∶ L ∩ g −1 i Lg i ] −2 p L ≥ r −2 p L for all i ∈ {1, . . . , n}. We showed that there is 0 < δ ∶= r −2 < 1 such that for all ε > 0 there is x ∶= [K∶L] n ∑ n i=1 p L u g i p K u * g i p L ∈ I and an invertible element y ∶= 1 n ∑ n i=1 p L u g i p L u * g i p L ∈ p L C * red (G)p L such that x − y < ε and σ(y) ⊂ [δ, 1]. Then σ(y −1 ) ⊂ [1, δ −1 ] and in particular y −1 ≤ δ −1 . So x ⋅ y −1 − p L ≤ x − y y −1 < εδ −1 . Choosing ε ≤ δ, we conclude that x ⋅ y −1 ∈ I is invertible in p L C * red (G)p L . So p L ∈ I, which proves the claim. Since G ↷ ∂T is minimal the set Σ 0 ∶= {x ∈ ∂T G x is open and G x ∩ G 0 contains a hyperbolic element} is dense in ∂T . Moreover, for every set Σ ⊂ Σ 0 such that Σ ≥ 3, the pointwise stabiliser G Σ fixes a point in T and is hence compact. By faithfulness of G ↷ ∂T , we conclude that the compact open subgroups G Σ with Σ ⊂ Σ 0 and Σ ≥ 3 form a neighbourhood basis of e in G. So the claim combined with Proposition 2.14 shows that p L ∈ C * red (G)p K C * red (G) for all compact open subgroups L ≤ G. By Proposition 2.13, (p K ), K ≤ G compact open, strictly converges to 1 in C * red (G), it follows that the closed two-sided ideal generated by all projections p K , K ≤ G compact open subgroup, is C * red (G). This finishes the proof of the proposition. C * -simplicity This section has two aims. First, we show that a C * -simple group must be totally disconnected, extending Proposition 4 of [BCdlH94]. Then we combine results from Sections 4 and 5 in order to prove that groups satisfying condition ( * ) are C * -simple. In connection with examples from Section 9, this gives rise to the first C * -simplicity result for non-discrete groups. Let G be a locally compact group and π a unitary representation of G. We denote byπ the *representation of the maximal group C * -algebra C * max (G) that is induced by π. If π, ρ are two unitary representations of G, then we say that π is weakly contained in ρ and write π ≺ ρ, if kerπ ⊃ kerρ. If π ≺ ρ and π ≻ ρ, we say that π and ρ are weakly equivalent and write π ∼ ρ. We refer to [BdlHV08, Appendix F] for more a good summary of basic properties of weak containment. It is clear from the definitions that G is C * -simple if and only if π ≺ λ G implies π ∼ λ G for every unitary representation π of G. We will use this characterisation of C * -simplicity in the proof of the following theorem. Theorem 6.1. Let G be a locally compact C * -simple group. Then G is totally disconnected. Proof. Let G be a locally compact group that is not totally disconnected. We have to show that there is a unitary representation π ≺ λ G such that π ∼ λ G . Since G is not totally disconnected, the connected component G 0 ≤ G of the identity is not trivial. Let K ⊴ G 0 the maximal compact normal subgroup. By the structure theorem for locally compact groups, G 0 K is a connected Lie group. If R denotes the inverse image in G 0 of the amenable radical of G 0 K, then H ∶= G 0 R is a connected semi-simple Lie group with trivial centre and hence it is linear. Note that R is amenable, since it is compact-by-amenable. Moreover, R is a characteristic subgroup of G 0 and hence normal in G. If R ≠ {e} then the quasi-regular representation λ G,R is weakly contained in λ G , but it is not injective on G. Hence λ G,R ∼ λ G , which finishes the proof So we may assume that R = {e} and hence H = G 0 is a connected semi-simple linear Lie group. Let π 0 be a principal series representation of H and let π = Ind G H (π 0 ) be the induced representation of G. Then π ≺ λ G , since π 0 ≺ λ H . We show that π is not weakly equivalent to λ G . By Mackey's subgroup theorem [Mac51, Theorem 12.1], we have Res G H (π) ∼ ⊕ α∈G π 0 ○α ≺ ⊕ α∈Aut(H) π 0 ○α, where we apply α ∈ G via its image in Aut(H). By the definition of the Fell topology, {ρ irrep of H ρ ≺ π 0 } is closed in the unitary dualĤ. Since every semi-simple Lie group has a finite outer automorphism group, by [Mur52, Corollary 2], and π 0 ○α depends up to unitary equivalence only on the image of α ∈ Aut(H) in Out(H), it follows that ⋃ α∈Aut(H) {ρ irrep of H ρ ≺ π 0 ○ α} is closed inĤ. So if ρ is an irreducible unitary representation of H such that ρ ≺ ⊕ α∈Aut(H) π 0 ○ α, then ρ ≺ π 0 ○ α for some α ∈ Aut(H). Since H is a CCR group by [Dix77, Theorem 15.5.6] and since π 0 ○ α is irreducible, we conclude that ρ ≅ π 0 ○ α. We showed that Res G H (π) weakly contains only finitely many irreducible representations of H. However λ H = Res G H (λ G ) weakly contains all principal series representations, of which there are infinitely many. This implies π ∼ λ G and finishes the proof of the theorem. Theorem 6.2. Let G be a group satisfying condition ( * ). Then C * red (G) is simple. Proof. Take G ≤ Aut(T ) as in the statement of the theorem. Let I ⊲ C * red (G) be a non-trivial closed two-sided ideal and take 0 ≠ x ∈ I positive. By Proposition 4.5 (i) there is a compact open subgroup K ≤ G and some number r ∈ N such that for all compact subsets F ⊂ G ∖ K, Powers property with control r holds with respect to (K, F ) inside G. We choose the Plancherel weight ϕ on C * red (G) satisfying ϕ(p K ) = 1. We may scale x so that ϕ(p K xp K ) = 1. Fix ε > 0 and find y 0 ∈ C c (G) such that x−y 0 < ε 2 . Since G has the Powers property with control r with respect to K and supp y 0 ∖K, Proposition 4.3 says that there are elements g 1 , . . . , g n ∈ G 0 for which Powers averaging gives 1 n n i=1 u g i (y 0 p K − p K )u * g i < ε 2 and [K ∶ K ∩ g i Kg −1 i ] ≤ r for all i ∈ {1, . . . , n}. We obtain that 1 n n i=1 u g i (x − p K )u * g i ≤ 1 n n i=1 u g i (y 0 − p K )u * g i + ε 2 < ε 2 + ε 2 . Then also 1 n n i=1 p K u g i xu * g i p K − 1 n n i=1 p K u g i p K u * g i p K < ε . Note that [K ∶ K ∩ g −1 i Kg i ] = [K ∶ K ∩ g i Kg −1 i ]∆(g −1 i ) ≤ r for all i ∈ {1, . . . , n}. By Lemma 5.1, we have p K u g p K u * g p K ≥ [K ∶ K ∩ g −1 Kg] −2 p K for all g ∈ G. This implies that 1 n ∑ n i=1 p K u g i p K u * g i p K ≥ r −2 p K . Summarising the proof up to now, we showed that there is 0 < δ ∶= r −2 < 1 such that for all ε > 0 there is a ∶= 1 n ∑ n i=1 p K u g i xu * g i p K ∈ I and an invertible element b ∶= 1 n ∑ n i=1 p K u g i p K u * g i p K ∈ p K C * red (G)p K such that a − b < ε and σ(b) ⊂ [δ, 1]. Then σ(b −1 ) ⊂ [1, δ −1 ] and in particular b −1 ≤ δ −1 . So a ⋅ y −1 − p K ≤ a − b b −1 < εδ −1 . Choosing ε ≤ δ, we conclude that a ⋅ b −1 ∈ I is invertible in p K C * red (G)p K . So p K ∈ I. By Lemma 5.2, we obtain I = C * red (G). This finishes the proof. Applications In this section we apply Theorem 6.2 to two problems of independent interest. We first give to the best of our knowledge the first non-trivial examples of simple reduced Hecke-C * -algebras and then show that certain groups acting on trees are not of type I. Corollary 6.3. Let T be thick tree and Γ ≤ Aut(T ) some not necessarily closed group acting without proper invariant subtree. Let Λ be some vertex stabiliser of in Γ and assume that there is a finite index subgroup Λ 0 ≤ Λ such that N Γ (Λ 0 ) Λ 0 contains an element of infinite order. Then C * red (Γ, Λ) is simple. Proof. Take Λ = Γ ρ ≤ Γ ≤ Aut(T ) as in the statement of the corollary. Let G = Γ and K = Λ. Then the natural bijection between G K and Γ Λ conjugates G and Γ Λ. So by [Tza03,Theorem 4.2], we have C * red (Γ, Λ) ≅ p K C * red (G)p K . In view of Theorem 6.2, it hence suffices to show that G satisfies condition ( * ). Since Γ acts on T without any proper minimal subtree, also G does so. So Proposition 2.5 shows that G ↷ ∂T is minimal. Since T is thick, ∂T is a Cantor space. So Then G is not a type I group. Proof. Take G ≤ Aut(T ) as in the statement of the corollary and assume that G. Then by Proposition 3.1, G satisfies property (*) and hence also G 0 satisfies (*). Assuming that G is a type I group, also its open subgroup G 0 is a type I group. So we may assume in addition that G is unimodular. By Theorem 6.2 G is C * -simple. Since it is also a type I group, λ is unitarily equivalent to a multiple of an irreducible representation. In particular, L(G) ≅ B(H) for some Hilbert space H. Since G is unimodular, the Plancherel weight ϕ on L(G) agrees with the unique tracial weight on B(H). So if K ≤ G denotes some compact open subgroup of G, the fact that ϕ(p K ) < ∞, implies that p k is a finite projection in B(H). This shows that p K L(G)p K is isomorphic to a finite type I factor, i.e. p K L(G)p K ≅ M n (C) for some n ∈ N. Take some hyperbolic element g ∈ G x . Then by p k u g p K satisfies ⟨p K u g n p K ξ, η⟩ → 0 for all ξ, η ∈ p K L 2 (G). This means that p K u g n p K converges to 0 weakly. However, p k u * g n p K u g n p K ≥ [K ∶ K ∩g n Kg −n ] −2 p K , which is bounded from below by Proposition 3.2. So p K u g p K does not converge strongly to 0, contradicting the isomorphism p K L(G)p K ≅ M n (C). This finishes the proof of the corollary. KMS-weights and von Neumann factors In this section we apply Powers group methods to prove uniqueness of certain natural KMS-weights on the reduced group C * -algebra of certain groups G satisfying condition ( * ). This uniqueness allows us to conclude factoriality of the associated group von Neumann algebra L(G) and to determine its type. Theorem 7.1. Let G be a group satisfying condition ( * ) and assume that some compact open subgroup of G is topologically finitely generated. Let ϕ be a KMS-weight for the natural one-parameter group (σ t ) t on C * red (G). Assume that ϕ(p K ) < ∞ for all compact open subgroups K ≤ G. Then ϕ is a Plancherel weight on C * red (G). Proof. Let ϕ be a KMS-weight as described in the statement of the theorem. By Lemma 2.23 it suffices to show that there is a left Haar measure µ on G such that for all compact open subgroups K ≤ G and for all g ∈ G we have ϕ(u g p K ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 µ(K) ,g ∈ K 0 ,otherwise. Claim. If K ≤ G is a compact open subgroup and g ∈ G ∖ K, then ϕ(u g p K ) = 0. Using Proposition 2.14 it suffices to show the claim for a neighbourhood base of e consisting of compact open subgroups K ≤ G. So fix a neighbourhood e ∈ N ⊂ G. By Proposition 4.5 we find a compact open subgroup K ≤ G that is contained in N such that G has the Powers property with respect to K and gK for all g ∈ G ∖ K. Proposition 4.3 says that there is r ∈ N such that for every ε > 0 there are elements g 1 , . . . g n ∈ G 0 such that 1 n n i=1 u g i u g p K u * g i < ε and [K ∶ K ∩ g i Kg −1 i ] ≤ r for all i ∈ {1, . . . , n}. Using a GNS-construction associated with ϕ, this gives rise to the following estimate for any averaging projection p L with L ≤ G compact open. ϕ(p L 1 n n i=1 u g i u g p K u * g i p L ) = ⟨ 1 n n i=1 u g i u g p K u * g i p L , p L ⟩ ϕ ≤ 1 n n i=1 u g i u g p K u * g i p L 2 2,ϕ ≤ εϕ(p L ) 2 . Since K is topologically finitely generated by Proposition 2.2, Lemma 2.1 says that there are only finitely many subgroups in K that have index bounded by r. The intersection L of all closed subgroups of K which have index bounded by r satisfies g −1 i Lg i ≤ K for all i ∈ {1, . . . , n}. Lemma 2.15 implies that p K u * g i p L = p K u * g i for all i ∈ {1, . . . , n}. Using the KMS-condition with Proposition 2.24 and ∆(g i ) = 1 for all i ∈ {1, . . . , n} we see that εϕ(p L ) 2 ≥ ϕ(p L 1 n n i=1 u g i u g p K u * g i p L ) = ϕ( 1 n n i=1 u g i u g p K u * g i ) = 1 n n i=1 ϕ(u g p K u * g i σ −i (u g i )) = 1 n n i=1 ∆(g i )ϕ(u g p K u * g i u g i ) = ϕ(u g p K ) . Since ε > 0 is arbitrary and the choice of L is independent of ε, we see that ϕ(u g p K ) = 0. This proves the claim. Fix a compact open subgroup L ≤ G and let µ be the left Haar measure of G satisfying µ(L) = 1 ϕ(p L ) . If K ≤ G is an arbitrary compact open subgroup, we can apply Proposition 2.14 and the claim to obtain 1 µ(L) = ϕ(p L ) = 1 [L ∶ K ∩ L] g(K∩L)∈L K∩L ϕ(u g p K∩L ) = 1 [L ∶ K ∩ L] ϕ(p K∩L ) = 1 [L ∶ K ∩ L] g(K∩L)∈K K∩L ϕ(u g p K∩L ) = [K ∶ K ∩ L] [L ∶ K ∩ L] ϕ(p K ) . We infer that ϕ(p K ) = 1 µ(K) , finishing the proof of the proposition. Since a Plancherel weight on L(G) restricts to the corresponding Plancherel weight on C * red (G), the previous theorem allows us to conclude factoriality of L(G). We are also able to compute its type. Theorem 7.2. Let G be a group satisfying condition ( * ). Further assume that some compact open subgroup of G is topologically finitely generated. Then L(G) is a factor and S(L(G)) = ∆(G). • If G is discrete, then L(G) is a type II 1 factor. • If G is unimodular but not discrete, then L(G) is a type II ∞ factor. • If ∆(G) = λ Z for some λ ∈ (0, 1), then L(G) is a type III λ factor. • If ∆(G) is not singly generated, then L(G) is a type III 1 factor. Proof. Let ϕ be a Plancherel weight on L(G). We first show that L(G) is a factor. Assume that this is not the case. Then there is a central projection z ∈ L(G)∖{0, 1}. Since z is central, ψ ∶= ϕ(z⋅) is a weight with the same modular automorphism group as ϕ. Hence ψ restricts to a KMS-weight on C * red (G). By Theorem 7.1 there is a scalar c ∈ R >0 such that ψ C * red (G) = c ⋅ ϕ C * red (G) . Since ϕ is faithful, we have ψ(1 − z) ≠ 0, which contradicts the definition of ψ. We have shown that L(G) is a factor. We next show that L(G 0 ) is a factor. Let K ≤ G be a compact open subgroup such that N G (K) K contains a element of infinite order. Since K is compact and open, N G (K) ≤ G 0 . So G 0 satisfies condition (*) and the first part of the proof implies that L(G 0 ) is a factor. We may hence apply Theorem 2.27, implying that S(L(G)) = ∆(G). According to Section 2.4.1, it only remains to determine the type of L(G) in case G is unimodular. If G is discrete, then L(G) is a type II 1 factor. If G is unimodular but not discrete, then L(G) is a factor with a faithful properly infinite trace. Hence L(G) is of type I ∞ or type II ∞ . Let K ≤ G be a compact open subgroup. We show that the finite factor p K L(G)p K is not of type I. Then it is of type II 1 and hence L(G) is of type II ∞ . By Propositions 3.2 and 3.4 there is a hyperbolic element g ∈ G such that [K ∶ K ∩ g n Kg −n ] is bounded for n ∈ Z. Since g is hyperbolic, g n → ∞ in G. So for all ξ, η ∈ L 2 (G) we have ⟨p K u n g p K ξ, η⟩ = ⟨u n g p K ξ, p K η⟩ → 0 . So p K u n g p K → 0 weakly. At the same time, Lemma 5.1 says that p K u −n g p K u n g p K ≥ [K ∶ K ∩ g n Kg −n ] −2 p K is bounded from below and cannot converge to 0. Put differently, the sequence (p K u n g p K ) n does not converge to 0 in the strong topology. Since on finite type I factors the strong and the weak topology coincide, we conclude that p K L(G)p K must be of type II 1 . Non-amenability The following theorem gives a non-amenability criterion for a group von Neumann algebra of a locally compact group. It is based on the fact that L(G) is amenable if and only if G is amenable, as long as G is supposed to be discrete. Proposition 8.1. Let G be a locally compact group containing some compact open subgroup K ≤ G such that N G (K) is not amenable. Then L(G) is not amenable. Proof. Take K ≤ G as in the statement of the proposition. Put H ∶= N G (K). Since H ≤ G is open, there is a normal conditional expectation L(G) → L(H). So it suffices to prove that L(H) is nonamenable. Since K ⊴ H, we have p K L(H)p K ≅ L(H K), which is a non-amenable von Neumann algebra. It follows that also L(H) is non-amenable, which finishes the proof. We can apply our non-amenability criterion to groups acting on trees. Theorem 8.2. Let G be a group satisfying condition ( * ) and assume that some compact open subgroup of G is topologically finitely generated. Then L(G) is not amenable. Proof. We show that there is a compact open subgroup L ≤ G and transverse hyperbolic elements g, h ∈ N G (L). Fix K ≤ G a compact open subgroup. By Lemma 4.4 there are pairwise transverse hyperbolic elements (g i ) i≥1 and r ∈ N such that [K ∶ K ∩ g l i Kg −l i ] ≤ r for all i ≥ 1 and l ∈ Z. For fixed i the subgroup K i ∶= ⋂ l∈Z g l i Kg −l i ≤ K is normalised by g i . Proposition 2.2 implies that K is topologically finitely generated. So by Lemma 2.1, there are only finitely many closed subgroups of index less or equal to r in K. Hence K i is an intersection of finitely many open subgroups of K. It is hence open in K and [K ∶ K i ] is bounded by a constant only depending on r and K. We hence find different indices i, j ≥ 1 such that K i = K j . Put L ∶= K i , g ∶= g i and h ∶= g j . Since g, h are transverse hyperbolic elements the ping-pong lemma implies that ⟨g n , h n ⟩ ≅ F 2 for some n ∈ N. Moreover, we may assume that each element of ⟨g n , h n ⟩ is hyperbolic. This implies that ⟨g n , h n ⟩ ∩ K = {e}. So H ∶= ⟨g, h, L⟩ is an open non-amenable subgroup of G and K ⊴ H is normal. We can hence apply Proposition 8.1. This finishes the proof. Schlichting completions of Baumslag-Solitar groups In this section we show that Schlichting completions G(m, n) of non-amenable Baumslag-Solitar groups satisfy condition ( * ). They are hence the first examples of non-discrete C * -simple groups. We further calculate the type of the factors L(G(m, n)). In unpublished work with C.Ciobotaru, we obtained factoriality of L(G(m, n)) and could calculate its type by different methods. It is possible to give a criterion for graphs of groups with finite index inclusions of edge groups into vertex group, so as to make sure that the Schlichting completion of its fundamental group satisfies condition ( * ). This method gives rise to further examples to which our main result Theorem 6.2 applies. Let 2 ≤ m ≤ n be natural numbers. Then the Baumslag-Solitar group BS(m, n) ∶= ⟨a, t ta m t −1 = a n ⟩ contains the commensurated subgroup ⟨a⟩. Denote by G(m, n) = BS(m, n) ⟨a⟩ ≥ K(m, n) = ⟨a⟩ the Schlichting completion of the pair BS(m, n) ≥ ⟨a⟩. Recall that G(m, n) ≤ Sym(BS(m, n) ⟨a⟩) is a closed subgroup. By Bass-Serre theory, BS(m, n) ⟨a⟩ has the structure of a m + n-regular tree T , were gt⟨a⟩ and g⟨a⟩ are connected by an edge for all g ∈ BS(m, n). It follows that G(m, n) ≤ Aut(T ) ≤ Sym(BS(m, n) ⟨a⟩). The next lemma describes the image of the modular function of G(m, n). Proof. If m = n , then G(m, n) is discrete and hence it is unimodular. We may hence assume that 1 ≤ m < n. Then BS(m, n) ⊂ G(m, n) is a dense subgroup. Since m n Z ⊂ R >0 is discrete it suffices to show that ∆(BS(m, n)) = m n Z . Since a ∈ N G(m,n) (K(m, n)), it follows that ∆(a) = 1. Further t −1 ⟨a n ⟩t = ⟨a m ⟩ showing that ∆(t) = m n . Since BS(m, n) is generated by a, t, this finishes the proof of the lemma. Theorem 9.2. Let 2 ≤ m ≤ n and consider the relative profinite completion G(m, n) of the Baumslag-Solitar group BS(m, n). Then the following statements are true. • L(G(m, n)) is a non-amenable factor. • If m = n, then G(m, n) is discrete and L(G(m, n)) is of type II 1 . • If m ≠ n, then L(G(m, n)) is of type III m n . • C * red (G(m, n)) is simple. Proof. First note that G(m, n) ≅ Z nZ * Z if m = n. So in this case G(m, n) is icc and nonamenable, showing that L(G(m, n)) is a non-amenable type II 1 factor. Moreover, G(m, n) is a Powers group in the sense of de la Harpe [dlH85]. So C * red (G(m, n)) is simple. In case m ≠ n we have BS(m, n) ≤ G(m, n). Note that G(m, n) acts transitively on T , so that G(m, n) ↷ ∂T is minimal by Proposition 2.5. The element atat −1 ∈ G(m, n) 0 is hyperbolic and normalises the closure of ⟨a n ⟩, which is open in K(m, n). So Proposition 3.2 shows that the fixed points of atat −1 have an open stabiliser in G. This verifies condition ( * ) for G(m, n). Since K(m, n) is topologically singly generated, Theorems 7.2, 6.2 and 8.2 apply. Since ∆(G(m, n)) = m n Z by Lemma 9.1, Theorem 7.2 shows that L(G(m, n)) is a type III λ factor for λ = m n . By Theorem 8.2, L(G(m, n)) is not amenable. Theorem 6.2 implies that C * red (G(m, n)) is simple. Proposition 2. 5 . 5Let G ≤ Aut(T ) be a non-amenable subgroup. If G ↷ T admits no proper invariant subtree then G ↷ ∂T is minimal. Vice versa, if G is not compact and G ↷ ∂T is minimal, then T admits no proper invariant subtree. Proof. If G ↷ ∂T is not minimal, then there is some G-invariant open set U ⊂ ∂T such that ∂T ∖ U contains an open subset. Let T ′ = ⋃ x,y∈U (x, y) ⊂ T be the subtree consisting of all vertices on geodesics joining points in U. Then T ′ ≠ T , since ∂T ∖ U contains an open set. So we have found a proper G-invariant subtree T ′ ⊂ T Let d = d(ρ, η) + 1 and take y ∈ ∂T ∖ {x} such that m ρ (x, y) ≥ d. Then the geodesic [ρ, y) passes through η in the same direction as [ρ, x), meaning that [ρ, y) ∩ (η, x) ≠ ∅. Denote by ξ the vertex in which the geodesic [ρ, y) and [ρ, x) split. Since ξ ∈ (η, x) ⊂ (x, x ′ ) lies on the axis of g, and x is the attracting fixed point of g, we have [ Con73 ] Con73Connes introduced the invariant S(M ) ⊂ R ≥0 of a factor M . He proved that 0 ∉ S(M ) if and only if M is of type I or II. Further, S(M ) ∩ R >0 is a closed subgroup. He then proceeds to the following definition. Definition 2 . 11 . 211Let M be a factor of type III. If S(M ) = {0, 1}, then M is of type III 0 . If S(M ) = λ Z ∪ {0} for some λ ∈ (0, 1), then we say that M is of type III λ . If S(M ) = R ≥0 , then M is of type III 1 . Lemma 3. 3 . 3Let G ≤ Aut(T ) be a closed subgroup fixing a point x ∈ ∂T . Let H be the set of all elliptic elements of G. Then H is an ascending union of compact open subgroups and every element of G ∖ H is hyperbolic. For every g ∈ G ∖ H whose attracting fixed point is x there is a compact open subgroup K ≤ H such that gKg −1 ≥ K and H = ⋃ n∈Z g n Kg −n . Lemma 3. 3 . 3Since left and right multiplication with g preserve the left Haar measure of G, this implies gKg −1 = K. So g normalises a compact open subgroup of G. Finally if g normalises some compact open subgroup K ≤ G, then Proposition 3.2 applies to g and g −1 , showing that both fixed points of g have an open stabiliser. Since an arbitrary hyperbolic element g ∈ G normalises a compact open subgroup of G if and only if g −1 does so, the last statement of the proposition follows. Lemma 4. 4 . 4Let G ≤ Aut(T ) be a closed non-amenable subgroup acting minimally on ∂T . Assume that there is x ∈ ∂T such that G x is open and G x ∩ G 0 contains a hyperbolic element. Then for every non-empty open set O ⊂ ∂T there is r ∈ N and a sequence of pairwise transverse hyperbolic elements (g i ) i≥1 in G 0 such that the fixed points of g i lie in O for all i ≥ 1 and for all compact open subgroups K ≤ G the indices [K ∶ K ∩ g l i Kg −l i ] are uniformly bounded in i ≥ 1 and l ∈ Z. m ]r 2 0 2=∶ r for all i ≥ 1 and all l ∈ Z. Before we finish the proof in case (i), let us consider the other cases of the proposition. Consider case (ii). Take a compact open subgroup L ≤ G and x ∈ ∂T such that G x is open. Put K ∶= L ∩ G x and F ∶= L ∖ K. Since L is compact and G x is open, F x is finite by Proposition 3.1. Moreover, x ∉ F x. By Proposition 2.9 there is an open K-invariant set O ⊂ ∂T such that f O ∩O = ∅ for all f ∈ F . G does not fix a point in V(T ) ∪ E(T ) ∪ ∂T . Now Proposition 3.5 implies that G is not amenable. The closure of any finite index subgroup of Λ is open in K. So the closure L of Λ 0 is a compact open subgroup and N G (L) L = N Γ (Λ 0 ) Λ 0 contains an element of infinite order. We verified condition ( * ), finishing the proof of the corollary. Corollary 6 . 4 . 64Let T be a thick tree and G ≤ Aut(T ) be a closed subgroup acting minimally on ∂T . Assume that there is x ∈ ∂T such that • Kx is finite for some compact open subgroup K ≤ G, and • there is some hyperbolic element in G 0 ∩ G x . Lemma 9 . 1 . 91Let m, n ∈ Z × . Then the image of the modular function of G(m, n) is m n Z . for any left Haar measure µ on G and any measurable set A ⊂ G with finite non-zero Haar measure. By van Dantzig's theorem [vD36, TG 39], a locally compact group G is totally disconnected if and only if e ∈ G admits a neighbourhood basis of compact open subgroups. The modular function ∆ of a totally disconnected locally compact group G with left Haar measure µ satisfies Proposition 2.2. Let G be a locally compact group. If some compact open subgroup of G is topologically finitely generated, then all compact open subgroups of G are topologically finitely generated. Proof. Assume that some compact open subgroup K ≤ G is topologically. Any other compact open subgroup of G is commensurated with K, that is any compact open finite for all compact open subgroups K ≤ G. Further, if Kx < ∞ for some compact open subgroup, then K x ≤ K is closed and has finite index and it is hence a compact open subgroup of G. Finally, if some compact open subgroup K fixes x, then K ≤ G x , showing that G x is open. Lemma 3.3 applies to show that F ∩ G x only contains hyperbolic elements fixing x. It is covered by cosets h 1 K, . . . , h m K where h 1 , . . . , h m are hyperbolic elements fixing x. Find ρ ∈ T fixed by K. Since x is a fixed point for all hyperbolic elements h 1 , . . . , h m , we can apply Proposition 2.8 to m ρ (x, ⋅ ). Wefind some d ∈ N such that O 1 ∶= {z ∈ ∂T m ρ (x, z) = d} satisfies h i O 1 ∩ O 1 = ∅ for all i ∈ {1, . . . , m}. Since K fixes x and m ρ is K-invariant, we see that kO 1 = O 1 for all k ∈ K. So (F ∩ G x )O 1 ∩ O 1 = ∅.Recall from Remark 2.7 that the sets {z ∈ ∂T m ρ (x, z) ≥ d ′ }, d ′ ∈ N, form a basis of compact open neighbourhoods of x. We hence may choose d big enough so as to assume that O 0 ∩ O 1 ≠ ∅. Putting O ∶= O 1 ∩ O 0 , we found a non-empty K-invariant open subset of ∂T such that f O ∩ O = ∅ for all f ∈ F . AcknowledgementsWe want to thank Alain Valette for his hospitality at the University of Neuchâtel, where part of this work was done. We are grateful to Pierre-Emmanuel Caprace for useful comments on groups acting on trees. We thank Siegfried Echterhoff for asking us whether C * -simple groups are totally disconnected and for a helpful discussion about this question. 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J Bernstein, Functional Anal. Appl. 8J. Bernstein. All reductive p-adic groups are of type I. Functional Anal. Appl. 8, 91-93, 1974. C * -simplicity and the unique trace property for discrete groups. E Breuillard, M Kalantar, M Kennedy, N Ozawa, arXiv:1410.2518E. Breuillard, M. Kalantar, M. Kennedy, and N. Ozawa. C * -simplicity and the unique trace property for discrete groups. arXiv:1410.2518, 2014. Une classification des facteurs de type III. A Connes, Ann. Sci. Éc. Norm. Supér. 64A. Connes. Une classification des facteurs de type III. Ann. Sci. Éc. Norm. Supér. (4) 6, 133-252, 1973. Classification des facteurs. A Connes, Operator Algebras and Applications, Part. Kingston, Ontario; Providence, RIAmerican Mathematical Society2A. Connes. Classification des facteurs. In Operator Algebras and Applications, Part 2 (Kingston, Ontario, 1980), volume 38 of Proc. Sympos. Pure Math., pages 43-109. Providence, RI: Ameri- can Mathematical Society, 1982. C * -algebras. 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The plasma wave instabilities in ballistic Field Effect Transistors (FETs) have a promise of developing sensitive THz detectors and efficient THz sources. One of the difficulties in achieving efficient resonant plasmonic detection and generation is assuring proper boundary conditions at the contacts and at the heterointerfaces and tuning the plasma velocity. We propose using the tunable narrow channel regions of an increased width, which we call "stubs" for optimizing the boundary conditions and for controlling the plasma velocity. We developed a compact model for THz plasmonic devices using the transmission line (TL) analogy. The mathematics of the problem is similar to the mathematics of a TL with a stub. We applied this model to demonstrate that the stubs could effectively control the boundary conditions and/or the conditions at interfaces. We derived and solved the dispersion equation for the device with the stubs and showed that periodic or aperiodic systems of stubs allow for slowing down the plasma waves in a controllable manner in a wide range. Our results show that the stub designs provide a way to achieve the optimum boundary conditions and could also be used for multi finger structures -stub plasmonic crystals -yielding better performance of THz electronic detectors, modulators, mixers, frequency multipliers and sources.I.

10.1109/ted.2018.2854869

1806.00682

eb30fe12934e85792c9e9f790a1bfb1ad1a704c3

Plasmons in ballistic nanostructures with stubs: transmission line approach G R Aizin Kingsborough College The City University of New York 11235BrooklynNYUSA J Mikalopas Kingsborough College The City University of New York 11235BrooklynNYUSA M Shur Rensselaer Polytechnic Institute 12180TroyNYUSA Plasmons in ballistic nanostructures with stubs: transmission line approach 1 The plasma wave instabilities in ballistic Field Effect Transistors (FETs) have a promise of developing sensitive THz detectors and efficient THz sources. One of the difficulties in achieving efficient resonant plasmonic detection and generation is assuring proper boundary conditions at the contacts and at the heterointerfaces and tuning the plasma velocity. We propose using the tunable narrow channel regions of an increased width, which we call "stubs" for optimizing the boundary conditions and for controlling the plasma velocity. We developed a compact model for THz plasmonic devices using the transmission line (TL) analogy. The mathematics of the problem is similar to the mathematics of a TL with a stub. We applied this model to demonstrate that the stubs could effectively control the boundary conditions and/or the conditions at interfaces. We derived and solved the dispersion equation for the device with the stubs and showed that periodic or aperiodic systems of stubs allow for slowing down the plasma waves in a controllable manner in a wide range. Our results show that the stub designs provide a way to achieve the optimum boundary conditions and could also be used for multi finger structures -stub plasmonic crystals -yielding better performance of THz electronic detectors, modulators, mixers, frequency multipliers and sources.I. INTRODUCTION Numerous existing and potential applications of THz technology require efficient electronic THz sources and detectors. [1] This is especially important for THz communications supporting beyond 5G WI FI that would require massive deployment of sub-THz and THz systems. Developing the THz electronic sources is the key challenge to be met for closing the famous THz gap. The existing sub-THz and THz electronic sources use Gunn [2] and IMPATT [3] diodes (directly or with frequency multiplication by Schottky diodes [4]), InP based High Electron Mobility Transistor [5] or Si CMOS [6] and BiCMOS [7] Integrated Circuits. The plasma wave instabilities in ballistic Field Effect Transistors (FETs) have a promise of enabling more efficient and tunable THz sources based on the Dyakonov-Shur (DS) instability [8] and the "plasmonic boom" instabilities [9,10]. The mechanism of the DS instability involves the plasma wave reflections from the source and drain channel edges. The DS instability has the largest increment when the boundary conditions at the source and drain edges of the device channel correspond to zero amplitude and to the largest amplitude of the THz electric field variation, respectively. It has not been clear how to realize such boundary conditions at the THz frequencies. The problem of controlling the boundary conditions and conditions at interfaces between different device sections becomes even more important for the periodic "plasmonic crystal" structures using the sections experiencing the DS instability [11] or "plasmonic boom" transitions. [9,10] In the latter structures, the plasmonic instability occurs when the electron drift velocity crosses the plasma wave velocity. This could be achieved by modulating the carrier concentration in the plasmonic device channel and/or the channel width. In either case, controlling the conditions at the interfaces of the channel sections are very important. The "plasmonic boom" mechanism requires the electron drift velocity to become higher than the plasma velocity. This requirement might be difficult to meet because the plasma velocity is typically quite high. As shown in this paper, using stubs would allow to slow down the plasma waves in a controllable fashion. The plasmonic detection/generation experiments reported so far relied on the asymmetry at the source and drain contact. Such asymmetry is sufficient, in principle, to achieve the THz detection or even the DS instability but the sensitivity of the detection or the instability increment are enhanced in the structure with the build-in asymmetry [12]. In this paper, we describe how to solve these problems by adding narrow protruding regions with tunable carrier concentration to the device channel (see Fig. 1). Following [13], we call these regions "stubs." This term emphasizes the analogy with the transmission line formalism that has been used for the analysis of the plasmonic structures. [14,15,16,17,18] [19] allows for an effective control of the stub width and length. The combination of the top and side gates could also suppress the narrow and/or short channel effects. Figure 2. Unmodulated stub and different stub modulation schemes using top and side gates. The stub introduces impedance that could vary from minus infinity to infinity at the plasma frequency depending on the stub effective area and other parameters. Hence, placing the stubs at the two edges of the channel should allow for setting the optimum boundary conditions for the DS instability. Introducing the stubs at the region interfaces should allow for controlling the transitions between the interfaces and phenomena such as choking [20]. Our results also show that stubs could slow down the plasma waves, which is important for achieving the plasmonic boom conditions [9,10]. BASIC EQUATIONS We consider plasma waves propagating between the source and the drain contacts (x-axis) in the gated 2D electron layer located at the = 0 plane. The hydrodynamic equations (the Euler equation and equation of continuity) for the local sheet electron density ( , ) and velocity ( , ) in the plasma wave are + = * ,(1)+ ( ) = 0,(2) where ( , = 0, ) is electric potential in the 2D layer, − and * are the electron charge and effective mass, respectively. The hydrodynamic approach is justified if the time of the electron-electron collisions is much less than the time of electron collisions with impurities and phonons and the electron transit time across the device. In the Euler equation, we omitted the pressure gradient term since in the gated 2D channels this term is typically much smaller than the field term in the above threshold regime as was discussed in [10]. We neglect effect of collisions with phonons and impurities on the electron transport, i.e. we assume the ballistic electron transport with the electron mean free path larger than the channel length and ≫ 1, is the plasmon frequency, is characteristic collision time. Equations (1, 2) could be linearized with respect to the small fluctuations of the electron density ( , ) and velocity ( , ) by assuming that ( , ) = + where is the equilibrium electron density [21] = 0 ln �1 + �(3) Here is the gate voltage swing, is the Boltzmann constant , is the ideality factor, T is temperature, 0 = 0 2 , 0 = 0 +∆ is the gate-to-channel capacitance per unit area, is the dielectric constant, 0 is the dielectric permittivity of vacuum, d is the channel-togate separation, and ∆ is the effective thickness of the 2DEG. We consider the strong gate screening limit when is linked with the fluctuation of electric potential as − = , where C is the differential capacitance given by is ( ) = 1 − 1 + 2 − 2 (5) ( ) = 1 � 1 − 1 − 2 − 2 �(6) where = − is the total current and V ω ω δϕ ≡ is the voltage distribution in the plasma wave of frequency propagating in the channel of width , 1,2 / p q v ω = ± are the wave vectors of the plasma waves propagating in two opposite directions in the channel, and = � * �1 + − � ln �1 + �(7) is the plasma velocity. [22]. Constants 1 and 2 are determined by the boundary conditions. As discussed in [14][15][16][17][18], description of the plasma waves in the 2D electron system within the hydrodynamic model is analogous to the description of the electromagnetic signals in the transmission line (TL). The linearized hydrodynamic equations for the plasma waves in the 2D electron channel are equivalent to the telegrapher's equations for the TL with the distributed inductance ℒ = * 2 0 , resistance ℛ = ℒ/ per unit channel length, and distributed capacitance, which depends on the gating conditions [14,17]. In the limit of a strong gate screening, the distributed capacitance per unit length equals to with defined in (4). The 2D electron channel can be viewed as a plasmonic waveguide supporting transverse electromagnetic (TEM) plasma modes. FET with the stubs with separately biased gates allows for tuning the stub impedances by adjusting the carrier concentration in the stubs and, therefore, the plasma velocity in the FET channel (see Fig. 2). Within the TL approach, the stub in Fig. 1 represents an open circuit TL stub maintaining its own standing plasma modes and characterized by the input stub impedance = − 01 cot 1(8) where 01 = 1/� 1 1 1 �is the characteristic impedance of the plasmonic TL in the stub, and 1 are the length and the width of the stub, 1 and 1 are the differential capacitance and plasma velocity in the stub defined by Eq. (4) and (7), respectively.This stub model is valid if the width of the stub 1 is much smaller than the plasmon wavelength in the 2D FET channel so that the junction of the stub to the channel is a well-defined point. Zeros of correspond to the excitations of the standing plasma modes in the stub with wavelengths = 4 (2 +1) , = 0,1, … [13]. Fig. 3 shows the equivalent TL electric circuit for the FET shown in Fig. 1 with the stub located close to the source edge of the channel (Fig. 3a) and at some arbitrary position in the channel at distances 1 and 2 from the source and the drain (Fig. 3b). In this Figure, we also introduced terminating impedances and describing the electric links between the gate and the source and drain contacts, respectively. These links determine the boundary conditions in the plasmonic cavity formed between the source ( = 0) and the drain ( = ) contacts in the FET as (0) = − (0) (9) ( ) = ( ) It follows from (8) that the value of at the plasma frequency can be tuned from minus infinity to infinity by varying the carrier concentration and the geometry of the stub. If the stubs are located near the source and/or the drain edge of the channel as shown in Fig. 3a the impedance is combined with and to provide a very effective way to control and optimize the boundary conditions for the plasmonic FET. To illustrate this conclusion, we used the TL formalism to calculate the input gate-tosource impedance in the stub FET assuming = ∞: Here 0 = 1/ is the characteristic impedance of the plasmonic TL in the FET 2D channel. Fig. 4 shows the calculated dependence of / 0 on the frequency / 0 , 0 = , for the stub FET with a stub located close to the source (L1 << L2) and 01 = 0 : Figure 4. Frequency dependence of the input source-to-gate impedance for the stub FET shown in Fig.1 at different stub lengths . Inset: equivalent TL electric circuit. As seen from (8) and Fig. 4, the input impedance is tunable in a wide range by changing the parameters of the stub, such as the electron density or effective dimensions of the stub using the stub modulation schemes shown in Fig. 2. The impedance derivative with respect to frequency could be also tuned as clearly seen from Fig. 4, where this derivative is zero at certain frequencies. This result is important for designing broad band plasmonic devices. In agreement with the results reported in [13], Fig. 4 shows that the system with a stub has two types of plasmonic resonances ( = 0): (1) the resonances associated with the channel and (2) the resonances associated with the stub. The latter resonances are tunable by varying the stub parameters. Anti-resonances ( = ∞) occur when the stub and the channel impedances (that are connected in parallel, see the inset in Fig. 4) cancel each other. Tuning the plasmonic frequencies and the input impedance in a wide range by varying the effective stub parameters enables the applications of the system with stubs for the terahertz spectroscopy by adjusting the resonance frequency to coincide with the frequency of the impinging THz signal. Another application is in the resonant THz interferometry. [23] Whereas the stubs located near the source and drain ends of the channel allow us to control the input and output impedances and, hence, the boundary conditions, the stubs attached to the channel at some distance from the source and the drain edges as shown in Fig. 3b control the plasmonic spectrum of the channel. To demonstrate this effect, we derived the plasmon dispersion relation for the plasmonic cavity with the stub shown in Fig. 3b. We used the TL formalism to solve (5) with the stub impedance given by (8) and the boundary conditions given by (9) with the result 2 2 = �1 − 0 � �1 − 0 − 0 � �1 + 0 � �1 + 0 + 0 � (11) = 0 cos 1 + 0 sin 1 0 cos 1 + sin 1 , In Fig. 5, we present the results of the numerical solution of the plasmon dispersion equation (11) in case when the stub is positioned at the center of the channel, 1 = 2 = /2, and = 1 , 1 = . Plasmonic spectrum is shown as a function of the stub length assuming = 0 and = ∞. As seen from Fig. 5, addition of the stub reduces the unperturbed resonant plasmonic frequencies = 2 (2 + 1) , = 0,1, … for the same channel length and, hence, for the same wave vector. This means the plasma waves in the device with a stub have a smaller velocity. The unique feature of the stub design is that this velocity is tunable by the stub side and/or top gates modulating the effective stub area and/or the carrier concentration in the stub (see Fig. 2). For the conventional design, controlling the carrier concentration in the channel simultaneously adjusts the plasma velocity and the input impedance. The addition of the independent stub gate control allows for an optimization of the plasmonic structures for THz electronics applications independently adjusting the plasma velocity and the input and output impedances. Inserting the stubs near the source and/or near the drain separates the channel design from the input/output impedance design. The source stub could control , The drain stub could control . One or more stubs attached to the channel could control the channel properties. This approach makes the stubs a perfect optimization tool box for the plasmonic design. The generalization of this approach are stub plasmonic crystals of different dimensionality and stub plasmonic crystals with complex (asymmetric) elementary cell shown in Figs. 6a, 6b, 6c and 6d, respectively. Fig. 6e shows the equivalent TL electric circuit for a 1D stub plasmonic crystal in Fig. 6a. where is the plasmon wave vector. In the first fundamental band at → 0 the plasmon dispersion law is linear with plasmon velocity given by = ± �1+ 0 01(13) It follows from (13) that plasmons are slowing down, and this process can be tuned by adjusting the effective width or length, or effective carrier concentration in the stubs. Fig. 7 shows the plasmon dispersion relations calculated from (12) for stubs of different length and width and = 1 illustrating this conclusion. Figure 7. Plasmonic band spectrum in the 1D stub plasmonic crystal with period at different lengths and widths 1 of the stub. is the width of the 2D channel. In this case, (13) reduces to = ± √1+(14) where = 1 / is the effective stub area equal to the ratio of the stub area to the area of the one period of the 2D channel. Fig. 8 shows variation of the plasma velocity with the effective stub area and the derivative � � that shows a high sensitivity of the plasmon velocity to the effective stub area that could be adjusted by the stub gate bias, see Fig. 2. This is very important for the optimization of the plasmonic THz devices. Varying the stub area in the sequence of stubs (as shown in Fig. 9) allows for a gradual change of the plasma velocity along the channel. For the most interesting case when ≫ 1 , (15) becomes cos = cos � − 2 � − 1 2 tan � − 2 � sin � − 2 �(16) As seen, the key parameter for the ballistic transport approximation is . This parameter has to exceed unity for the quasi ballistic approximation to be valid. For typical room temperature values of τ 10 -13 s and vp ~ 10 6 m/s, we obtain L < 100 nm. This is a very realistic value that could be exceeded by one or even two orders of magnitude at cryogenic temperatures and in advanced materials, such as graphene. The proposed approach enables a large variety of the tunable THz devices ranging from sources, to spectrometers, interferometers, mixers, frequency multipliers, modulators, and detectors. In the latter case, the design with asymmetric stubs (as schematically shown in Fig. 6d is beneficial. CONCLUSIONS The device impedance in the presence of stubs has a wide range tunability that can be used for optimizing the plasmonic detectors and sources. The solutions of the dispersion equation for the device with the stubs demonstrated that the stubs can adjust and tune the plasma velocity. This feature makes the plasmonic boom devices feasible. This new device design also allows to implement 1D, 2D, and 2D plasmonic crystals for achieving higher powers and better radiation extraction. The proposed TL modeling approach support compact design modeling of the future generation of plasmonic THz devices for applications in THz sensing, imaging, and communications. ACKNOWLEDGMENTS The work at RPI and CUNY was supported by the US Army Research Office (Project Manager Dr. Joe Qiu). Figure 1 . 1Stub configurations and notations. Fig. 2 2shows the designs with different implementations of tuning the effective electric properties of the stub. The top gate controls the carrier concentration in the stub. The side gate implementation case, the solution of the linearized equations for the Fourier harmonics ( Figure 3 . 3Equivalent TL electric circuits of the FET channel with a stub. Figure 5 . 5Plasmon frequencies in the stub FET plasmonic cavity of length with a stub at the center of the cavity as a function of the stub length . Figure 6 . 6Schematic diagram of the stub plasmonic crystals: (a) 1D; (b) 2D, (c) 3D (d) 1D asymmetric stub plasmonic crystal; (e) equivalent TL electric circuit for the 1D stub plasmonic crystalThe "Kronig-Penney" analysis yields the following plasmon dispersion equation for the 1D stub plasmonic crystal with period : Figure 8 . 8Plasmon velocity (red) and � / � (blue) as a function of the effective stub area, , in the 1D plasmonic stub crystal Fig. 9 . 9Sequence of stubs enabling a gradual change of the plasma velocity along the channel The above results for the plasmonic crystal had been derived assuming ballistic electron transport when scattering of electrons on phonons and impurities can be neglected. 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The world is moving into a new era with the deployment of 5G communication infrastructure. Many new developments are deployed centred around this technology. One such advancement is 5G Vehicle to Everything communication. This technology can be used for applications such as driverless delivery of goods, immediate response to emergencies and improving traffic efficiency. The concept of Intelligent Transport Systems (ITS) is built around this system which is completely autonomous. This paper studies the Distributed Denial of Service (DDoS) attack carried out over a 5G network and analyses security threats. The aim is to implement a machine learning model capable of classifying different types of DDoS attacks and predicting the quality of 5G latency.The initial steps of implementation involved the synthetic addition of 5G parameters into the dataset. Subsequently, the data was label encoded, and minority classes were oversampled to match the other classes. Finally, the data was split as training and testing, and machine learning models were applied. Although the paper resulted in a model that predicted DDoS attacks, the dataset acquired significantly lacked 5G related information. Furthermore, the 5G classification model needed more modification. The research was based on largely quantitative research methods in a simulated environment. Hence, the biggest limitation of this research has been the lack of resources for data collection and sole reliance on online data sets. Ideally, a Vehicle to Everything (V2X) project would greatly benefit from an autonomous 5G enabled vehicle connected to a mobile edge cloud. However, this project was conducted solely online on a single PC which further limits the outcomes. Although the model underperformed, this paper can be used as a framework for future research in Intelligent Transport System development.

2202.00005

1a5eb7b56d86b0f5da26fc36c05eaba7b99b98f5

5G enabled Mobile Edge Computing security for Autonomous Vehicles DrDaryll Ralph D&apos;costa [email protected] Macquarie University Robert Abbas [email protected] Macquarie University 5G enabled Mobile Edge Computing security for Autonomous Vehicles 1 The world is moving into a new era with the deployment of 5G communication infrastructure. Many new developments are deployed centred around this technology. One such advancement is 5G Vehicle to Everything communication. This technology can be used for applications such as driverless delivery of goods, immediate response to emergencies and improving traffic efficiency. The concept of Intelligent Transport Systems (ITS) is built around this system which is completely autonomous. This paper studies the Distributed Denial of Service (DDoS) attack carried out over a 5G network and analyses security threats. The aim is to implement a machine learning model capable of classifying different types of DDoS attacks and predicting the quality of 5G latency.The initial steps of implementation involved the synthetic addition of 5G parameters into the dataset. Subsequently, the data was label encoded, and minority classes were oversampled to match the other classes. Finally, the data was split as training and testing, and machine learning models were applied. Although the paper resulted in a model that predicted DDoS attacks, the dataset acquired significantly lacked 5G related information. Furthermore, the 5G classification model needed more modification. The research was based on largely quantitative research methods in a simulated environment. Hence, the biggest limitation of this research has been the lack of resources for data collection and sole reliance on online data sets. Ideally, a Vehicle to Everything (V2X) project would greatly benefit from an autonomous 5G enabled vehicle connected to a mobile edge cloud. However, this project was conducted solely online on a single PC which further limits the outcomes. Although the model underperformed, this paper can be used as a framework for future research in Intelligent Transport System development. INTRODUCTION The world is advancing into a new era of autonomous self-driving vehicles. A lot of research is currently underway to improve the functionality of these vehicles and make them safe. With the emergence of 5G cellular networks, the autonomous self-driving industry can leverage this infrastructure to improve autonomous driving technology. Human mistake is responsible for over 94 percent of all car accidents whereas mechanical failure is responsible for only 2% of all accidents [1]. While taking a closer look, it is found that speeding is responsible for one out of every four deaths [1]. The figures are stunning, revealing that a quarter of all traffic fatalities and most traffic accidents are avoidable. As a result, connected vehicles will be a revolutionary in terms of traffic safety. Engineers are trying to implement an Intelligent Transport System (ITS) where vehicles communicate with each other and their surroundings completely on their own. This requires the vehicles to be equipped with high grade sensors and reliable wireless technology. Autonomous vehicles are usually equipped with sensors like Light Detection and Ranging (LiDAR), Global Positioning System (GPS), radar and odometry that collect data and send it to the onboard processing unit. Additionally, the vehicles are also equipped with high-resolution cameras record everything. Subsequently, these vehicles then use image processing and computer vision to understand their surroundings and make decisions using machine learning. If the automobile cannot recognize a specific movement or pattern, it can send the data to the cloud. The cloud is a collection of vast to nearly unlimited resources for computing and storage. 5G networks have a component called mm Waves which enable Ultra-Reliable Low Latency Communication (URLLC) which allows signaling speeds of less than one millisecond. Additionally, cellular networks employ the concept of network slicing that reserves a dedicated frequency spectrum for Vehicle to Everything (V2X) based communication. Malicious attackers can use these low latency mm Wave frequencies for wrong reasons. Security is the most critical aspect in self driving since human life is involved. Attackers try to manipulate, modify, corrupt, or block communication to the intended vehicle which can cause mishaps to happen on the road. One such attack is Distributed Denial of Service (DDoS) which aims to overload the receiving interface at the car and render it blind. This could potentially disrupt a specific vehicle or a group of vehicles and cause accidents. A possible solution is to employ machine learning principles to detect an attack and mitigate it. BACKGROUND AND RELATED WORK This section explores the different aspects of the 5G V2X environment. It explains how vehicles communicate within themselves and to the infrastructure and environment. It also explains the importance of mobile edge cloud and the enabling aspects. Since this research discusses vehicular 5G security, it studies all types of attacks. Since the attack scenario is DDoS attacks, the thesis explains the Distributed Denial of Service attacks and their types. In addition, it discusses a brief overview of the various machine learning models. Finally, the thesis explains information on the dataset and other details of its implementation. Mobile Edge Computing Cloud computing is an IT paradigm that allows users global access to shared pools of configurable system resources and higher-level services deployed rapidly and with less administrative effort over the Internet. Cloud computing, similar to a public resource, relies on shared resources to achieve continuity and profits. MEC is a group of edge servers that are a type of edge device that serves as a network entrance point. Additionally, routers and network switches are examples of other edge devices. Edge devices frequently positioned within Internet exchange points (IxPs) allow multiple networks to join and share bandwidth [2]. The final aspect of MEC is customer premises equipment (CPE). The CPE is a part of the CSP's wide-area network (WAN) infrastructure provided to its enterprise users. It is located at a client location and ends WAN links into that location. Its goal is to provide stable and secure connectivity between WAN connected devices and the local area network (LAN) at that specific location. As a result, a CPE usually includes switching and smart routing technology for navigating traffic between the WAN and the LAN. In addition, these smart devices can perform load balancing between WAN links when multiple links are present. The security system installed provides industry-standard security services to traffic moving into and out of the organization. The switching, routing and firewall functions are usually a group of policies that offer Quality of Service (QoS) distinction and integration with enterprise policy systems and other features [3]. Many cloud-based companies and vendors take part in the Mobile Edge Computing ecosystem. Google, Amazon Web Services (AWS) and Microsoft Azure are examples of cloud companies that provide the necessary resources and services. They are Hyperscale Cloud Providers (HCP's) and actively compete to take over the edge computing market. Enabling Technologies In recent years various technologies devised have further improved existing hardware. These technologies enable even faster communication of vehicular networks. In addition, they are abstracted on top of existing hardware and use their resources and work as individual software instances. It is a powerful tool, as it allows multiple separate instances of the software to run completely different tasks without being dependent on additional resources. Consequently, the hardware configured allocates a set amount of each technology. Some of the technologies mentioned are listed below. Network Function Virtualization (NFV) By employing full-blown virtualization technology, NFV improves how network operators construct their infrastructure. It is carried out by decoupling software instances from hardware platforms and decoupling functionality from the location for faster networking service provisions. In other words, NFV virtualizes network functions and runs them on existing hardware using software virtualization techniques (i.e., industry servers, switches, and storage units. Furthermore, virtual appliances launched on-demand do not require the need for additional hardware [4]. While in use, NFV technology in 5G V2X networks provides the availability of the IOS that makes it simple and easy to update the system by simply updating the software. The feasibility and viability of Vehicle-to-vehicle (V2V) collaboration in 5G V2X networks is due to Virtual Network Functions (VNF) reuse and migration. Additionally, VNF reuse enables efficient cluster-based networking. Communication delay is reduced by allowing adjacent vehicles to act as local VNF managers instead of faraway service centers [5]. Software Defined Networks SDN (software-defined networking) is a new network concept that combines logic, centralised control, and programmability in one package. The SDN controller used to manage network resources flexibly centralises network data and integrates diverse network protocols and standards [6]. Smart Collaborative Networking To create smart Internet architectures, Smart Collaborative Networking (SCN) was introduced. SCN believes that the future Internet narrative should switch from the existing service provisioning through controlled ownership of infrastructures to a future of unified management systems full of dynamic control and coordination between users, networks, and facilities through virtualization technology and programmability [7]. Blockchain In 2008, blockchain technology was developed to enable the creation of Bitcoin. It's essentially a collection of rules that govern how the virtual currency works. The cryptocurrency was developed and traded further, and the community that developed the technology began to work on fixing bugs in the system. It is a communal registration system that consists of a chain of blocks, thus the name. This implies that rather than being kept in a single location, all blockchain data is spread across the numerous computing units linked to it [8]. DDoS attack types There are many forms of DDoS attacks. Each attack type aims to exploit a particular weakness in the networking protocols. The overall goal of DDoS attacks is to overload networking equipment with traffic causing congestion and disabling the correct entity from accessing information. This attack can be even more dangerous when coupled with 5G. The super-fast speed of 5G networks will make it even harder for network equipment to handle DDoS traffic. NTP based attack An attacker exploits Network Time Protocol (NTP) server functionality to flood a specific client-server or other networks with an increasing amount of User Datagram Protocol (UDP) data traffic in this DDoS attack feedback. This attack has the potential to render the destination and its network infrastructure inaccessible to regular traffic [9]. DNS based attack An attacker uses a Botnet to create a high number of resolution requests to a designated Internet Protocol (IP) address in this DDoS assault based on reflection [9]. Modern Domain Name System (DNS) applications rely on the DNS Security Extensions' sophisticated security protocols. Vehicular DNS communications require validation of the source and legitimacy of DNS resource entries. A solution to this attack is employing multi hop Vehicle-to-Vehicle communications to reach a name server, as well as a new Bloom filter-based technique to record verification [10]. LDAP based attack It is a reflection-based DDoS attack. Here the attacker sends requests to a publicly accessible insecure Lightweight Directory Access Protocol (LDAP) server to create amplified answers mirrored to a target server [9]. MSSQL based attack It is a reflection-based DDoS attack. Here the attacker uses the Microsoft SQL Server Resolution Protocol (MC-SQLR) to execute scripted requests with a faked IP address to appear as it was coming from the target server [9]. NetBIOS based attack It is a reflection-based DDoS attack. Here the attacker delivers a victim machine fraudulent "Name Release" or "Name Conflict" notifications to block all Network Basic Input/Output System (NetBIOS) network traffic [9]. SNMP based attack It is a massive DDoS attack that leverages the Simple Network Management Protocol (SNMP) to produce attack volumes of thousands of gigabits per second to clog the target's network pipes [9]. SNMP is used to configure and gather information from network devices such as servers, switches, routers, and printers. Additionally, it applies to SNMP reflected amplification attacks. The attacker uses SNMP to provide a flood of answers to the victim, like other reflection attacks. The offender sends many SNMP queries with a fabricated IP address (the target's) to numerous connected devices, which respond to the fraudulent address [11]. SSDP based attack Simple Service Discovery Protocol (SSDP) is a reflection-based DDoS attack. Here the attacker uses Universal Plug and Play (UPnP) networking protocols to transmit an amplified amount of traffic to a targeted victim [9]. UDP Lag based attack The goal of this attack is to use IP packets containing UDP datagrams to slow down or interrupt the targeted host [9]. Web DDoS based attack The goal of this attack is to hack into a Web server or application using malware takes advantage of valid HyperText Transfer Protocol (HTTP) GET or POST queries [9]. SYN based attack This attack makes use of the standard Transmission Control Protocol (TCP) three-way handshake (i.e., transmitting SYN (synchronize), SYN-ACK (synchronize-acknowledge), and responding with an ACK (acknowledge)) to consume resources and render the targeted network server unavailable [9]. TFTP based attack This attack uses TFTP servers connected to the internet to exploit the Trivial File Transfer Protocol (TFTP). Specifically, an attacker sends a file request to the victim TFTP server, which returns the data to the requesting target host [9]. Port Scan based attack This attack conducts a network security audit by scanning ports on a single machine or a network. The Queries in use detect which services are running on a remote server during the scanning process [9]. DATA ANALYSIS AND VISUALIZATION The dataset was downloaded from [12]. The dataset consisted of a ZIP file which in turn consisted of twelve folders. Each folder was a simulation of a specific kind of DDoS attack. After unpacking the data, the first step involved loading the data into a python Jupyter notebook. Since each dataset was too large, a total of 2200000 samples from each dataset were imported to not overload the operating PC environment. The twelve unique datasets were then merged to form a single dataset that contained all the DDoS attacks. On initial inspection of the dataset, it is observed that there was a total of 88 columns representing a distinct feature. The resultant dataset consisted of 5500000 unique data points. The dataset consisted of a Label column which described the type of DDoS attack. The initial unique Label counts are as follows: The dataset reveals a good distribution of most forms of attacks. The only attack which has a smaller number of recorded observations is Web DDoS attacks. Also, since most attacks are focused around simulating the attack, there are also very few recorded observations of Benign data. Consequently, the lack of these observations is handled in data preprocessing stage and the number of recorded observations is synthetically increased. The dataset is a pure simulation of only DDoS attacks only. This means that there are no features corresponding to 5G. Since this thesis focuses on the aspects of 5G networks, and the goal is to simulate a machine learning base in a 5G Mobile Edge Computed environment, the dataset was updated to include information of 5G based parameters as well. This was done by making three new feature columns namely, 5G_RSRP, 5G_RSRQ and 5G_Latency. Data visualizations To better understand the distribution of the DDoS attack types, a pie chart is created to see the percentage of each attack. Figure 1 is the resultant pie chart showing the necessary information with respect to DDoS attacks. The analysis shows that most of the DDoS attacks are nicely balanced. Each once consists of an average of 9 percent of unique data. Except of Web DDoS attack which consists of 0.01 percent. Web DDoS attacks are generally based off the application layer [13], so it is common for the number of samples to be lesser. The pie chart analysis also shows that there is less benign data since most of the unique samples are simulated to be a part of DDoS attack. Of all the data, the NetBIOS, NTP, TFTP and Syn attacks have the highest distribution of samples. Since there is a total of over eighty features, only a few of them are explained. The explained features are in accordance with the K best features selected. Source Port The source port is the point of origin of an IP address. It is uniform across the TCP/IP and UDP protocol stacks. Figure 2 shows the distribution of source ports and the number ports being used. Analysis of the plot shows that the common ports for application layer access are used in benign. Generally, port 80 for HTTP access is used. It is observed that the other attacks use port numbers starting from roughly 5000 and end up using almost all the ports till over 60,000. TFTP in particular has higher counts of port usage. Destination Port Similar to source ports, the destination port is the final destination the IP packed is supposed to reach. This is also uniform cross the TCP/IP and UDP protocols. Figure 3 shows the distribution of destination ports and the number of them being used. From the plot, it is evident that almost all the ports are being occupied by the various attack types. The count for each port is also very high. This means that from one source port, the attack type tries to attack multiple destination port as evident in the plot. Only benign data is being transported over the pre-defined application layer ports. 5G Latency Label The 5G latency was synthetically added. The latency was further classified as good or bad with respect to attack type. The label good is given to latencies if the latency value is below 30 ms. All other latencies are classified as bad as the DDoS attacks may be affecting the 5G radio system. Figure 4 shows the counts of good 5G latency vs bad 5G latency. This 5G latency label is one of the output labels that will predicted using machine learning. From the counts we can see that most of the counts come under the bad category. Machine learning models will become biased if the current dataset was to be passed. The minority is balanced out in a later stage to give the models a fair chance to analyze the data. DATA PRE-PROCESSING Label encoding Machine learning models struggle to process anything other than numbers. The goal is to predict the DDoS attack and 5G Labels but is not possible if they are of categorical form. Until this stage, each DDoS attack has a name associated such as DDoS_NTP. Label encoding converts these labels into a unique number. The process of label encoding is applied to both the DDoS labels as well as to 5G Latency Labels. The figure 10 shows the code used to convert the labels into numerical labels. Handling categorical, infinite, and unwanted data Once the encoding process is done the column with categorical labels for both DDoS and 5G Latency values must be dropped. The data set also consists of infinite values which need to be handled properly. A small function is written which takes in the data frame columns as values processes at and returns the columns which have infinite values. There are also some categorical labels that need to be dropped. Additionally, there are some other features like unnamed:0, source port and destination port which are added do a final list of columns that need to be dropped. Consequently, all the unwanted columns are grouped together and dropped using the pandas drop method. Performing oversampling to rectify minority classes As discussed in earlier sections there are a minority of labels corresponding to benign and Web DDoS. Here the process of synthetic minority oversampling technique (SMOTE) [14] is used to rectify the minorities and make their distribution equal to the other classes. This process is carried out separately for both DDoS and 5G input features. The Figures 5 and 6 below shows the new distribution of the classes and the total number of values associated to them. It is evident that all the minorities are rectified. Feature Selection Machine learning is a delicate process. If the data is not preprocessed properly, it will affect the performance of the models. Since the dataset is very large with over 80 features, there is a high chance that feeding all the features as inputs will confuse the model and the predictions will be drastically affected. For this reason, a two-step feature selection process is considered. Feature selectionstep 1: K Best Features As stated, the first stage of feature selection employs the K best features approach. This method is adopted as a crude form of feature selection to get the 40 best features. This is method is first adopted because it is faster in processing the data and returning the output. K best features is class of the scikit learn feature selection library. The score function used is the f_regression method. It is a univariate linear regression test that return F-statistic and p-values. It ranks the features in the same order if they are positively correlated to the target. Feature selectionstep 2: Recursive Feature Estimation (RFE) The second stage of feature selection employs the Recursive Feature Estimation technique. This is used as the last step of feature selection to get the refined final 20 features that will directly be fed to the machine learning models. The RFE is also a part of the scikit learns feature estimation library. The recursive method is relatively time consuming and has a slight overhead. For this reason, it was adopted as the second stage to reduce the time taken to process the data. The estimator passed to the RFE was the Decision Tree Regressor. The final features were then stored in a final variable called X DDoS and 5G train/test and y DDoS and 5G train/test. RESULTS AND ANALYSIS This section deciphers the performance of all the models after training was completed. First, a list is made consisting of all the names of the classifiers that were used in the modeling process. Next a list named scores_ddos is created which holds a list accuracy score, precision score, f1 score and recall score of all the classifiers used for DDoS predictions. Similarly, a list named scores_5G is created which holds information of scores of all the classifiers used for 5G predictions. The plots are visualized in the next section. Model evaluation The plots are created with the axes showing all the classifiers and how they performed. Figure 7 show the plot for all the classifiers for DDoS. From the above figure it is observed that the Random Forest classifier performs the best with the accuracy peaking at around 74 percent. The overall performance of all the other scores of the classifier are also relatively good. This means that adopting the random forest is a good choice in predicting the output of the type of DDoS attack. The Decision Tree, Random Forest and TensorFlow classifiers are not far behind. The models can be tweaked further to improve the accuracy and other scores of the models. Adding extra hidden layers in the deep neural net could be beneficial. In particular, the random forest estimators can be increased to get higher accuracy. Running recursive estimation could help identify even better features to feed to the model. Similarly, the figure 8 below shows the plot of all the 5G label classifiers and their performance. From the above figure it is observed that the AdaBoost classifier performs the best with the accuracy peaking at around 53 percent. The overall performance of all the other scores of the classifier are also relatively good. This means that adopting the AdaBoost classifier gives the highest chance in predicting the output of the type of 5G Latency Label. The other classifiers are not far behind. The models can be tweaked further to improve the accuracy and other scores of the models. Adding extra hidden layers in the deep neural net or increasing the nearest neighbors could be beneficial. In particular, the AdaBoost estimators can be increased to get higher accuracy. Running recursive estimation could help identify even better features to feed to the model. Similarly, the figure 14 below shows the plot of all the classifiers and their performance. The classifier selection process is important as each classifier algorithm behavior is different. A lot of real-world testing must be conducted before any other decisions are made. CONCLUSION The main goal this thesis implementation was to understand the behavior of 5G latency with respect to an attack. This implementation started out with exploring and analyzing the initial dataset and its contents. A visual analysis of the DDoS attack types and the 5G components is displayed and described in brief. Some of the important features are also explored and visualized. It is observed that the flow duration, ACK flag count, total forward packets, flow packets per second, total backward packets, protocol, source, and destination ports are some of the main features. After visualizing the features, they are split into inputs and outputs for machine learning. Since the 5G labels and benign data was not equally balanced, the observations were oversampled to match the other majority classes. The data was then scaled and normalized to make it easier for the machine learning models to interpret. Finally, the data was split into training and testing. A total of eight machine learning models were applied and the best ones were visualized. Therefore, a machine The research was based on largely quantitative research methods in a simulated environment. Hence, the biggest limitation of this research was the lack of resources for data collection and sole reliance on online data sets. Ideally a V2X project would greatly benefit from an autonomous 5G enabled vehicle connected to a mobile edge cloud. However, this project was conducted solely online on a single PC which further limits the outcomes. 5G and cloud service providers are working hand in hand to try and make a resilient 5G vehicular network. One of the main challenges they face is in securing these networks. The aim of this thesis project is helping the infrastructure providers secure the 5G V2X system by applying machine learning for attack detection and prediction of 5G latency. This can be a steppingstone to unlock the full potential of an intelligent transport system that can prevent road accidents. Figure 1 :Figure 2 : 12Pie chart of DDoS attacks Source ports Figure 3 :Figure 4 : 34Destination 5G Latency Label (Good vs Bad) Figure 4 :Figure 4 : 44Oversampling DDoS labels results Oversampling 5G latency label results Figure 7 :Figure 8 : 78DDoS attack classifier comparison 5G Latency label classifier comparison learning model to correctly predict the impact of an attack on the 5G latency of a vehicle was developed. 5G IoT and the Future of Connected Vehicle. J Locke, accessed 04 NovemeberJ. Locke. "5G IoT and the Future of Connected Vehicle." DIGI. https://www.digi.com/blog/post/5g-iot-and- the-future-of-connected-vehicle (accessed 04 Novemeber, 2021). 5G enabled Mobile Edge Computing security for V2X. D Dcosta, unpublishedD. DCosta, "5G enabled Mobile Edge Computing security for V2X," unpublished, 2021. Multi-Access Edge Computing in Action. D Sabella, A Reznik, R Frazao, Taylor & Francis GroupMilton, UNITED KINGDOMD. Sabella, A. Reznik, and R. Frazao, Multi-Access Edge Computing in Action. Milton, UNITED KINGDOM: Taylor & Francis Group, 2019. Network function virtualization: Challenges and opportunities for innovations. B Han, V Gopalakrishnan, L Ji, S Lee, 10.1109/MCOM.2015.7045396IEEE communications magazine. 532B. Han, V. Gopalakrishnan, L. Ji, and S. Lee, "Network function virtualization: Challenges and opportunities for innovations," IEEE communications magazine, vol. 53, no. 2, pp. 90- 97, 2015, doi: 10.1109/MCOM.2015.7045396. Average Service Time Analysis of a Clustered VNF Chaining Scheme in NFV-Based V2X Networks. Y Han, X Tao, X Zhang, S Jia, 10.1109/ACCESS.2018.2882248IEEE Access. 6Y. Han, X. Tao, X. Zhang, and S. Jia, "Average Service Time Analysis of a Clustered VNF Chaining Scheme in NFV-Based V2X Networks," IEEE Access, vol. 6, pp. 73232-73244, 2018, doi: 10.1109/ACCESS.2018.2882248. Edge Computing-Based Privacy-Preserving Authentication Framework and Protocol for 5G-Enabled Vehicular Networks. J Zhang, H Zhong, J Cui, M Tian, Y Xu, L Liu, 10.1109/TVT.2020.2994144IEEE transactions on vehicular technology. 697J. Zhang, H. Zhong, J. Cui, M. Tian, Y. Xu, and L. Liu, "Edge Computing-Based Privacy-Preserving Authentication Framework and Protocol for 5G- Enabled Vehicular Networks," IEEE transactions on vehicular technology, vol. 69, no. 7, pp. 7940- 7954, 2020, doi: 10.1109/TVT.2020.2994144. Enabling Collaborative Edge Computing for Software Defined Vehicular Networks. K Wang, H Yin, W Quan, G Min, 10.1109/MNET.2018.1700364IEEE network. 325K. Wang, H. Yin, W. Quan, and G. Min, "Enabling Collaborative Edge Computing for Software Defined Vehicular Networks," IEEE network, vol. 32, no. 5, pp. 112-117, 2018, doi: 10.1109/MNET.2018.1700364. Security and Trust Issues in Internet of Things : Blockchain to the Rescue. S K Sharma, B Bhushan, B Unhelkar, Taylor & Francis GroupMilton, UNITED KINGDOMS. K. Sharma, B. Bhushan, and B. Unhelkar, Security and Trust Issues in Internet of Things : Blockchain to the Rescue. Milton, UNITED KINGDOM: Taylor & Francis Group, 2020. Deep Learning-Based Intrusion Detection for Distributed Denial of Service Attack. M A Ferrag, L Shu, H Djallel, K.-K R Choo, 10.3390/electronics10111257102021in Agriculture 4.0," ElectronicsM. A. Ferrag, L. Shu, H. Djallel, and K.-K. R. Choo, "Deep Learning-Based Intrusion Detection for Distributed Denial of Service Attack in Agriculture 4.0," Electronics, vol. 10, no. 11, 2021, doi: 10.3390/electronics10111257. Secure and Privacypreserving V2X multicast DNS. A Atif, J Arieltan, Glossary of Common DDoS AttacksA. Atif and J. Arieltan, "Secure and Privacypreserving V2X multicast DNS," ed, 2020. [11] allot. "Glossary of Common DDoS Attacks." allot. https://www.allot.com/ddos-attack- glossary/#snmp_reflected_amplification_attack (accessed 04 November, 2021). Developing Realistic Distributed Denial of Service (DDoS) Attack Dataset and Taxonomy. I Sharafaldin, A H Lashkari, S Hakak, A A Ghorbani, IEEEI. Sharafaldin, A. H. Lashkari, S. Hakak, and A. A. Ghorbani, "Developing Realistic Distributed Denial of Service (DDoS) Attack Dataset and Taxonomy," ed: IEEE, 2019, pp. 1-8. Analysis of 5G mobile technologies and DDOS defense. F Pathan, K Shringare, International Research Journal of Engineering and Technology. 66F. Pathan and K. Shringare, "Analysis of 5G mobile technologies and DDOS defense," International Research Journal of Engineering and Technology, 6 (6), pp. 2973-2979, 2019. Improving Imbalanced Land Cover Classification with K-Means SMOTE: Detecting and Oversampling Distinctive Minority Spectral Signatures. J Fonseca, G Douzas, F Bacao, 10.3390/info12070266Information (Basel). 1272021J. Fonseca, G. Douzas, and F. Bacao, "Improving Imbalanced Land Cover Classification with K- Means SMOTE: Detecting and Oversampling Distinctive Minority Spectral Signatures," Information (Basel), vol. 12, no. 7, p. 266, 2021, doi: 10.3390/info12070266.

Particles fullfill several distinct central roles in the Japanese language. They can mark arguments as well as adjuncts, can be functional or have semantic funtions. There is, however, no straightforward matching from particles to functions, as, e.g., ga can mark the subject, the object or an adjunct of a sentence. Particles can cooccur. Verbal arguments that could be identified by particles can be eliminated in the Japanese sentence. And finally, in spoken language particles are often omitted. A proper treatment of particles is thus necessary to make an analysis of Japanese sentences possible. Our treatment is based on an empirical investigation of 800 dialogues. We set up a type hierarchy of particles motivated by their subcategorizational and modificational behaviour. This type hierarchy is part of the Japanese syntax in VERBMOBIL.

cs/9906003

391b5d333a155dc83271454b1cdb317935eb92f4

THE SYNTACTIC PROCESSING OF PARTICLES IN JAPANESE SPOKEN LANGUAGE Melanie Siegel Department of Computational Linguistics University of the Saarland Postfach 151150 D-66041SaarbriickenGermany THE SYNTACTIC PROCESSING OF PARTICLES IN JAPANESE SPOKEN LANGUAGE Particles fullfill several distinct central roles in the Japanese language. They can mark arguments as well as adjuncts, can be functional or have semantic funtions. There is, however, no straightforward matching from particles to functions, as, e.g., ga can mark the subject, the object or an adjunct of a sentence. Particles can cooccur. Verbal arguments that could be identified by particles can be eliminated in the Japanese sentence. And finally, in spoken language particles are often omitted. A proper treatment of particles is thus necessary to make an analysis of Japanese sentences possible. Our treatment is based on an empirical investigation of 800 dialogues. We set up a type hierarchy of particles motivated by their subcategorizational and modificational behaviour. This type hierarchy is part of the Japanese syntax in VERBMOBIL. Introduction The treatment of particles is essential for the processing of the Japanese language for two reasons. The first reason is that these are the words that occur most frequently. The second reason is that particles have various central functions in the Japanese syntax: case particles mark subcategorized verbal arguments, postpositions mark adjuncts and have semantic attributes, topic particles mark topicalized phrases and no marks an attributive nominal adjunct. Their treatment is difficult for three reasons: 1) despite their central position in Japanese syntax the omission of particles occurs quite often in spoken language. 2) One particle can fulfill more than one function. 3) Particles can cooccur, but not in an arbitrary way. In order to set up a grammar that accounts for a larger amount of spoken language, a comprehensive investigation of Japanese particles is thus necessary. Such a comprehensive investigation of Japanese particles was missing up to now. Two kinds of solutions have previously been proposed: (1) the particles are divided into case particles and postpositions. The latter build the heads of their phrases, while the former do not (cf. [6], [12]). (2) All kinds of particles build the head of their phrases and have the same lexical structure (cf. [1]). Both kinds of analyses lead to problems: if postpositions are heads, while case particles are nonheads, a sufficient treatment of those cases where two or three particles occur sequentially is not possible, as we will show. If on the other hand there is no distinction of particles, it is not possible to encode their different behaviour in subcategorization and modification. We carried out an empirical investigation of cooccurrences of particles in Japanese spoken language. As a result, we could set up restrictions for 25 particles. We show that the problem is essentially based at the lexical level. Instead of assuming different phrase structure rules we state a type hierarchy of Japanese particles. This makes a uniform treatment of phrase structure as well as a differentiation of subcategorization patterns possible. We therefore adopt the 'all-head' analysis, but extend it by a type hierarchy in order to be able to differentiate between the particles. Our analysis is based on 800 Japanese dialogues of the VERBMOBIL data concerning appointment scheduling. 2 The Type Hierarchy of Japanese Particles Japanese noun phrases can be modified by more than one particle at a time. There are many examples in our data where two or three particles occur sequentially. On the one hand, this phenomenon must be accounted for in order to attain a correct processing of the data. On the other hand, the discrimination of particles is Particle case-particle complementizer modifying particle wa ga ni-case to noun-modifying particle verb-modif ing particle no topic-article adverbial particle postpositions wa ga-top mo kola ni-adv-p to-adv-p de Figure 1: Type Hierarchy of Japanese Particles. Postpositions include e, naNka, sonota, tomo, kara, made, soshite, nado, bakari, igai, yori, toshite, toshimashite, nitsuite, nikaNshite and nikakete motivated by their modificational and subcategorizational behaviour. We carried out an empirical analysis, based on our dialogue data. Table 1 shows the frequency of cooccurrence of two particles in the dialogue data. There is a tendency to avoid the cooccurrence of particles with the same phonology, even if it is possible in principal in some cases. The reason is obvious: such sentences are difficult to understand. [2] the part-of-speech class P contains only ga, wo and ni. [12] defines postpositions and case particles such that postpositions are the Japanese counterpart of prepositions in English and cannot stand independently, while case particles assign case and can follow postpositions. Her case particles include ga, wo, ni, no and wa. [7] divides case markers (ga, wo, ni and wa) from copula forms (ni, de, na and no). He argues that ni, de, na and no are the infinitive, gerund and adnominal forms of the copula. In the class of particles, we include case particles, complementizers, modifying particles and conjunctional particles. We thus assume a common class of the several kinds of particles introduced by the other authors. But they are further divided into subclasses, as can be seen in figure 1. We assume not only a differentiation between case particles and postpositions, but a finer graded distinction that includes different kinds of particles not mentioned by the other authors. de is assumed to be a particle and not a copula, as [7] proposes. It belongs to the class of adverbial particles. One major motivation for the type hierarchy is the observation we made of the cooccurrence of particles. Case particles (ga, wo, ni) are those that attach to verbal arguments. A complementizer marks complement sentences. Modifying particles attach to adjuncts. They are further divided into noun-modifying particles and verb-modifying particles. Verb modifying particles can be topic particles, adverbial particles, or postpositions. Some particles can have more than one function, as for example ni has the function of a case particle and an adverbial particle. Figure 1 shows the type hierarchy of Japanese particles. The next sections examine the individual types of particles. Case Particles There is no number nor gender agreement between noun phrase and verb. The verbs assign case to the noun phrases. This is marked by the case particles. Therefore these have a syntactic function, but not a semantic one. Unlike in English, the grammatical functions cannot be assigned through positions in the sentence or (2) naNji what time c-command-relations, since Japanese exhibits no fixed word position for verbal arguments. The assignment of the grammatical function is not expressed by the case particle alone but only in connection with the verbal valency. There are verbs that require ga-marked objects, while in most cases the ga-marked argument is the subject: (1) nantoka yotei somehow time ga toreru UA can take COP SAP §aAr. (Somehow (I) can find some time.) Japanese is described as a head-final language. [1] therefore assumes only one phrase structure rule: Mother Daughter Head. However, research literature questions whether this also applies to nominal phrases and their case particles. [9]:45 assume Japanese case particles to be markers. On the one hand, there are several reasons to distinguish case particles and modifying particles. On the other hand, I doubt whether it is reasonable to assume different phrase structures for NP+case particle and NP+modifying particle. The phrase-structural distinction of case particles and postpositions leads to problems, when more than one particle occur. The following example comes from the Verbmobil corpus: If one now assumes that the modifying particle kara is head of naNji as well as of the case particle ga, the result for naNji kara ga with the head-marker structure described in [9] 1 would be as shown in figure 2. The case particle ga would have to allow nouns and modifying particles in SPEC. The latter are however normally adjuncts that modify verbal projections. Therefore the head of kara entails the information that it can modify a verb. This information is inherited to the head of the whole phrase by the Head-Feature Principle as is to be seen in the tree above. As a result, this is also admitted as an adjunct to a verb, which leads to wrong analyses for sentences like the following one: desu ka COP QUE (3) *naNji kara ga what time from GA you 6aA j ikaN time UA toremasu ka can take QUE If, on the other hand, case particles and topic markers are heads, one receives a consistent and correct processing of this kind of example too. This is because the head information [MOD none] is given from the particle ga to the head of the phrase naNji kara ga. Thus this phrase is not admitted as an adjunct. Instead of assuming different phrase structure rules, a distinction of the kinds of particles can be based on lexical types. HPSG offers the possibility to define a common type and to set up specifications for the different types of particles. We assume Japanese to be head-final in this respect. All kinds of particles are analysed as heads of their phrases. The relation between case particle and nominal phrase is a 'Complement-Head' relation. The complement is obligatory and adjacent 2 . Normally the case particle ga marks the subject, the case particle wo the direct object and the case particle ni the indirect object. There are, however, many exceptions. We therefore use predicate-argument-structures instead of a direct assignment of grammatical Figure 2: NaNji kara ga with Head-Marker Structure functions by the particles (and possibly transformations). The valency information of the Japanese verbs does not only contain the syntactic category and the semantic restrictions of the subcategorized arguments, but also the case particles they must be annotated with3. In most cases the ga-marked noun phrase is the subject of the sentence. However, this is not always the case. Notably stative verbs subcategorize for ga-marked objects. An example is the stative verb dekimasu4: (4) kano jo she 6aA oyogi swimming aA dekimasu can (She can swim.) These and other cases are sometimes called 'double-subject constructions' in the literature. But these gamarked noun phrases do not behave like subjects. They are neither subject to restrictions on subject honorification nor subject to reflexive binding by the subject. This can be shown by the following example: (5\ gogo no hou yukkuri hanashi tla dekimasu ne ) afternoon NO side 6aA at ease talking can SAP (We can talk at ease in the afternoon.) hanashi does not meet the semantic restriction [+animate] stated by the verb dekimasu for its subject. There are even ga-marked adjuncts. [5] assumes these 'double-subject constructions' to be derived from genitive relations. But this analysis seems not to be true for example 5), because the following sentence is wrong: (6) *gogo no hou no yukkuri hanashiga dekimasu afternoon NO side NO at ease talk GA can The case particle wo normally marks the direct object of the sentence. In contrast to ga, no two phrases in one clause may be marked by wo. This restriction is called 'double-wo constraint' in research literature (see, for example, [12]:249ff.). Object positions with wo-marking as well as subject positions with ga-marking can be saturated only once. There are neither double subjects nor double objects. This restriction is also valid for indirect objects. Arguments found must be assigned a saturated status in the subcategorization frame, so that they cannot be saturated again (as in English). The verbs subcategorize for at most one subject, object and indirect object. Only one of these arguments may be marked by wo, while a subject and an object may both be marked by ga. These attributes are determined by the verbal valency. The wo-marked argument is not required to be adjacent to the verb. It is possible to reverse NP-ga and NP-wo as well as to insert adjuncts between the arguments and the verb. The particle ni can have the function of a case particle as well as that of an adjunct particle modifying the predicate. [10] also identify homophoneous ni that can mark adjuncts or complements. They use the notion of 'affectedness' to distinguish them. This is however not useful in our domain. [8] suggest testing the possibility of passivization. Some verbs subcategorize for a ni-marked object, as for example nark: (7\ raigetsu ni naru N desu / next month NI become COP SAP (It will be next month.) ni-marked objects cannot occur twice in the same clause, just as ga-marked subjects and wo-marked objects. The 'double-wo constraint' is neither a specific Japanese restriction nor a specific peculiarity of the Japanese direct object. It is based on the wrong assumption that grammatical functions are assigned by case particles. There are a lot of examples with double NP-ni, but these are adjuncts. The lexical entries of case particles get a case entry in the HEAD. Possible values are ga, wo, ni and to. They are neither adjuncts nor specifiers and thus get the entries [MOD none] and [SPEC none]. They subcategorize for an adjacent object. This can be a noun, a postposition or an adverbial particles. The Complementizer to to marks adjacent complement sentences that are subcategorized for by verbs like omon, in or kaku. 3 [8] investigates the particles ni, ga and wo and also states that grammatical functions must be clearly distinguished from surface cases 4 see [4] for a semantic classification of verbs that take ga-objects 5 A fundamental difference between Japanese grammar and English grammar is the fact that verbal arguments can be optional. For example, subjects and objects that refer to the speaker are omitted in most cases in spoken language. The verbal arguments can freely scramble. Additionally, there exist adjacent verbal arguments. To account for this, our subcategorization contains the attributes SAT and VAL. In SAT it is noted, whether a verbal argument is already saturated (such that it cannot be saturated again), optional or adjacent. VAL contains the agreement information for the verbal argument. Adjacency must be checked in every rule that combines heads and arguments or adjuncts. Some verbs subcategorize for a to marked object. This object can be optional or obligatory with verbs like kuraberu. (9) kono hi that day MO too chotto hito to somewhat people TO meet A meet plan V gozaimasu exist (That day too, there is a plan to meet some people.) to in these cases is categorized as a complementizer. Another possibility is that to marks an adjunct to a predicate, which qualifies to as a verb modifying particle: (10) shimizu seNsei to teNjikai WO go-issho sasete itadaku Shimizu Prof. TO exhibition WO together do HON (I would like to organize an exhibition with Prof. Shimizu.) Finally, the complementizer to can be an NP conjunction (which will not be considered at the moment, see [4]). The complementizer gets a case entry, because its head is a subtype of case-particle-head. It subcategorizes for a noun, a verb, an utterance, an adverbial particle or a postposition. Modifying Particles An essential problem is to find criteria for the distinction of case particles and modifying particles. On the semantic level they can be distinguished in that modifying particles introduce semantics, while case particles have a functional meaning. According to this, the particle no is a modifying one, because it introduces attributive meaning, as opposed to ( [12]:134), who classifies it as a case particle. Another distinctive criterion that is introduced by [12]:135 says that modifying particles 6 are obligatory in spoken language, while case particles can be omitted. Case particles are indeed suppressed more often, but there are also cases of suppressed modifying particles. These occur mainly in temporal expressions in our dialogue data: (11) Finally [12] gives the criterion that case particles can follow modifying particles while modifying particles cannot follow case particles. This criterion in particular implies that a finer distinction is necessary, as we have shown that it is not that easy. This can be realized with HPSG types. According to this criterion, no behaves like a modifying particle, while according to the criterion on meaning, it behaves like a case particle. Our first distinction is thus a functional one: modifying particles differ from case particles in that their marked entities are not subcategorized for by the verb. Case particles get the head information [CASE case] that controls agreement between verbs and their arguments. Modifying particles do not get this entry. They get the information in MOD that they can become adjuncts to verbs (verb modifying particles) or nouns (the noun modifying particle no) and semantic information. They subcategorize for a noun, as all particles do. The modifying particles share the following features in their lexical entries. Verb Modifying Particles The verb modifying particles specify the modification of the verb in MOD. The postpositions modify a (nonauxiliary) verb as an adjunct and subcategorize for a nominal object. [7] treats ni and de as the infinitive and the gerund form of the copula. ni is similar to the infinitive form to the extend that it can take an adverb as its argument (gogo wa furii ni nat-te i-masu -afternoon -WA -free -become). But the 6He calls them `postpositions'. infinitive is clearly distinct from the characteristics of ni, that cannot be used with N desu, cannot mark a relative sentence ( *John ga furii ni koto) and cannot be marked with the complementizer to ( *John ga furii ni to omou). The adjunctive form 'de' has both qualities of a gerundive copula and qualities of a particle. But there is some data that shows different behaviour of de and other gerundives. Firstly, it concerns the cooccurrence possibilities of de and other particles, compared to gerundive forms and particles: de wa -V-te wa de mo -V-te mo de no -V-te no de ni -*V-te ni de ga -*V-te ga de wo -*V-te wo de de -*V-te de Secondly, a gerund may modify auxiliaries, e.g. shite kudasai, shite orimasu, but de may not. Additionally there is something which distinguishes de of a copula: it may not subcategorize for a subject. A word that is an adjunct to verbs, subcategorizes for an unmarked noun or a postpositional phrase and is subcategorized for by several particles (see above) fits well into our description of a verb modifying particle. The adverbial particles ni, de and to subcategorize for a noun or a postposition. As already described, to behaves like an adverbial particle, too. The Noun Modifying Particle NO no is a particle that modifies nominal phrases. This is an attributive modification and has a wide range of meanings. ? [12]:134ff. assigns no to the class of case particles. However, the criteria she sets up to distinguish between case particles and postpositions do not apply to this classification of no: firstly, Tsujimura's postpositions have their own semantic meaning. Case particles have a functional meaning. no however has a semantic, namely attributive meaning. Secondly, Tsujimura's postpositions are obligatory in spoken language, case particles are optional. no is as obligatory as kara and made. Finally, Case particles can -as Tsujimura states -follow postpositions, but postpositions cannot follow case particles. According to this criterion, no behaves like a case particle. no combines qualities of case particles with those of modifying particles (which Tsujimura calls `postpositions'). This means that a special treatment of this particle is necessary. The particle no subcategorizes for a noun, as the other particles do. It also modifies a noun. This separates it from the other modifying particles. The particle no modifies a noun phrase and occurs after a noun or a verb modifying particle. Particles of Topicalization The topic particle wa can mark arguments as well as adjuncts. In the case of argument marking it replaces the case particle. In the case of adjunct marking it can replace the verb modifying particle or it can occur after it. On the syntactic level, it has to be decided, whether the topic particle marks an argument or an adjunct, when it occurs without a verb modifying particle. This is difficult because of the optionality of verbal arguments in Japanese. If it marks an argument, it has to be decided which grammatical function this argument has. This problem can often not be solved on the purely syntactic level. Semantic restrictions for verbal arguments are necessary: Subject and object of the verb shimashou are suppressed in this example. The sentence can be interpreted as having a topic adjunct, but no surface subject and object, when using semantic restrictions for the subject (agentive) and the object (situation). [2] analyses Japanese topicalization with a trace that introduces a value in SLASH and the 'Binding Feature Principle' that unifies the value of SLASH with a wa-marked element 8 . This treatment is similar to the one introduced by [9] for the treatment of English topicalization. However, Japanese topicalization is fundamentally different from English one. Firstly, it occurs more frequently. Up to 50% of the sentences ? See also [11] 8 The Binding Feature Principle says: The value of a binding feature of the mother is identical to the union of the values of the binding feature of the daughters minus the category bound in the branching. [ Figure 5: Topic Particle AVM are concerned ( [15]). Secondly, there are examples where the topic occurs in the middle of the sentence, unlike the English topics that occur sentence-initially. Thirdly, suppressing of verbal arguments in Japanese could be called more a rule than an exception in spoken language. The SLASH approach would introduce traces in almost every sentence. This, in connection with scrambling and suppressed particles, could not be restricted in a reasonable way. If one follows Gunji's interpretation of those cases, where the topic-NP can be interpreted as a noun modifying phrase, a genitive gap has to be assumed. But this leads to assuming a genitive gap for every NP that is not modified. Further, genitive modification can be iterated. Finally, two or three occurences of NP-wa are possible in one utterance. Thus, we decided to assign topicalized sentences the same syntactic structure as non-topicalized sentences and to resolve the problem on the lexical level. The topic particle is, on the syntactic level, interpreted as a verbal adjunct. The binding to verbal arguments is left to the semantic interpretation module in VERBMOBIL, see figure 5. mo is similar to wa in some aspects. It can mark a predicative adjunct and can follow de and ni. But it can also follow wa, an adjective and a sentence with question mark: mo is a particle that has the head of a topic-adverbial particle, but a different subcategorization frame than wa. koso is another topic particle that can occur after nouns, postpositions or adverbial particles. Omitted Particles Some particles can be omitted in Japanese spoken language. Here is an example from the Verbmobil corpus: This phenomenon can be found frequently in connection with pronouns and temporal expressions in the domain of appointment scheduling. [3] assumes that exclusively wa can be suppressed. [14] however shows that there are contexts, where ga, wo or even e can be omitted. He assigns it as 'phonological deletion'. [5] analyses omitted wo particles and explains these with linearization: a particle wo can only be omitted, when it occurs directly before a verb. [14] however gives examples to prove the opposite. It can be observed that NPs without particles can fulfill the functions of a verbal argument or of a verbal adjunct (ex. 14). We decided to interpret these NPs as verbal adjuncts and to leave the binding to argument positions to the semantic interpretation. NPs thus get a MOD value that allows them to modify nonauxiliary verbs. ga-Adjuncts One can find several examples with ga marked adjuncts in the Verbmobil data. On the level of information structure it is said that ga marks neutral descriptions or exhaustive descriptions (c.f. [1], [4]). Gunji analyses these exhaustive descriptions syntactically in the same way as he analyses his 'type-I topicalization'. They build adjuncts that control gaps or reflexives in the sentence. He views ga marked adjuncts without control relations as relying on a very specialized context. However, his treatment leads to problems. Firstly, in all cases, where ga marks a constituent that is subcategorized as ga-marked by the verb, a second reading is analysed that contains a ga marked adjunct controlling a gap. This is not reasonable. The treatment of the different meaning of ga marking and wa marking belongs to the semantics and not into the phrase structure. Secondly, this treatment assumes gaps. We already criticized this in connection with topicalization. Therefore, we do not need reflexive control at the moment. However, it contains mostly examples with ga marked adjuncts without syntactic control relation to the rest of the sentences. At the level of syntax, we do not decide whether a ga-marked subject or object is a neutral description or an exhaustive listing. This decision must be based on context information, where it can be ascertained whether the noun phrase is generic, anaphoric or new. We distinguish occurrences of NP+ga that are verbal arguments from those that are adjuncts. The examples for ga-marked adjuncts in the Verbmobi/ dialogues either describe a temporal entity or a human. All cases found are predicate modifying. To further restrict exhaustive interpretations, we introduced selectional restrictions for the marked NP, based on observations in the data. Conclusion The syntactic behaviour of Japanese particles has been analysed based on the Verbmobil dialogue data. We observed 25 different particles in 800 dialogues on appointment scheduling. It has been possible to set up a type hierarchy of Japanese particles. We have therefore adopted a lexical treatment instead of a syntactic treatment based on phrase structure. This is based on the different kinds of modification and subcategorization that occur with the particles. We analysed the Japanese particles according to their cooccurrence potential, their modificational behaviour and their occurrence in verbal arguments. We clarified the question which common characteristics and differences between the individual particles exist. A classification in categories was carried out. After that a model hierarchy could be set up for an HPSG grammar. The simple distinction into case particles and postpositions was proved to be insufficient. The assignment of the grammatical function is done by the verbal valency and not directly by the case particles. The topic particle is ambiguous. Its binding is done by ambiguity and underspecification in the lexicon and not by the Head-Filler Rule as in the HPSG for English ( [9]). what time would you like to start?) Figure 3 : 3.OBJ.LOCAL.CAT.HEAD noun or postposition or adv-p Head and Subcat of Case Particles Figure 4 : 4Head and Subcat of Modifying Particles the 13th of June suit you?) Table 1 : 1Cooccurrence of 2 Particles in the 800 Dialogues[4] treats wa, ga, wo, ni, de, to, made, kara and ya as 'particles'. They are divided into those that are in the deep structure and those that are introduced through transformations. An example for the former is kara, examples for the latter are ga(SBJ), wo(OBJ), ga(OBJ) and ni(0 BJ2).[1] assigns all particles the part-ofspeech P. Examples are ga, wo, ni, no, de, e, kara and made. All particles are heads of their phrases. Verbal arguments get a grammatical relation [GR OBJ/SBJ]. In 1 The Marking Principle says: In a headed phrase, the MARKING value is token-identical with that of the MARKER-DA UGHTER if any, and with that of the HEAD-DAUGHTER otherwisef9j.2 0bligatory Japanese arguments are always adjacent, and vice versa.[ HEAD 131 SUBCAT MARKING [1 jga HEAD [31 HEAD.SPEC [21 121 SUBCAT SUBCAT MARKING unmarked MARKING [1 jga COMPLEMENT HEAD ga [HEAD 141 151 SUBCAT dc, MARKING unmarked HEAD 131 SUBCAT <151> MARKING unmarked naNji kara SUBCAT VAL.OBJ.LOCAL.CAT.HEAD noun or vmod-p or comp or verb[te] or idiom -2] HEAD Pos HEAD MOD.LOCAL.CAT.HEAD nonaux_verb [[C +] SPEC none [ SAT.OBJ adjacent I don't know if I can)dekiru ka mo QUE MO shiremaseN do not know ( The approach presented here is part of the syntactic analysis of Japanese in the Verbmobi/ machine translation system. It is implemented in the PAGE parsing system[13]. It has been proved to be essential for the processing of a large amount of Japanese dialogue data. Further research concerning coordinating particles (to, ya, toka, yara, ka etc.) and sentence end particles (ka, node, yo, ne etc.) is necessary. Japanese Phrase Structure Grammar. Takao Gunji, Dordrecht: Reidel. Takao Gunji. Japanese Phrase Structure Grammar. Dordrecht: Reidel., 1987. An overview of JPSG: A constraint-based descriptive theory for Japanese. Takao Gunji, Proceedings of Japanese Syntactic Processing Workshop. Japanese Syntactic Processing WorkshopDuke UniversityTakao Gunji. An overview of JPSG: A constraint-based descriptive theory for Japanese. In Proceedings of Japanese Syntactic Processing Workshop. Duke University, 1991. Particle deletion in Japanese and Korean. John Hinds, Linguistic Inquiry. 84John Hinds. Particle deletion in Japanese and Korean. Linguistic Inquiry, 8(4):602-604, 1977. The Structure of Japanese Language. Susumo Kuno, MIT PressCambridge, MassSusumo Kuno. The Structure of Japanese Language. Cambridge, Mass.: MIT Press., 1973. Japanese Syntax and Semantics. Collected Papers. S.-Y Kuroda, Natural Language and Linguistic Theory. DordrechtKluwer Academic Publishers22S.-Y. Kuroda. Japanese Syntax and Semantics. Collected Papers., volume 22 of Studies in Natural Language and Linguistic Theory. Dordrecht: Kluwer Academic Publishers,_ 1992. Predication and numeral quantifiers. Shigeru Miyagawa, Papers from the Second International Workshop on Japanese Syntax. William J. PoserCSLIShigeru Miyagawa. Predication and numeral quantifiers. In William J. Poser, editor, Papers from the Second International Workshop on Japanese Syntax, pages 157-191. CSLI, 1986. An HPSG Account of the Japanese Copula and Related Phenomena. Stephen Nightingale, University of EdinburghPhD thesisStephen Nightingale. An HPSG Account of the Japanese Copula and Related Phenomena. PhD thesis, University of Edinburgh, 1996. Annularity in the distribution of the case particles ga. Kiyoharu Ono, Japanese. Theoretical Linguistics. 201Kiyoharu Ono. Annularity in the distribution of the case particles ga, o and ni in Japanese. Theoretical Linguistics, 20(1):71-93, 1994. Head-Driven Phrase Structure Grammar. C Pollard, I A Sag, University of Chicago PressChicagoC. Pollard and I.A. Sag. Head-Driven Phrase Structure Grammar. Chicago: University of Chicago Press., 1994. On the nature of the "dative. Kumi Sadakane, Masatoshi Koizumi, 33particle ni in Japanese. LinguisticsKumi Sadakane and Masatoshi Koizumi. On the nature of the "dative" particle ni in Japanese. Linguistics, 33:5-33, 1995. Semantics and pragmatics of adnominal particle no in Quixote. Hiroshi Tsuda, Yasunari Harada, Takao GunjiUniversality of Constraint-Based Structure Grammars. Osaka.Hiroshi Tsuda and Yasunari Harada. Semantics and pragmatics of adnominal particle no in Quixote. In Takao Gunji, editor, Studies in the Universality of Constraint-Based Structure Grammars. Osaka., 1996. An Introduction to Japanese Linguistics. Natsuko Tsujimura, Blackwell, CambridgeNatsuko Tsujimura. An Introduction to Japanese Linguistics. Blackwell, Cambridge, 1996. . Hans Uszkoreit, Rolf Backofen, Stephan Busemann, Abdel Kader Diagne, Elizabeth A Hinkelman, Walter Kasper, Bernd Kiefer, Hans-Ulrich Krieger, Klaus Netter, Gunter Neumann, Stephan Oepen, Stephen P , Hans Uszkoreit, Rolf Backofen, Stephan Busemann, Abdel Kader Diagne, Elizabeth A. Hinkelman, Walter Kasper, Bernd Kiefer, Hans-Ulrich Krieger, Klaus Netter, Gunter Neumann, Stephan Oepen, and Stephen P. DISCO-an HPSG-based NLP system and its application for appointment scheduling. Spackman, Proceedings of COLING-94. COLING-94Spackman. DISCO-an HPSG-based NLP system and its application for appointment scheduling. In Proceedings of COLING-94, pages 436-440, 1994. Scrambling and Japanese Phrase Structure. Shoichi Yatabe, Stanford University.PhD thesisShoichi Yatabe. Scrambling and Japanese Phrase Structure. PhD thesis, Stanford University., 1993. Tense and Aspect in Japanese and English. Kei Yoshimoto, Universitdt StuttgartPhD thesisKei Yoshimoto. Tense and Aspect in Japanese and English. PhD thesis, Universitdt Stuttgart, 1997.

We study the Gram matrix determinants for the groups S n , O n , B n , H n , for their free versions S + n , O + n , B + n , H + n , and for the half-liberated versions O * n , H * n . We first collect all the known computations of such determinants, along with complete and simplified proofs, and with generalizations where needed. We conjecture that all these determinants decompose as D = π ϕ(π), with product over all associated partitions.2000 Mathematics Subject Classification. 46L54 (15A52, 33C80).

10.1063/1.3511332

1009.4036

72e2989c4ec0a4c2e7d62679d536f9b73f60296c

DECOMPOSITION RESULTS FOR GRAM MATRIX DETERMINANTS 30 Sep 2010 Teodor Banica Stephen Curran DECOMPOSITION RESULTS FOR GRAM MATRIX DETERMINANTS 30 Sep 2010arXiv:1009.4036v2 [math.QA] We study the Gram matrix determinants for the groups S n , O n , B n , H n , for their free versions S + n , O + n , B + n , H + n , and for the half-liberated versions O * n , H * n . We first collect all the known computations of such determinants, along with complete and simplified proofs, and with generalizations where needed. We conjecture that all these determinants decompose as D = π ϕ(π), with product over all associated partitions.2000 Mathematics Subject Classification. 46L54 (15A52, 33C80). Introduction We discuss in this paper the computation of certain advanced representation theory invariants, for the main examples of "easy quantum groups". These are the groups S n , O n , B n , H n , their free versions S + n , O + n , B + n , H + n , and the half-liberated versions O * n , H * n . Here S n , O n are the permutation and orthogonal groups, B n is the bistochastic group consisting of orthogonal matrices whose rows and columns sum to 1, and H n = Z 2 ≀ S n is the hyperoctahedral group. For a global introduction to these groups and quantum groups, we refer to our previous papers [4], [5], [6]. According to a paper of Weingarten [35], further processed and generalized by Collins [12], then Collins andŚniady [14], a number of advanced representation theory invariants of the quantum group are encoded in a certain associated matrix G kn , called Gram matrix. For instance the inverse of the Gram matrix G kn is the Weingarten matrix W kn , whose knowledge allows the full computation of the Haar functional. See [35], [12], [14]. Among these invariants, the central object is the Gram matrix determinant, det(G kn ). For instance the roots of det(G kn ) are the poles of the Weingarten function W kn , and the knowledge of these numbers clarifies the invertibility assumptions in [4], [5], [6]. The quantity det(G kn ) appears in fact naturally in relation with many other questions, and its exact computation a well-known problem. A first purpose of the present work is to collect all the available formulae from the literature, and to write them down by using our unified "easy quantum group" formalism, along with complete, simplified proofs. The basic example of such a formula is that for S n , H n , H * n . Here the Gram matrix is upper triangular, up to a simple determinant-preserving operation. The determinant, computed by Jackson [23] and Lindström [29], is as follows: det(G kn ) = π∈P(k) n! (n − |π|)! Here P(k) is the set of partitions associated to the quantum group, namely all partitions for S n , all partitions with even blocks for H n , and all partitions with blocks having the same number of odd and even legs for H * n , and |.| is the number of blocks. In the general case the situation is much more complicated. However, one may still wonder for a general decomposition result, of the following type: det(G kn ) = π∈P(k) ϕ(π) This question is of course quite vague, depending on how explicit we would like our functions ϕ to be. For instance a natural requirement would be that in the case of a liberation G n → G + n , the functions ϕ are related by an induction/restriction procedure. This kind of specialized question appears to be quite difficult. In this paper we will obtain some preliminary decomposition results of the above type, as follows: (1) For O n , B n , O * n a formula comes from the work of Collins-Matsumoto [13] and Zinn-Justin [38]. The natural decomposition here is over Young diagrams, and in principle one can pass to partitions by applying a certain surjective map. (2) For O + n , B + n , S + n , H + n we use the work of Di Francesco, Golinelli and Guitter [17], [18], [19], [20], Tutte [33] and Dahab [16]. We will obtain some evidence towards the existence of contributions ϕ(π), of "trigonometric" nature. We will make as well a number of speculations in relation with quantum group/planar algebra methods, and with spectral measure/orthogonal polynomial interpretations. As a conclusion, there is still a lot of work to be done, mostly towards the conceptual understanding, at the level of Gram determinants, of the operation G n → G + n . The paper is organized as follows: 1-2 are preliminary sections, in 3-4 we discuss the classical and half-liberated cases, and in 5-6 we discuss the free case. The final sections, 7-9, contain a number of speculations on the formulae, and a few concluding remarks. Acknowledgements. This work was started at the Bedlewo 2009 workshop "Noncommutative harmonic analysis", and we are highly grateful to M. Bożejko for the invitation, and for several stimulating discussions. S.C. would like to thank the Paul Sabatier University and the Cergy-Pontoise University, where another part of this work was done. The work of T.B. was supported by the ANR grants "Galoisint" and "Granma", and the work of S.C. was partially supported by an NSF Postdoctoral Fellowship. Easy quantum groups Let P s be the category of all partitions. That is, P s (k, l) is the set of partitions between an upper row of k points and a lower row of l points, and the categorical operations are the horizontal and vertical concatenation, and the upside-down turning. A category of partitions P ⊂ P s is by definition a collection of sets P(k, l) ⊂ P s (k, l), which is stable under the categorical operations. We have the following examples. [36], [37] to the easy quantum group G × = (G × n ), with the notations in [4], [7]. We use the notation P(k) = P(0, k). We denote by ∨ and ∧ the set-theoretic sup and inf of partitions, always taken with respect to P s , and by |.| the number of blocks. Definition 1.2. Associated to any category of partitions P and to any numbers k, n ≥ 0 are the following matrices, with entries indexed by π, σ ∈ P(k): (1) Gram matrix: G kn (π, σ) = n |π∨σ| . (2) Weingarten matrix: W kn = G −1 kn . In order for G kn to be invertible, n must be big enough, and n ≥ k is known to be sufficient. The precise bounds depend on the category of partitions, and can be deduced from the various explicit formulae of det(G kn ), to be given later on in this paper. The interest in the above matrices comes from the fact that in the case P = P × g , they describe the integration over the corresponding easy quantum group G × n . Theorem 1.3. We have the Weingarten formula G × n u i 1 j 1 . . . u i k j k du = π,σ∈P × g (k) δ π (i)δ σ (j)W kn (π, σ) where the δ symbols are 0 or 1, depending on whether the indices fit or not. Proof. This follows by using a classical argument from [35], [14]. See [4], [7]. The exact computation of the Weingarten matrix is a quite subtle problem. A precise result is available only in the finite group case, where the formula is given in terms of the Möbius function µ on P as follows. Proposition 1.4. For S n , H n the Weingarten function is given by W kn (π, σ) = τ ≤π∧σ µ(τ, π)µ(τ, σ) (n − |τ |)! n! and satisfies W kn (π, σ) = n −|π∧σ| (µ(π ∧ σ, π)µ(π ∧ σ, σ) + O(n −1 )). Proof. The first assertion follows from the Weingarten formula: in that formula the integrals on the left are known, and this allows the computation of the right term, via the Möbius inversion formula. The second assertion follows from the first one. In the general case we have the following result, which is useful for applications. Proposition 1.5. For π ≤ σ we have the estimate W kn (π, σ) = n −|π| (µ(π, σ) + O(n −1 )) and for π, σ arbitrary we have W kn (π, σ) = O(n |π∨σ|−|π|−|σ| ). Proof. Once again this follows by using a classical argument, see [4]. Gram determinants In this paper, we will be mainly interested in the computation of det(G kn ). Let us being with some simple observations, coming from definitions. (1) D k is monic, of degree s k = π∈P(k) |π|. (2) We have n b k |D k , where b k = #P(k). Proof. (1) This follows from |π ∨ σ| ≤ |π|, with equality if and only if σ ≤ π. Indeed, from the inequality we get deg(D k ) ≤ s k . Now the coefficient of n s k is the signed number of permutations f : P(k) → P(k) satisfying f (π) ≤ π for any π, and since there is only one such permutation, namely the identity, we obtain that this coefficient is 1. (2) This is clear from the definition of D k , and from |π ∨ σ| ≥ 1. The above result raises the question of computing the numbers b k = #P(k) and s k = π∈P(k) |π|. It is convenient here to introduce as well the related numbers m k = s k /b k and a k = 2s k − kb k = (2m k − k)b k , which will appear several times in what follows. Proposition 2.2. The numbers b k , s k , m k , a k are as follows: ( 1) O n , O * n , O + n : b 2l = (2l)!!, l!, 1 l+1 2l l , s 2l = lb 2l , m 2l = l, a 2l = 0. (2) S n : b k = Bell, s k = b k+1 − b k , m k = b k+1 b k − 1, a k = 2b k+1 − (k + 2)b k . (3) S + n : b k = 1 k+1 2k k , s k = 1 2 2k k , m k = k+1 2 , a k = b k . (4) H + n : b 2l = 1 2l+1 3l l , s 2l = 3l−1 l−1 , m 2l = 2l+1 3 , a 2l = −2 3l−1 l−2 . Proof. All these results are well-known. For the remaining quantum groups, namely B n , B + n , H n , H * n , the numbers b k , s k , m k , a k are given by quite complicated formulae. The best approach to their computation is via the trace of the Gram matrix, and its analytic interpretations. So, let us first reformulate Proposition 2.1, in the following way. Proposition 2.3. With D k (n) = det(G kn ) and T k (t) = T r(G kt ), we have: (1) D k (n) = n s k (1 + O(n −1 )) as n → ∞, where s k = T ′ k (1). (2) D k (n) = O(n b k ) as n → 0, where b k = T k (1). Proof. This is indeed just a reformulation of Proposition 2.1, using a variable t around 1. Note that in (2) we regard the variable n as a formal parameter, going to 0. The trace can be understood in terms of the associated Stirling numbers. Proposition 2.4. We have the formula T k (t) = k r=1 S kr t r where S kr = #{π ∈ P(k) : |π| = r} are the Stirling numbers. Proof. This is clear from definitions. Another interpretation of the trace, analytic this time, is as follows. Proposition 2.5. For any t ∈ (0, 1] we have the formula T k (t) = lim n→∞ G × n χ k t where χ t = [tn] i=1 u ii are the truncated characters of the quantum group. Proof. As explained in [7], [4], this follows from the Weingarten formula. In general, the Stirling numbers S kr and the trace T k (t) are given by quite complicated formulae, unless we are in the situation of one of the quantum groups in Proposition 2.2. Here these invariants are well-known in the O, S cases, and for H + we have: T 2l (t) = l r=1 1 r l − 1 r − 1 2l r − 1 t r See [1]. In general now, the conceptual result concerns the asymptotic measures of truncated characters, i.e. the probability measures µ t satisfying T k (t) = x k dµ t (x). Theorem 2.6. The asymptotic measures of truncated characters are as follows: (1) S n /S + n : Poisson/free Poisson. (4) B n /B + n : shifted Gaussian/shifted semicircular. (5) O * n /H * n : symmetrized Rayleigh/squeezed ∞-Bessel. Proof. The one-parameter measures in the statement are best found via a direct computation, by using classical and free cumulants. See [7], [4], [5]. The basic formula We discuss now the explicit computation of the Gram determinants. The basic formula here, coming from the work of Jackson [23] and Lindström [29], is as follows. Theorem 3.1. For S n , H n , H * n we have det(G kn ) = π∈P(k) n! (n − |π|)! where |.| is the number of blocks. Proof. We use the fact that the partitions have the property of forming semilattices under ∨. The proof uses the upper triangularization procedure in [29] together with the explicit knowledge of the Möbius function on P(k) as in [23]. Consider the following matrix, obtained by making determinant-preserving operations: G ′ kn (π, σ) = π≤τ µ(π, τ )n |τ ∨σ| It follows from the Möbius inversion formula that we have: G ′ kn (π, σ) = n(n − 1) . . . (n − |σ| + 1) if π ≤ σ 0 if not Thus the matrix is upper triangular, and by computing the product on the diagonal we obtain the formula in the statement. A first remarkable feature of the above result is that the determinant for S n , H n , H * n can be computed from the trace: indeed, the Gram trace gives the Stirling numbers, which in turn give the Gram determinant. However, the connecting formula is quite complicated, so let us just record here an improvement of the first estimate in Proposition 2.3. Proposition 3.2. With D k (n) = det(G kn ) and T k (t) = T r(G kt ) we have D k (n) = n s k 1 − z k 2 n −1 + O(n −2 ) where s k = T ′ k (1) and z k = T ′′ k (1). Proof. In terms of Stirling numbers, the formula in Theorem 3.1 reads: D k (n) = k r=1 n! (n − r)! S kr We use now the following basic estimate: n! (n − r)! = n r r−1 s=1 1 − s n = n r 1 − r(r − 1) 2 n −1 + O(n −2 ) Together with T k (t) = k r=1 S kr t r , this gives the result. The above discussion raises the general question on whether the Gram determinant can be computed or not from the Gram trace, or from the measures in Theorem 2.6. Since the connecting formula for S n , H n , H * n is already quite complicated, let us formulate for the moment a more modest conjecture, as follows. with the "contributions" being given by an explicit function ϕ : P(k) → Q(n). This statement is of course quite vague, depending of the meaning of the above word "explicit". As already mentioned, one would expect ϕ to come from the Gram trace, or from the Stirling numbers, or, even better, from the measures in Theorem 2.6. Such a decomposition could potentially clarify the behavior of the Gram determinants under the "liberation" procedure G → G + . This kind of general question appears to be quite difficult. In what follows we will obtain some evidence towards such general decomposition results. The orthogonal case We discuss now the cases O, B, O * . Here the combinatorics is that of the Young diagrams. We denote by |.| the number of boxes, and we use quantity f λ , which gives the number of standard Young tableaux of shape λ. Theorem 4.1. For O n we have det(G kn ) = |λ|=k/2 f n (λ) f 2λ where f n (λ) = (i,j)∈λ (n + 2j − i − 1). Proof. This follows from the results of Collins and Matsumoto [13] and Zinn-Justin [38]. Indeed, it is known from [38] that the Gram matrix is diagonalizable, as follows: G kn = |λ|=k/2 f n (λ)P 2λ Here 1 = ΣP 2λ is the standard partition of unity associated to the Young diagrams having k/2 boxes, and the coefficients f n (λ) are those in the statement. Now since we have T r(P 2λ ) = f 2λ , this gives the result. Theorem 4.2. For B n we have det(G kn ) = n a k |λ|≤k/2 f n (λ) ( k 2|λ| )f 2λ where a k = π∈P(k) (2|π| − k), and f n (λ) = (i,j)∈λ (n + 2j − i − 2). Proof. We recall from [7] that we have an isomorphism B n ≃ O n−1 , given by u = v + 1, where u, v are the fundamental representations of B n , O n−1 . We get: F ix(u ⊗k ) = F ix (v + 1) ⊗k = F ix k r=0 k r v ⊗r Now if we denote by det ′ , f ′ the objects in Theorem 4.1, we obtain: det(G kn ) = n a k k r=1 det ′ (G r,n−1 ) ( k r ) = n a k k r=1   |λ|=r/2 f ′ n−1 (λ) f 2λ   ( k r ) This gives the formula in the statement. Theorem 4.3. For O * n we have det(G kn ) = |λ|=k/2 f n (λ) f λ 2 where f n (λ) = (i,j)∈λ (n + j − i). Proof. We use the isomorphism of projective versions P O * n = P U n , established in [8]. This isomorphism shows that the Gram matrices for O * n are the same as those for U n . But for U n it is known from [38] that the Gram matrix is diagonalizable, as follows: G kn = |λ|=k/2 f n (λ)P λ Here 1 = ΣP λ is the standard partition of unity associated to the Young diagrams having k/2 boxes, and the coefficients f n (λ) are those in the statement. Now since we have T r(P λ ) = f λ 2 , this gives the result. Observe that the above results provide a kind of answer to Conjecture 3.3, but with the Young diagrams contributing to the determinant, instead of the partitions. The remaining problems are to find the relevant surjective map from diagrams to partitions, and to see if the above formulae further simplify by using this surjective map. Meander determinants In this section we discuss the computation of the Gram matrix determinant, in the free cases O + n , B + n , S + n , H + n . Let P r be the Chebycheff polynomials, given by P 0 = 1, P 1 = n and P r+1 = nP r − P r−1 . Consider also the following numbers, depending on k, r ∈ Z: f kr = 2k k − r − 2k k − r − 1 We set f kr = 0 for k / ∈ Z. The following key result was proved in [19]. Theorem 5.1. For O + n we have det(G kn ) = [k/2] r=1 P r (n) d k/2,r where d kr = f kr − f k+1,r . Proof. As already mentioned, the result is from [19]. We present below a short proof. The result holds when k is odd, all the exponents being 0, so we assume that k is even. Step 1. We use a general formula of type G kn (π, σ) =< f π , f σ >. Let Γ be a locally finite bipartite graph, with distinguished vertex 0 and adjacency matrix A, and let µ be an eigenvector of A, with eigenvalue n. Let L k be the set of length k loops l = l 1 . . . l k based at 0, and H k = span(L k ). For π ∈ P o + (k) define f π ∈ H k by: f π = l∈L k i∼πj δ(l i , l o j )γ(l i ) l Here e → e o is the edge reversing, and the "spin factor" is γ = µ(t)/µ(s), where s, t are the source and target of the edges. The point is that we have G kn (π, σ) =< f π , f σ >. We refer to [30], [24], [21] for full details regarding this formula. Step 2. With a suitable choice of (Γ, µ), we obtain a fomula of type G kn = T kn T t kn . Indeed, let us choose Γ = N to be the Cayley graph of O + n , and the eigenvector entries µ(r) to be the Chebycheff polynomials P r (n), i.e. the orthogonal polynomials for O + n . In this case, we have a bijection P o + (k) → L k , constructed as follows. For π ∈ P o + (k) and 0 ≤ i ≤ k we define h π (i) to be the number of 1 ≤ j ≤ i which are joined by π to a number strictly larger than i. We then define a loop l(π) = l(π) 1 . . . l(π) k , where l(π) i is the edge from h π (i − 1) to h π (i). Consider now the following matrix: T kn (π, σ) = i∼πj δ(l(σ) i , l(σ) o j )γ(l(σ) i ) We have f π = σ T kn (π, σ) · l(σ), so we obtain as desired G kn = T kn T t kn . Step 3. We show that, with suitable conventions, T kn is lower triangular. Indeed, consider the partial order on P o + (k) given by π ≤ σ if h π (i) ≤ h σ (i) for i = 1, . . . , k. Our claim is that σ ≤ π implies T kn (π, σ) = 0. Indeed, suppose that σ ≤ π, and let j be the least number with h σ (j) > h π (j). Note that we must have h σ (j − 1) = h π (j − 1) and h σ (j) = h π (j) + 2. It follows that we have i ∼ π j for some i < j. From the definitions of T kn and l(σ), if T kn (π, σ) = 0 then we must have h σ (i − 1) = h σ (j) = h π (j) + 2. But we also have h π (i − 1) = h π (j), so that h σ (i − 1) = h π (i − 1) + 2, which contradicts the minimality of j. Step 4. End of the proof, by computing the determinant of T kn . Since T kn is lower triangular we have: det(T kn ) = π T kn (π, π) = π i∼πj P h π(i) P h π(i)−1 = k/2 r=1 P e kr /2 r Here the exponents appearing on the right are by definition as follows: e kr = π i∼πj δ hπ(i),r − δ hπ(i),r+1 Our claim now, which finishes the proof, is that for 1 ≤ r ≤ k/2 we have: π i∼πj δ hπ(i)r = f k/2,r Indeed, note that the left term counts the number of times that the edge (r, r + 1) appears in all loops in L k . Define a shift operator S on the edges of Γ by S(s, t) = (s + 1, t + 1). Given a loop l = l 1 . . . l k and 1 ≤ s ≤ k with l s = (r, r + 1), define a path S r (l s ) . . . S r (l k )l o s−1 . . . l o 1 . Observe that this is a path in Γ from 2r to 0 whose first edge is (2r, 2r + 1) and first reaches r − 1 after k − s + 1 steps. Conversely, given a path f 1 . . . f k in Γ from 2r to 0 whose first edge is (2r, 2r + 1) and first reaches r − 1 after s steps, define a loop f o k . . . f o s S −r (f 1 ) . . . S −r (f s−1 ). Observe that this is a loop in Γ based at 0 whose k − s + 1 edge is (r, r + 1). These two operations are inverse to each other, so we have established a bijection between k-loops in Γ based at 0 whose s-th edge is (r, r + 1) and k-paths in Γ from 2r to 0 whose first edge is (2r, 2r + 1) and which first reach r − 1 after k − s + 1 steps. It follows that the left hand side is equal to the number of paths in Γ = N from 2r to 0 whose first edge is (2r, 2r + 1). By the usual reflection trick, this is the difference of binomials defining f k/2,r , and we are done. We use the notation a k = π∈P(k) (2|π| − k), which already appeared in section 2. Theorem 5.2. For B + n we have: det(G kn ) = n a k [k/2] r=1 P r (n − 1) [k/2] l=1 ( k 2l )dlr Proof. We have B + n ≃ O + n−1 , see e.g. [32], so we can use the same method as in the proof of Theorem 4.2. By using prime exponents for the various O + n -related objects, we get: det(G kn ) = n a k [k/2] l=1 det ′ (G 2l,n−1 ) ( k 2l ) = n a k [k/2] l=1 l r=1 P r (n − 1) d lr ( k 2l ) Together with Theorem 5.1, this gives the formula in the statement. Theorem 5.3. For S + n we have: det(G kn ) = ( √ n) a k k r=1 P r ( √ n) d kr Proof. Let π → π be the "cabling" operation, obtained by collapsing neighbors. According to the results of Kodiyalam-Sunder [27] and Chen-Przytycki [11], we have: |π ∨ σ| = k/2 + 2| π ∨ σ| − | π| − | σ| In terms of Gram matrices we get G kn = D kn G ′ det(G kn ) = ( √ n) a k [k/2] r=1 P r ( √ n) 2d ′ k/2,r with d ′ sr = f ′ sr − f ′ s,r+1 , where f ′ sr = 3s s−r − 3s s−r−1 for s ∈ Z, f ′ sr = 0 for s / ∈ Z. Proof. According to [2], the diagrams for H + n are the "cablings" of the Fuss-Catalan diagrams [10], so we can use the same method as in the previous proof. So, by using the above formula from [27], [11], we have G kn = D kn G ′ 2k, √ n D kn , where D kn = diag(n | π|/2−k/4 ), and where G ′ is the Gram determinant for the Fuss-Catalan algebra. But this latter determinant was computed by Di Francesco in [18], and this gives the result. Algebraic manipulations In this section we perform some algebraic manipulations on the formulae found in the previous sections. Consider the quantity a k = π∈P(k) (2|π| − k), which already appeared, several times. Then n a k is a true "contribution", in the sense of Conjecture 3.3. We will prove here that a n a k factor appears naturally, in all the 10 formulae of Gram determinants. This can be regarded as a piece of evidence towards Conjecture 3.3. In the classical and half-liberated cases there is no need for supplementary work in order to isolate this n a k factor, and the unified result is as follows. Theorem 6.1. In the classical and half-liberated cases, we have S n , H n , H * n : det(G kn ) = n a k π∈P(k) n k−2|π| n! (n − |π|)! O n : det(G kn ) = n a k |λ|=k/2 f n (λ) f 2λ B n : det(G kn ) = n a k |λ|≤k/2 f ′ n (λ) ( k 2|λ| )f 2λ O * n : det(G kn ) = n a k |λ|=k/2 f ′′ n (λ) f λ 2 where f • n (λ) = (i,j)∈λ (n + j − i + ϕ • ), with ϕ = j − 1, ϕ ′ = j − 2, ϕ ′′ = 0. Proof. This is a reformulation of the results in section 4, by using a k = 0 for O n , O * n . In order to process the formulae in section 5, we need the following technical result. Proposition 6.2. The Chebycheff polynomials P r have the following properties: (1) P r (n − 1) = Q r (n), with Q 0 = 1, Q 1 = n − 1 and Q r+1 = (n − 1)Q r − Q r−1 . (2) P 2l (n) = R 2l (n 2 ), with R 0 = 1, R 2 = n − 1 and R 2l+2 = (n − 2)R 2l − R 2l−2 . (3) P 2l+1 (n) = nR 2l+1 (n 2 ), with R 1 = 1, R 3 = n − 2 and R 2l+3 = (n − 2)R 2l+1 − R 2l−1 . (4) P 2l (n) = n −l S 2l (n 2 ) 1/2 , with S 0 = 1, S 2 = n(n − 1) 2 and so on. (5) P 2l+1 (n) = n −l S 2l+1 (n 2 ) 1/2 , with S 1 = n, S 3 = n 2 (n − 2) 2 and so on. Proof. This is routine. As pointed out in section 7 below, Q r are the orthogonal polynomials for B + n , and R 2l are the orthogonal polynomials for S + n . As for the polynomials S r , these are some technical objects, introduced in relation with the H + n computation. Theorem 6.3. In the free cases, we have O + n : det(G kn ) = n a k [k/2] r=1 P r (n) d 1 k/2,r B + n : det(G kn ) = n a k [k/2] r=1 Q r (n) [k/2] l=1 ( k 2l ) d 1 lr S + n : det(G kn ) = n a k k r=1 R r (n) d 1 kr H + n : det(G kn ) = n a k [k/2] r=1 S r (n) d 2 k/2,r where d i kr = f i kr − f i k,r+1 , with f i kr = (i+1)k k−r − (i+1)k k−r−1 for k ∈ Z, and f i kr = 0 for k / ∈ Z. Proof. The O + n formula is the one in Theorem 5.1, with a n a k = 1 factor inserted. The B + n formula is the one in Theorem 5.2, with P r (n − 1) replaced by Q r (n). For the S + n formula, we use Theorem 5.3. By replacing the Chebycheff polynomials P 2l , P 2l+1 by the polynomials R 2l , R 2l+1 from Proposition 6.2, we get: det(G kn ) = ( √ n) a k k r=1 P r ( √ n) d 1 kr = ( √ n) a k √ n [(k+1)/2] l=1 d 1 k,2l−1 k r=1 R r (n) d kr Now recall from Proposition 2.2 that a k = 1 k+1 2k k . On the other hand a direct computation gives [(k+1)/2] l=1 d 1 k,2l−1 = 1 k+1 2k k , so we get the formula in the statement. For the H + n formula we use a similar method. With k = 2l, Theorem 5.4 gives: det(G 2l,n ) = ( √ n) a 2l l r=1 P r ( √ n) 2d 2 lr = ( √ n) a 2l ( √ n) −2 l s=2 [s/2]d 2 ls l r=1 S r (n) d 2 lr Now recall from Proposition 2.2 that a 2l = −2 3l−1 l−2 . On the other hand a direct computation gives l s=2 [s/2]d 2 ls = 3l−1 l−2 , so we get the formula in the statement. As a conclusion, the formulae in Theorem 6.1 and Theorem 6.3 are an intermediate step towards a general decomposition result of type det(G kn ) = π∈P(k) ϕ(π). We will come back to the question of finding such a general decomposition result in section 8 below. Orthogonal polynomials We present here a speculation in the free case, in relation with orthogonal polynomials. As we will see, this speculation works for S + n , O + n , B + n , but doesn't work for H + n . Definition 7.1. The orthogonal polynomials for a real probability measure µ are the polynomials Q 0 , Q 1 , Q 2 , . . . satisfying the following conditions: (1) Q k (n) = n k + a 1 n k−1 + . . . + a k−1 n + a k , with a i ∈ R. (2) For any k = l we have Q k (n)Q l (n) dµ(n) = 0. The orthogonal polynomials can be constructed by using a recursive formula, of type Q k+1 = (n − α k )Q k − β k Q k−1 . Here the parameters α k , β k ∈ R are uniquely determined by the linear equations coming from the fact that Q k+1 must be orthogonal to n k−1 , n k . More precisely, by solving these two equations we obtain the following formulae, where the integral sign denotes the integration with respect to µ: α k = n k+1 Q k n k Q k − n k Q k−1 n k−1 Q k−1 , β k = n k Q k n k−1 Q k−1 The numbers α k , β k are called Jacobi parameters of the sequence {Q k }. Since Q 0 = 1, in order to describe {Q k } we just need to specify Q 1 , and the Jacobi parameters. The orthogonal polynomials for an easy quantum group are by definition those for the asymptotic measure of the main character, given in Theorem 2.6. Proposition 7.2. The basic orthogonal polynomials are as follows: (1) O n : here Q 1 = n and Q k+1 = nQ k − kQ k−1 . (2) B n : here Q 1 = n − 1 and Q k+1 = (n − 1)P k − kQ k−1 . (3) O * n : here Q 1 = n and Q k+1 = nQ k − [(k + 1)/2]Q k−1 . (4) S n : here Q 1 = n − 1 and Q k+1 = (n − k − 1)Q k − kQ k−1 . (5) O + n : here Q 1 = n and Q k+1 = nQ k − Q k−1 . (6) B + n : here Q 1 = n − 1 and Q k+1 = (n − 1)Q k − Q k−1 . (7) S + n : here Q 1 = n − 1 and Q k+1 = (n − 2)Q k − Q k−1 . Proof. This result is well-known, and easy to deduce from definitions. Note that all the polynomials in the above statement are versions of the polynomials appearing in (1,3,4,5), which are respectively the Hermite, Charlier and Chebycheff polynomials. Let us go back now to the considerations in section 6. The polynomials R 2l appearing in Proposition 6.2 are the orthogonal polynomials for S + n , and it is natural to call {R n |n ∈ N} the family of "extended orthogonal polynomials" for S + n . Proof. This follows from Theorem 6.3 and Proposition 7.2. Regarding now H + n , the combinatorics here is that of the Fuss-Catalan algebra [10], see [1], [2]. Since µ is symmetric, the orthogonal polynomials are given by Q 1 = n and Q k+1 = nQ k − β k Q k−1 , where β k = γ k /γ k−1 , with γ k = n k P k . The data is as follows: This suggests the following general formula: β k =            3(3k − 1)(3k + 2) 4(2k − 1)(2k + 1) (k even) 3(3k − 2)(3k + 1) 4(2k − 1)(2k + 1) (k odd) The problem can be probably investigated by using techniques from [22], [28], [31]. Our main problem is of course: what is the analogue of Theorem 7.3 for H + n ? Let us also mention that the computation of the orthogonal polynomials for H n , H * n looks like a quite difficult problem. Probably the good framework here is that of the quantum groups H (s) n from [4], because at s = 2, ∞ we have H n , H * n . We have as well the following question: is there a quantum group/planar algebra proof of Theorem 7.3, in the cases B + n , S + n ? For O + n this was done in Theorem 5.1. More manipulations We have seen in the previous section that the quantum group H + n is somehow of a more complicated nature than the other quantum groups under consideration. In this section we restrict attention to O + n , B + n , S + n , and we further rearrange the formulae in Theorem 6.3. The idea comes from the formula of O + n . Indeed, the numbers f kr for O + n count the P o + diagrams with 2r upper points and 2k lower points, with the property that each upper point is paired with a lower point. This kind of diagrams, called "epi" in the paper of Jones, Shlyakhtenko and Walker [26], have the following generalization. Definition 8.1. Let P be a category of partitions, and let 0 ≤ r ≤ k. (1) We let P r (k) be the set of partitions σ ∈ P(r, k), with 0 ≤ r ≤ k, such that each upper point is connected to lower points only, and to at least one of them. (2) The elements of P r (k) are called "epi". We let P + (k) = ∪ k r=0 P r (k). For an epi σ ∈ P r (k), we denote by r(σ) = r the number of its upper legs. With these notations, we can now state and prove our main result. This is a global formula for the Gram determinants associated to the quantum groups O + n , B + n , S + n . F r(σ) F r(σ)−1 where F r = P r/2 , Q r/2 , R r are the corresponding extended orthogonal polynomials. Proof. Observe first that the F r quantities in the statement make indeed sense. This is because the epi for O + n , B + n must have an even number of upper legs. (1) For O + n we have f 1 sr = #P 2r (2s), so the formula in Theorem 6.3 becomes: det(G kn ) = n a k [k/2] r=0 P r (n) f k/2,r −f k/2,r+1 = n a k [k/2] r=0 P r (n) #P 2r (k)−#P 2r+2 (k) Now since we have P + (k) = ∪ [k/2] r=0 P 2r (k), we obtain the formula in the statement: det(G kn ) = n a k [k/2] r=0   σ∈P 2r (k) Q r (n) P r−1 (n)   = n a k σ∈P + (k) P r(σ)/2 (n) P r(σ)/2−1 (n) (2) For B + n the epi have, according to our definitions, singletons only in the lower row. Thus these epi can be counted as function of those for O + n , and we get: det(G kn ) = n a k A similar manipulation as in (1) gives now the formula in the statement. (3) For S + n the epi are in standard bijection (via fatenning/collapsing of neighbors) with the epi for O + n . Thus the formula in Theorem 6.3 becomes: det(G kn ) = n a k k r=0 R r (n) d 1 kr = n a k k r=0 R r (n) #P r (k)−#P 2r+1 (k) Once again, a similar manipulation as in (1) gives the formula in the statement. Observe that the quantum group H + n cannot be included into the above general theorem, and this for 2 reasons: first, because the orthogonal polynomial interpretation of the polynomials appearing in Theorem 6.3. fails, cf. the previous section, and second, because the epi interpretation of the exponents appearing in Theorem 6.3 seems to fail as well. Concluding remarks We have seen in this paper that the Gram matrix determinants have a natural interpretation in the easy quantum group framework, developed in [7], [4], [5], [6]. The known computations, that we partly extended, simplified, or rearranged in this paper, provide a complete set of formulae for the main examples of easy quantum groups. Our conjecture is that these Gram determinants should have general decompositions of type det(G kn ) = π∈P(k) ϕ(π). More precisely, the situation here is as follows: (1) For S n , H n , H * n the conjecture holds, with ϕ(π) = n!/(n − |π|)!. (2) For O n , B n , O * n we have a decomposition result, but over Young diagrams. (3) For O + n , B + n , S + n we have a decomposition result, but over the associated epi. The remaining problem is to find the correct surjective maps for (2,3), i.e. the correct surjections from diagrams/epi to partitions. Of course, this question is not very clearly formulated. The main problem is probably to understand the behavior of the Gram matrix determinants in relation with the liberation operation G n → G + n . Indeed, we expect in this situation the contributions ϕ to be related by a kind of induction/restriction procedure. In addition to the concrete computations performed in this paper, let us mention that there are as well some quite heavy, abstract methods, that we haven't really tried yet. First, the inclusion G n ⊂ G + n gives rise to a planar algebra module in the sense of Jones [25], and our above "liberation conjecture" can be understood as saying that the Gram matrix combinatorics behaves well with respect to this planar module structure. And second, modulo the orthogonal polynomial issues discussed in the previous section, some useful tools should come from the analytic theory of the Bercovici-Pata bijection [9]. Proposition 2 . 1 . 21Let D k (n) = det(G kn ), viewed as element of Z[n]. ( 2 ) 2O n /O + n : Gaussian/semicircular. (3) H n /H + n : Bessel/free Bessel. Conjecture 3 . 3 . 33For any easy quantum group we have a formula of type det(G kn ) = n D kn , where D kn = diag(n | π|/2−k/4 ), and where G ′ is the Gram matrix for O + n , so the result follows from Theorem 5.1. Theorem 5.4. For H + n we have the formula Theorem 7 . 3 . 73In the O + n , B + n , S + n cases we have a formula of typedet(G kn ) = n a k k r=1 Q r (n) d kr with d kr ∈ N,where Q r (n) are the corresponding extended orthogonal polynomials. P * h : partitions with blocks having the same number of odd and even legs. Proof. This is clear from definitions. Note that P × g corresponds via Tannakian dualityProposition 1.1. The following are categories of partitions: (1) P o /P + o : all pairings/all noncrossing pairings. (2) P * o : pairings with each string having an odd leg and an even leg. (3) P b /P + b : singletons plus pairings/noncrossing pairings. (4) P s /P + s : all partitions/all noncrossing partitions. 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[email protected] S.C.: Department of Mathematics, University of [email protected].: Department of Mathematics, Cergy-Pontoise University, 95000 Cergy-Pontoise, France. [email protected] S.C.: Department of Mathematics, University of California, Los Angeles, CA 90095, USA. [email protected]

We show that a quantum state transfer, previously studied as a continuous time process in networks of interacting spins, can be achieved within the model of discrete time quantum walks with position dependent coin. We argue that due to additional degrees of freedom, discrete time quantum walks allow to observe effects which cannot be observed in the corresponding continuous time case. First, we study a discrete time version of the engineered coupling protocol due to Christandl et. al. [Phys. Rev. Lett. 92, 187902 (2004)] and then discuss the general idea of conversion between continuous time quantum walks and discrete time quantum walks. arXiv:1103.4185v1 [quant-ph]

10.1103/physreva.83.062315

1103.4185

b8b445ed54156a8343ed695122662805ab79a5ea

Discrete Time Quantum Walk Approach to State Transfer Pawe L Kurzyński Centre for Quantum Technologies National University of Singapore 3 Science Drive 2117543SingaporeSingapore Faculty of Physics Adam Mickiewicz University Umultowska 8561-614PoznańPoland Antoni Wójcik Faculty of Physics Adam Mickiewicz University Umultowska 8561-614PoznańPoland Discrete Time Quantum Walk Approach to State Transfer (Dated: January 15, 2013) We show that a quantum state transfer, previously studied as a continuous time process in networks of interacting spins, can be achieved within the model of discrete time quantum walks with position dependent coin. We argue that due to additional degrees of freedom, discrete time quantum walks allow to observe effects which cannot be observed in the corresponding continuous time case. First, we study a discrete time version of the engineered coupling protocol due to Christandl et. al. [Phys. Rev. Lett. 92, 187902 (2004)] and then discuss the general idea of conversion between continuous time quantum walks and discrete time quantum walks. arXiv:1103.4185v1 [quant-ph] INTRODUCTION Lattice models appear in quantum theory in various contexts. Just like most theoretical models in physics, they were first used to describe the properties of natural materials, however nowadays we are reaching the stage when we are no longer bounded by nature, since our ability to engineer our own new systems having desired physical properties is developing very rapidly. This allows testing of abstract theoretical lattice models in the laboratory with a help of new extraordinary systems like graphene [1], cold atoms in optical lattices [2], photons in arrays of linear optical elements [3], or compounds of thin layers of different materials known as heterostructures [4]. The new field of quantum information, which strongly relies on the capability of engineering and manipulating of quantum systems, has also rekindled interest in lattice models. In particular, it has been shown that lattice models offer the possibility of universal quantum computation [5,6]. In this work we consider two related fields of research: quantum walks on graphs [7,8] and quantum information transfer in spin systems [9,10]. In particular we are interested in the discrete time quantum walk realization of a quantum state transfer on a spin chain with position dependent couplings [11]. It is known that special cases of spin lattice models with one magnetic excitation can be interpreted as continuous time quantum walks [7]. Also, it has been shown that the relation between the two types of quantum walks is not trivial [12,13], because discrete time quantum walks require an additional degree of freedom known as a coin. We show for the first time how one can convert the position dependence of couplings into the position dependence of coins. We also argue that discrete time quantum walks provide a more general framework for the description of quantum diffusion on regular lattices, since due to the extra degree of freedom one gains more control over the walk than is allowed in a standard continuous time scenario. Moreover, the evolution of discrete time quantum walks is much easier to simulate on a classical computer. Our approach is based on the relation between discrete time quantum walks and the Dirac-like equation to which the underlying quantum walk is transformed in the continuous limit of both time and space [14][15][16]. BASIC CONCEPTS State transfer in spin networks Consider a network for which every vertex corresponds to a spin 1/2. We say that there is an edge between vertex i and j if the corresponding spins interact. Most authors choose to consider an XX model of spin-spin interactions [9,10], we follow them and set the Hamiltonian of the network to be of the form H = {i,j}∈E J ij σ i x σ j x + σ i y σ j y ,(1) where E denotes the set of edges and σ i x is a Pauli X matrix for the i'th spin. The above Hamiltonian conserves the total spin number along Z axis i σ i z , therefore one may restrict studies to a subspace with a fixed number of excitations, where by excitation we mean a spin pointing up along Z axis. In particular, we are interested in an evolution restricted to a subspace with no excitations, which is invariant under the evolution generated by the Hamiltonian, and to a subspace with only one excitation. In the second case, the single excitation can be considered as a single particle walking on an underlying network. Next, imagine that there are two marked spins, a sender and a receiver, and that the whole network is in a ground state, i.e. in a subspace with no excitations. Then, the sender prepares his spin in a superposition α| ↑ + β| ↓ , so that the whole network is in the state (α| ↑ + β| ↓ ) s ⊗ | ↓↓ . . . ↓ net ⊗ | ↓ r .(2) The goal is to employ the evolution generated by the Hamiltonian to transfer the quantum state from s to r in finite time T | ↓ s ⊗ | ↓↓ . . . ↓ net ⊗ (α| ↑ + β| ↓ ) r ,(3) which is equivalent to transporting the excitation from spin s to spin r. There are many protocols which allow for a perfect, or a nearly perfect state transfer over spin chains. The basic techniques are: engineered couplings, wave packet encoding, and active control (see [9,10]). Here we concentrate on engineered couplings. In particular, we will consider two protocols, namely the protocol of Christandl et. al. [11] and a weakly coupled spin protocol [17,18]. Continuous time quantum walks The state in continuous time quantum walks (CTQW) [7] is described by the position of a particle on a graph. The continuous evolution, which is governed by the Schrödinger equation, is determined by the Hamiltonian H which is an n × n Hermitian matrix proportional to the adjacency matrix of the underlying graph G which has n vertices H ij = 0, iff (i, j) ∈ E(G); = 0, else,(4) where E(G), as before, denotes the set of edges of G. The probability that at time t the walker, whose state is |ψ(t) = e −iHt |ψ(0) , is located at vertex x is P x (t) = | x|ψ(t) | 2 , where {|x } is the orthonormal basis spanning the space of vertices. For CTQW an a graph all of whose edges have equal weight, the corresponding Hamiltonian is simply the Laplacian of the underlying graph L = D − A, which is a diagonal matrix D minus the graph adjacency matrix A (whose entries A ij are either zero or one if vertices i and j are disconnected or connected, respectively). The diagonal entries D jj = d j denote the degree of vertex j. As a result, the sum of all elements in each column and each row of the Laplacian is zero. When the Laplacian plays the role of the Hamiltonian, the diagonal entries are important only in the case of irregular graphs. For regular graphs they are all equal, and since one can freely add or subtract from the Hamiltonian multiples of identity without changing the dynamics of the system, they can be neglected. In case of a general CTQW on a chain the only nonzero elements of the Hamiltonian are H j,j+1 , H j,j−1 and H jj . In case of a single excitation subspace of a spin network, the elements of the corresponding CTQW Hamiltonian are given exactly by the strengths of spin couplings H ij = J ij . Discrete time quantum walks In discrete time quantum walks (DTQW) [8] the state of the system |x, c is described by the position of a walker on a graph x and by the state of an auxiliary system c which determines the direction the walker is going to take in the next step. This auxiliary system is often referred to as the coin, which for a walk on a one-dimensional graph like a chain or cycle is simply a two-level system c =←, →. In such cases, one step of the evolution is given by a unitary operator U = SC, which is a product of the coin operator C|x, → = cos θ|x, → − sin θ|x, ← ,(5) C|x, ← = sin θ|x, → + cos θ|x, ← , and the conditional shift operator S|x, → = |x + 1, → ,(7)S|x, ← = |x − 1, ← .(8) Therefore one step is given by U |x, → = cos θ|x + 1, → − sin θ|x − 1, ← ,(9) U |x, ← = sin θ|x + 1, → + cos θ|x − 1, ← . (10) The parameter θ denotes the coin flip rate. Equivalently, the unitary evolution operator of one step can be written as U = e ipσz e iσyθ ,(11) where p is the momentum operator and σ j are Pauli matrices acting on the coin space. In DTQW on chains and even cycles, interference occurs only between positions separated by a distance of two (second nearest neighbors), therefore one can consider two independent walks: one on even vertices and another on odd vertices. Due to this fact even vertices can be treated as edges as in Fig. 1, which will be crucial for our work. Since we are interested in the walk on vertices, not on edges, in our case one step of evolution will be given by U 2 and the initial state will be always supported on the vertex space only, as in the Ref. [12]. Moreover, throughout this work we will consider DTQW on finite chains, which can be simulated by walks on cycles with a reversing coin at one vertex, i.e. a coin for which coin flip rate θ = π/2 -see Eqs. (5,6) and CONTINUOUS LIMIT OF QUANTUM WALKS The continuous limit of DTQW on a chain results in the one-dimensional Dirac equation [14][15][16]. The operator (11) can be rewritten as U = U (t) = e ipσzvt e iσyωt ,(12) where v is the velocity and ω is the angular frequency. The two parameters are chosen so that for a unit time step vt = 1 and ωt = θ, thus recovering Eq. (11). Taking infinitesimal time steps and applying the Trotter theorem one obtains lim n→∞ U (t/n) n = e i(pσzv+σyω)t .(13) The term in the bracket above corresponds to the Hamiltonian of the system, which has exactly the form of the one-dimensional Dirac Hamiltonian H = cpσ z + mc 2 σ y .(14) The correspondence between (13) and (14) is fully established when one identifies v with the speed of light and ω with the mass of the particle. In particular, ωt = θ, however we take t = 1, therefore the mass of the quantum walker corresponds to the coin flip rate θ. However, due to the discrete nature of time and the periodicity of coin operator e iσyθ , one is unable to differentiate between θ and θ + k2π. On the other hand, in the limit of infinite mass the particle should be almost immobile and the relation between the Dirac equation and DTQW requires θ → π 2 , hence one often takes m = tan θ (see [14]). Note that for small angles tan θ ≈ θ. It is also important to notice, that due to the structure of the conditional translation operator the continuization of time is intrinsically combined with the continuization of space, i.e. either both space and time are continuous, or both are discrete. The Hamiltonian of CTQW on a chain with uniform couplings is the following: H i,i+1 = H i+1,i = −J and H i,i = 2J, however for regular graphs diagonal elements can be neglected and put equal to zero. The action of the Hamiltonian on a state localized at x is H|x = −J (|x + 1 − 2|x + |x − 1 ) .(15) The above is a discrete version of the Laplacian. The continuization of the above leads to H = −J d 2 dx 2 .(16) Obviously, this Hamiltonian describes a free particle in one dimension with J = 1 2m , therefore the coupling constant J can be interpreted as an inverse of the walker's mass. In the continuous limit, the quantum walker can be considered as a relativistic (DTQW), or as a nonrelativistic free particle (CTQW). Main parameters governing the dynamics of the walk, namely coins and couplings, are related to the particle's mass. Strauch [12] showed that continuous quantum walk can be obtained from discrete one in the limit θ → π 2 . Heuristically, this limit corresponds to a very heavy particle, for which it is much harder to observe relativistic effects, therefore it corresponds to a relativistic to non-relativistic transition. Another important feature of quantum walks is that in both models space is discrete, but only for DTQW time is also discrete. Once again, a heuristic explanation of this fact can be given as follows. Let the two quantum walks be discrete versions of Dirac and Schrödinger free particle, respectively. In non-relativistic model time and space are treated separately, therefore there is nothing strange in discretizing only one of them. However, in relativistic physics time and space are combined and one has to discretize the whole spacetime at once, since it would be problematic to define consistent Lorentz transformations on spacetime which is only partially discrete. PERFECT STATE TRANSFER IN DTQW Let us apply the analogy between mass, coins and couplings to show that a perfect state transfer from one position to another can be realized within DTQW. We concentrate on the acclaimed perfect state transfer protocol introduced by Christandl et. al. [11]. This protocol relies on properly engineered couplings, i.e. properly chosen terms of the Hamiltonian governing the corresponding CTQW. These couplings depend on position, therefore by our analogy, the corresponding particle has position dependent mass. On the other hand, mass is related to coin flip rate and so the corresponding DTQW should have position dependent coin. Below, we examine how the choice of such coins affects the dynamics of DTQW. The chain coupling between nodes n and n + 1 in the protocol is given by J n = λ 2 n(N − n), where n = 1, 2, . . . , N − 1 and λ is a real parameter common for all couplings. As a result, the effective mass of the particle for the transfer between positions n and n + 1 is m n = 1 2J n = 1 λ n(N − n) .(17) Next, consider a step operator of the form U = U (n) = e ipσz e iσyθn ,(18) where θ n depends on position. The matrix form of the coin operator is given by C n = cos θ n sin θ n − sin θ n cos θ n .(19) Recall that we consider a double step operator U 2 , since our walk on a chain of length N/2 is simulated by a walk on an even N-cycle for which even positions correspond to edges and we are interested in a walk on positions corresponding to vertices. Coins associated with edge positions correspond to mass, but there are also coins associated with vertices -odd positions of the cycle. Here, we set all of them to be identities (θ 2k−1 = 0), although we note that a different choice (equal for all vertices) leads to similar results. The edge coins (even positions) are chosen according to the analogy between couplings and mass (m = tan θ) θ 2k = arctan 1 2λ k N 2 − k ,(20) where k = 1, 2, . . . , N/2 − 1. Our studies show that the transfer strongly depends on parameter λ. For small values of λ (large mass limit) the transfer is perfect. Around λ = π N it starts to drop down and it recovers back to perfect transfer for λ of the order O(1) and greater (small mass limit). We show that the nature of transport for the two limiting cases is drastically different. In the small mass limit perfect transfer occurs due to dispersion-less nature of the operator e ipσz whose spectrum is linear, whereas in the large mass limit the generator of the operator U 2 is effectively given by a Hermitian operator whose action is the same as the action of the Hamiltonian introduced by Christandl et. al. [11]. Interestingly, in both cases we found that the transfer fidelity does not depend on the initial coin state. Moreover, we checked numerically that a number of initial coin states, among which were eigenvectors of the three Pauli matrices, are always perfectly transfered. This fact allows us to conjecture that in DTQW case not only particle, but also its intrinsic coin state is perfectly transfered: α|1, → + β|1, ← → α|N − 1, → + β|N − 1, ← . In Figure 3 we present the transfer of particle from position 1 to position N −1 for the corresponding DTQW with respect to evolution operator U 2 . In this particular case N = 30 and λ = 0.03. As one can see transfer is perfect, and moreover the dynamics is periodic because after twice the transfer time the system goes back to the initial state, as in the continuous time case. The main idea of the authors of the protocol [11] was to find a mirror symmetric Hamiltonian with harmonic spectrum, which would provide periodicity of evolution and a perfect transfer between mirror symmetric positions for times equal to half of a period. Numerical simulations confirm that the spectrum of U 2 is also harmonic, yet it posses an additional relativistic property (see Fig. 4). The arguments of eigenvalues of U 2 can be interpreted as quasi-energies. They split into two bands, which can be related to positive and negative energy bands just like in the Dirac equation. Moreover, the two bands are separated by a band gap corresponding to the rest energy 2mc 2 . Due to the discrete nature of time for quasi-energies the band gap occurs twice, the first centered around zero and the second centered around π. Numerical simulations show that the width of the band gap is related to the inverse of λ: in the limit λ → 0 the band gap goes to π, and for λ → ∞ the band gap goes to zero. This is as expected, since the corresponding mass scales as 1 λ -see Eq. (17). Large mass limit Recall the formulae for position dependent coin (19) and coin flip rate (20) and recall that arctan θ can be expressed as arctan θ = arcsin θ √ 1 + θ 2 = arccos 1 √ 1 + θ 2 .(21) The entries of the coin matrix can be rewritten as cos θ 2k = 2λγ k 1 + 4λ 2 γ 2 k , sin θ 2k = 1 1 + 4λ 2 γ 2 k ,(22) where γ k = k N 2 − k . In the large mass limit λ → 0, which allows us to keep only the highest terms of Taylor expansion cos θ 2k ≈ 2λγ k , sin θ 2k ≈ 1.(23) The above limit resembles the one of Strauch [12] θ 2k → π 2 . The action of the double step operator U 2 on positions corresponding to vertices (odd positions) can be approximated by U 2 |2k − 1, → ≈ 2λγ k |2k + 1, → + |2k − 1, ← ,(24)U 2 |2k − 1, ← ≈ 2λγ k−1 |2k − 3, ← + |2k − 1, → .(25) Let us also introduce the following almost normalized states |ψ ± (k) = 1 √ 2 (|2k − 1, → ± i|2k − 1, ← ±i2λγ k |2k + 1, → − 2λγ k−1 |2k − 3, ← ) ,(26) and normalized states |φ ± (k) = 1 √ 2 (|2k − 1, → ± i|2k − 1, ← ) .(27) Note that |φ ± (k) and |ψ ± (k) are almost equal up to the factor of O(λ). The action of double step operator on |ψ ± (k) yields U 2 |ψ ± (k) = i (|φ ± (k) − i2λγ k |φ ± (k + 1) − i2λγ k−1 |φ ± (k − 1) ) + O(λ 2 ) (28) Now, it is enough to recognize that 2λγ k = J k,k+1 from the protocol [11] and that for short times the action of the unitary operator can be approximated by U ≈ 1 1 − iHt, where H is the Hamiltonian, therefore U 2 |ψ ± (k) ≈ i(1 1 − iH)|φ ± (k) .(29) The above Hamiltonian is exactly the Hamiltonian of Christandl et. al., therefore the CQTW behavior is restored. Moreover, the action of H does not mix ± states, which is the reason why we observe not only the transfer of the walker, but also the transfer of its intrinsic coin state. Small mass limit Let us consider λ → ∞. In this case a dispersionless movement dominates the evolution, since Eq. (22) is approximately cos θ 2k ≈ 1, sin θ 2k ≈ 0,(30) therefore the double step operator simplifies to U 2 = e −2ipσz , except at position N , where we apply the reversing coin which also causes a π phase shift of the particle coming from the left. The transfer occurs in N/2 double steps. The initial state α|1, → + β|1, ← evolves in the following way. The first part of this state, which is the right moving part, moves to the right, is reflected at position N and goes back to position N − 1 to finish in state −α|N − 1, ← . The second left moving part goes left to 0 ≡ N mod N , gets reflected and moves right to position N − 1 to finish in state β|N − 1, → . This also works for any initial state localized at x, for which the destination point is N − x. In order to restore the final coin state to the initial state one has to apply the σ y operation. Transition between the two types of transfer The unitary evolution operator U is the product of two operators S and C. Transfer properties rely on the interplay between this pair. In the small mass limit C does not play any significant role, since it is close to identity. On the other hand, in the large mass limit C is crucial, because it makes the evolution similar to the one of the protocol [11]. As we pointed out in the beginning of this section, we observed that the transition between the two types of behavior occurs around λ = π N . Here, we give the heuristic explanation for this value. The time of transfer over a chain of length N for the protocol [11] is T = π λ . In our case we consider chains which are effectively of length N/2, that is why we have to multiply λ by two, which is evident in Eq. (20). As a result, the corresponding time of transfer should be given by T = π 2λ and indeed this is approximately what we observe. Next, let us recall that DTQW is related to relativistic quantum mechanics via its similarity to the Dirac equation. In relativistic physics a distance traveled by a particle in time t can be at most ct. Similarly, in DTQW in one step the particle can move only to neighboring positions, that is why in order to travel the distance from one end of the chain to another the particle needs to take at least N/2 steps. This gives the critical value of λ cr = π N above which the particle would move with velocity greater than the speed of light. Since this is impossible, a relativistic behavior has to dominate the evolution and as a consequence the transition between the two types of behavior can be interpreted as a non-relativistic to relativistic transition. Other techniques for perfect transfer As mentioned earlier, nearly perfect transfer in spin networks can be obtained in many ways. In principle, all transfer techniques can be employed in DTQW. For example, wave packed encoding (see [9] and references therein) is straightforward since one can prepare the same spatial wave packet in DTQW as the one considered in CTQW. However, position dependent coupling method is the one for which the correspondence between DTQW and CTQW is the most nontrivial. Bellow we would like to mention another example based on our coin-masscoupling analogy. We present DTQW version of a protocol in which two spins are weakly coupled to ends of a spin chain with uniform couplings [17,18]. The CTQW Hamiltonian of a chain with weakly coupled ends is determined by coupling constants J i,i+1 = J for i = 2, . . . , N − 2 and J 1,2 = J N −1,N = aJ, where a 1. Due to weak coupling to the rest of the chain an interaction of the first and the last spin with the middle part can be treated as a perturbation. Without interaction the two spins would be in a degenerate state. Weak coupling breaks the degeneracy and perturbation theory predicts that there are new eigenstates which are even superpositions of two states: a state in which the first spin is up and one in which the last spin is up, or three states: the first spin up, the last spin up, or the middle of the chain being in one of its unperturbed eigenstates. The second case happens when the degenerated eigenvalue of the two spins is the same as one of the eigenvalues of the chain. Nearly perfect transfer is possible because initial and final states are almost completely supported on these two/three states only (for the detailed discussion see [17,18]). In DTQW the above protocol can be realized in the following way. As before, we can simulate N/2-chain on N -cycle with one reversing edge. All coins are the same θ k = θ, except θ 2 = θ N −2 = π 2 − ε. We observe that DTQW behaves exactly like the corresponding CTQW. Moreover, we also observe the transfer of the coin state. In Fig. 5 we show the eigenstate population for N = 30, θ = π 4 and initial state |1, → . In case ε 1 the initial state is almost entirely supported on four eigenstates (two positive energy eigenstates and two negative energy eigenstates). In Fig. 6 we present the population of positions 1 and N − 1 in time. The time of transfer scales as O 1 ε 2 . We also observe another interesting phenomenon. If one changes the coin operator at position 2 from e iσy θ 2 2 to e iσx θ 2 2 , the transfer is suppressed and the state is essentially localized at position 1 for all time. The reason for this behavior is that the initial energy at position 1 differs from the energy the particle would have at position N − 1, therefore the transport is forbidden due to energy conservation. In the next section we will explain this in further detail. HOW TO CONVERT BETWEEN CTQW AND DTQW? The conversion of CTQW into DTQW was considered by Childs [13], however his approach was purely mathematical and did not take into account the physical properties of the underlying quantum walks. On the other hand, our approach is based on physical aspects of quantum walks, and hence by studying the DTQW version of a continuous process one can learn many facts about the physics governing the corresponding system. In this section we show how to, in general, convert an arbitrary CTQW on a chain into DTQW. Once again, we are going to implement walk on a N/2-chain using N -cycle with vertex and edge positions. Recall that in general CTQW on a chain the only nonzero elements of the Hamiltonian are H j,j+1 , H j,j−1 and H jj . Diagonal terms, which are necessarily real, correspond to the potential energy u(j) = H jj .(31) Potential energy has been already considered in DTQW, mostly in the context of localization (see for example [19]). The kinetic energy for the transition between positions j and j + 1, which is proportional to the second power of momentum divided by the mass, corresponds to the real part of the off-diagonal terms H j,j+1 and H j+1,j . In this case Re[H j,j+1 ] = 1 2mj,j+1 . It was studied in the previous sections. The mass can be expressed as m j,j+1 = 1 2Re[H j,j+1 ] .(32) Finally, let us consider the imaginary part of H j,j+1 . First of all, note that operator proportional to −i|j j + 1| + i|j + 1 j|, which gives rise to nonzero imaginary part of H j,j+1 , can be approximated as a first power of momentum operator for positions j and j + 1 (see [20]). Indeed,   j −i|j j + 1| + i|j + 1 j|   2 = j 2|j j| − |j + 2 j| − |j j + 2|.(33) The above operator resembles the Laplacian, which is the kinetic energy operator proportional to the square of momentum. The first power of momentum can appear in the Hamiltonian if there is a vector potential A acting on the particle. In this case we have to substitute ( p) 2 → p − e c A 2 , where e is the charge of the particle. As before, in the following, we assume c = 1 and we also set e = 1. Using this analogy one arrives at Im[H j,j+1 ] = A j,j+1 m j,j+1 ,(34) where in general both, the vector potential and the mass, are position dependent. The vector potential alone can be estimated as follows A j,j+1 = 1 2 Im[H j,j+1 ] Re[H j,j+1 ] .(35) Our approach is based on the analogy between DTQW and the Dirac equation, whose general time-independent one-dimensional form is −i ∂ ∂x − A(x) σ z ψ + m(x)σ y ψ = i ∂ ∂t − u(x) ψ. (36) At this point we can go to DTQW via application of the Trotter formula. One step of the corresponding DTQW is given by U = SCV = e ipσz e i(m(j)σy−A(j)σz) e iu(j) ,(37) where j denotes position. To include the account of the scalar potential, we introduced a new operator V = e iu(j) . Moreover, the new coin operator C contains the vector potential term. It is worth noting that vector potentials allow us to implement Hadamard coins using relativistic analogy. Relativistic limit of Hadamard DTQW was considered in [16], where the mass term in the corresponding Dirac equation violated Lorentz covariance. Before we plug Eqs. (31), (32) and (35) into Eq. (37), let us emphasize that in some cases it might be convenient to consider inverse tangent of both potentials, like we did previously with the mass, because of the periodicity of the arguments in the exponents. Moreover, since we consider division of positions into edges and vertices, scalar potential terms should act on vertices, whereas mass and vector potential terms should act on edges. As a result the double step operator can be taken as A similar conversion can be done for higherdimensional quantum walks. Moreover, it is clear that using the above analogy one can convert the corresponding DTQW back into CTQW. However, there are certain kinds of DTQW which cannot be easily converted into CTQW, therefore let us now concentrate on the reverse problem of conversion of DTQW into CTQW. An example of a case in which the conversion is problematic is a DTQW with position dependent coin, where not only the coin flip ratio θ j depends on position, but also the generating operator σ j changes from position to position. In the previous section we considered DTQW version of a weakly coupled spin protocol and observed that the transfer can be suppressed due to a change of the first and the last coin flip generators from σ y to σ x . The coin degree of freedom is a qubit, therefore the coin generator is a linear combination of σ x , σ y and σ z . In the Bloch sphere picture, coin operation is a rotation of a qubit about an axis n, where the generator of rotation is n · σ. In general, n can point in any direction on a surface of a three-dimensional sphere, however in the case of the Dirac equation (36) it is bounded to the yz-plane. Due to this fact, simple conversion from DTQW into CTQW is not possible. As a result, DTQW offers a greater possibility of control over the transport than is allowed in the CTQW case. U 2 = SCSV = e ipσz e i Up to now we have considered only one-dimensional graphs, like chains. In two, and higher, dimensions one encounters even more complicated obstacles. Firstly, the typical formulation of a DTQW on a d-dimensional grid uses a 2d-dimensional coin. Each coin dimension corresponds to a different direction for the walker to take in the next step. This is in contrast to the Dirac formulation of relativistic motion, where for example in the three-dimensional case the coin space is only fourdimensional. The conditional shift operator of the corresponding DTQW is of the form Exp   i j p j ⊗ γ j   ,(39) where j = 1, 2, . . . , d enumerates different orthogonal directions, γ j = |2j 2j| − |2j − 1 2j − 1| acts on the coin space and p j is the corresponding momentum operator for direction j. Note that [γ j , γ k ] = 0 for all values of j and k, whereas the corresponding Dirac matrices anticommute {γ j ,γ k } = 2δ jk . Moreover,γ 2 j = 1 1 for all j, whereas for DTQW j γ 2 j = 1 1. Due to this reason, one is unable to recover the relativistic energy relation E 2 = | p| 2 +m 2 , but rather obtains E 2 = j |α j | 2 p 2 j +β 2 , where j |α j | 2 = 1 and β 2 is a parameter related to properties of the relevant coin operator, which can be interpreted as the mass. Since our method of conversion is closely related to the similarity of quantum walks to the Schrödinger and the Dirac equations, it is somehow expected that problems with conversion of DTQW into CTQW appear when the description of the DTQW departs from the Dirac one. It is important to mention here the research on quantum walk search protocols on d-dimensional grids. The efficiencies of the two protocols, the DTQW [21,22] and the CTQW [23], are quite similar. Only in dimensions d = 2, 3 the DTQW protocol is faster than the corresponding CTQW protocol. In [20], the authors of the CTQW protocol improved the efficiency of their model, however this was done at the cost of an additional coin degree of freedom, which had to be included -departure from the standard CTQW structure was inevitable. The heart of the DTQW protocol [21] is the so called Grover coin, which for d = 2 is of the following form For the two-dimensional Grover walk the probability distribution remains localized at the initial position for all time. This effect was explained by Inui et. al. [24], who noticed that eigenvalues of the Grover walk are highly degenerated. More than half of the eigenvalues are ±1. There is no CTQW analogy of the Grover walk. Since the Grover walk obeys translational symmetry, the corresponding CTQW version should also possess this property. However, for a translational symmetric CTQW on a N × N grid with periodic boundary conditions, coupling constant J = |J|e iϕ and diagonal term A, the set of eigenvalues is given by A + 2|J| (cos(ϕ − 2πk x /N ) + cos(ϕ − 2πk y /N )), where k x , k y = 0, 1, . . . , N − 1, and it is clear that there is no possibility of such a high degeneracy. CONCLUSIONS In this work we studied the DTQW version of the quantum state transfer, which is originally described as a continuous time process. While in the continuous time scenario the transfer properties depend on couplings between neighboring positions, in the discrete time case it is the coin operator which is responsible for the perfect transmission from one position to another. We applied the analogy between the DTQW and the Dirac equation and between the CTQW and the Schrödinger equation to show that both coins and couplings can be interpreted as the mass of a particle, which allowed us to transform tje CTQW into the DTQW. We examined the DTQW versions of perfect state transfer protocols studied in [11,17,18]. We found that in DTQW versions not only the particle, but also its intrinsic coin state is perfectly transfered. The general method of the CTQW transformation into the DTQW has been also discussed. Finally, we argued that some DTQW's do not have the corresponding CTQW versions due to the fact that in some cases the dynamics of DTQW is much richer than the CTQW one. One of the biggest advantages of CTQW is that it can be easily defined on any graph, whereas DTQW is natural on regular graphs only, where the same coin degree of freedom can be used for all vertices. However, many problems in physics, as well as in computer science, are defined on regular graphs. The above discussion shows that in many cases DTQW is more general than CTQW, since it allows to observe some effects which cannot be observed in CTQW. It would be interesting to find phenomena which can be observed in CTQW only. FIG. 1 : 1Due to the fact that interference phenomenon occurs for the second nearest neighbors even vertices can be interpreted as edges. FIG. 2 : 2DTQW on a finite chain is effectively simulated by a walk on a cycle with one reversing edge. FIG. 3 :FIG. 4 : 34Perfect state transfer from position 1 to position N − 1 for DTQW with position dependent coin (N = 30 and λ = 0.03). Thin blue line -probability of occupying position 1; thick purple line -pobability of occupying position N − 1; dashed line -probability of occupying remaining vertices of the chain. Arguments ϕ of eigenvalues e iϕ of a position dependent coin DTQW double step operator U 2 for N = 30 and λ = 0.03. Each dot corresponds to a doubly degenerated eigenvalue. FIG. 5 :FIG. 6 : 56Eigenstate population for N = 30, θ = π/4 and initial state |1, → . Left:θ1 = θN−1 = θ = π/4. Right: θ2 = θN−2 = π/2 − ε, where ε = π 2N; the initial state is almost entirely supported on four eigenstates only. Probability of population of positions 1 (solid) and N − 1 (dashed) in time. N = 30, θ = π 4 and ε = π 2N . ACKNOWLEDGEMENTSPK would like to thank Jiannis Pachos for stimulating discussions and Ravishankar Ramanathan for help with preparation of this manuscript. 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A novel technique for divided-pulse amplification is presented in a proof-of-principle experiment. A pulse burst, cut out of the pulse train of a mode-locked oscillator, is amplified and temporally combined into a single pulse. High combination efficiency and excellent pulse contrast are demonstrated. The system is mostly fiber-coupled, enabling a high interferometric stability. This approach provides access to the amplitude and phase of the individual pulses in the burst to be amplified, potentially allowing the compensation of gain saturation and nonlinear phase mismatches within the burst. Therefore, this technique enables the scaling of the peak power and pulse energy of pulsed laser systems beyond currently prevailing limitations.

10.1364/oe.25.013494

2101.08719

8d2267d1fab7b41c08998f0de57ffff831d230ba

Electro-optically controlled divided-pulse amplification Henning Stark *[email protected] Institute of Applied Physics Abbe Center of Photonics Friedrich-Schiller-Universität Jena Albert-Einstein-Straße 1507745JenaGermany Michael Müller Institute of Applied Physics Abbe Center of Photonics Friedrich-Schiller-Universität Jena Albert-Einstein-Straße 1507745JenaGermany Marco Kienel Institute of Applied Physics Abbe Center of Photonics Friedrich-Schiller-Universität Jena Albert-Einstein-Straße 1507745JenaGermany Helmholtz-Institute Jena Fröbelstieg 307743JenaGermany Currently with Active Fiber Systems GmbH Wildenbruchstraße 1507745JenaGermany Arno Klenke Institute of Applied Physics Abbe Center of Photonics Friedrich-Schiller-Universität Jena Albert-Einstein-Straße 1507745JenaGermany Helmholtz-Institute Jena Fröbelstieg 307743JenaGermany Jens Limpert Institute of Applied Physics Abbe Center of Photonics Friedrich-Schiller-Universität Jena Albert-Einstein-Straße 1507745JenaGermany Helmholtz-Institute Jena Fröbelstieg 307743JenaGermany Fraunhofer Institute for Applied Optics and Precision Engineering Albert-Einstein-Str. 707745JenaGermany Andreas Tünnermann Institute of Applied Physics Abbe Center of Photonics Friedrich-Schiller-Universität Jena Albert-Einstein-Straße 1507745JenaGermany Helmholtz-Institute Jena Fröbelstieg 307743JenaGermany Fraunhofer Institute for Applied Optics and Precision Engineering Albert-Einstein-Str. 707745JenaGermany Electro-optically controlled divided-pulse amplification OCIS codes (1403298) Laser beam combining(3207090) Ultrafast lasers(1403510) Lasers, fiber(1403280) Laser amplifiers A novel technique for divided-pulse amplification is presented in a proof-of-principle experiment. A pulse burst, cut out of the pulse train of a mode-locked oscillator, is amplified and temporally combined into a single pulse. High combination efficiency and excellent pulse contrast are demonstrated. The system is mostly fiber-coupled, enabling a high interferometric stability. This approach provides access to the amplitude and phase of the individual pulses in the burst to be amplified, potentially allowing the compensation of gain saturation and nonlinear phase mismatches within the burst. Therefore, this technique enables the scaling of the peak power and pulse energy of pulsed laser systems beyond currently prevailing limitations. References and links [1] W. Leemans, W. Chou, and M. Uesaka, "White Paper of the ICFA-ICUIL Joint Task Force -High Power Laser Technology for Accelerators," ICFA Beam Dyn. Newsl. 56, 11-88 (2011). [2] M. Lewenstein and P. Salières, "Generation of ultrashort coherent XUV pulses by harmonic conversion of intense laser pulses in gases: towards attosecond pulses," Meas. Sci. Technol. 12(11), 1818-1827 (2001). [3] D. Strickland and G. Mourou, "Compression of Amplified Chirped Optical Pulses," Opt. Commun. 55(6), 447-449 (1985). [4] F. Stutzki, F. Jansen, H.-J. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, "Designing advanced verylarge-mode-area fibers for power scaling of fiber-laser systems," Optica 1(4), 233-242 (2014). [5] T. Y. Fan, "Laser Beam Combining for High Power, High-Radiance Sources," IEEE J. Sel. Top. Quantum Eletron. 11(3), 567-577 (2005). T. Zhou, J. Ruppe, C. Zhu, I.-N. Hu, J. Nees, and A. Galvanauskas, "Coherent Pulse Stacking Amplification using Low-finesse Gires-Tournois Interferometers," Opt. Express 23(6), 7442-7462 (2015). [10] S. Szatmari and P. Simon, "Interferometric multiplexing scheme for excimer amplifiers," Opt. Commun. 98(1-3), 181-192 (1993). [11] S. Podleska, "Verfahren und Vorrichtung zum Strecken und Rekomprimieren von optischen Impulsen, insbesondere von Laserimpulsen hoher Intensität," DE Patent DE102006060703A1, 2006. [12] S. Zhou, D. G. Ouzounov, and F. W. Wise, "Divided-pulse amplification of ultrashort pulses," Opt. Lett. 32(7), 871-873 (2007). [13] Y. Zaouter, F. Guichard, L. Daniault, M. Hanna, F. Morin, C. Hönninger, E. Mottay, F. Druon, P. Georges, "Femtosecond fiber chirped-and divided-pulse amplification system," Opt. Lett. 38(2), 106-108 (2013). [14] M. Kienel, A. Klenke, T. Eidam, S. Hädrich, J. Limpert, and A. Tünnermann, "Energy scaling of femtosecond amplifiers using actively controlled divided-pulse amplification," Opt. Lett. 39(4), 1049-1052 (2014). [15] M. Kienel, M. Müller, A. Klenke, J. Limpert, and A. Tünnermann, "12 mJ kW-class ultrafast fiber laser system using multidimensional coherent pulse addition," Opt. Lett. 41(14), 3343-3346 (2016). [16] M. Kienel, M. Müller, A. Klenke, T. Eidam, J. Limpert, and A. Tünnermann, "Multidimensional coherent pulse addition of ultrashort laser pulses," Opt. Lett. 40(4), 522-525 (2015). [17] T. M. Shay, "Theory of electronically phased coherent beam combination without a reference beam," Opt. Express 14(25), 12188-12195 (2006 Introduction Ultrafast high-power lasers pose unique and indispensable tools for a plethora of medical, industrial and scientific applications and open up, due to their consistently increasing brightness, ever new application fields such as, for instance, wakefield particle acceleration [1] and highharmonic generation [2]. However, further brightness scaling is hampered by physical and technological challenges. For example, high pulse peak intensities induce detrimental nonlinear effects such as, for instance, self-phase modulation and self-focusing, impairing the pulse and beam quality and, ultimately, resulting in optical damage of the media involved. One of today's standard techniques to mitigate nonlinear effects, chirped-pulse amplification (CPA) [3], allows reaching formerly unattainable peak powers, but any further scaling is limited by the grating size available. In the case of fiber lasers, the mode-area scaling of small-core high-NA fibers to very-large-mode-area low-NA fibers [4] allowed for a tremendous increase of peak power. However, as with CPA, a further power scaling following this approach is limited, mostly due to the tight manufacturing tolerances required to make such fibers. Therefore, novel approaches for further peak power scaling are needed; a prominent example of them being coherent beam combining (CBC) [5,6]. This technique exploits parallelization of the amplification process with multiple amplifiers. The subsequent coherent superposition of all the beams from these amplifiers leads to a significant pulse peak power and average power enhancement. For this approach, especially fiber lasers have proven their suitability, since their simple architecture allows for compact and stable setups, while their reproducible and outstanding beam quality enables a highly efficient combining process. The amplification of temporally separated pulses and their subsequent combination into a single pulse can be interpreted as the time-domain counterpart of CBC. Thus, this technique allows mitigating nonlinear effects in the optical medium. Regarding this strategy, besides cavity enhancement [7,8] and coherent pulse stacking [9], divided-pulse amplification (DPA) [10][11][12][13] has especially shown its potential. In DPA, the temporally separated pulses are usually generated from and stacked in a series of optical delay lines. Using actively controlled DPA (ADPA) [14], where the setups for pulse division and combining are separated from one another, in conjunction with CBC, an amplification of four-pulse bursts with subsequent combining generated pulses with an energy of 12 mJ and a duration of 262 fs, which corresponds to 35 GW peak power [15]. Still, in this experiment, the energy-scaling potential of the individual fibers was not fully harvested, since a significant part of the energy stored in the fibers was not extracted. However, a further extraction of this residual energy by means of an increase of the number of pulses in the burst is hampered. In ADPA, the amount and sizes of the optical delay lines make the setup significantly more complex and sophisticated. Additionally, gain saturation occurs (since a considerable part of the energy stored in the active fiber is extracted), leading to the individual pulses of the burst experiencing different amplification levels. This, in turn, results in a mismatch of the amplitude and nonlinear phases among the pulses, which impairs their subsequent combination. In ADPA amplitude preshaping allows compensating for amplitude and also phase mismatches to a certain extent [16]. However, since in this approach amplitude and nonlinear phase cannot be shaped independently so far, a residual phase and/or amplitude mismatch remains, substantially reducing the overall performance of the combining. Therefore, the key to scale DPA to even more pulse replicas lies in an extensive control of the amplitude and phase of the individual pulses and in a reduction of the size and complexity of the optical setup. In this contribution, a novel concept for creating and amplifying a burst of pulses (as a power of two) is introduced. This technique allows stacking the pulses in an arbitrarily long series of optical delay lines. The approach potentially allows for an independent and individual control of amplitude and phase of each pulse while significantly reducing the complexity of the optical setup as it is largely fiber-coupled. We call this approach electro-optically controlled dividedpulse amplification (EDPA) as its key components are electro-optic phase modulators (EOM). This paper is organized as follows. First, the fundamental principle of EDPA is described. Then, the setup of this proof-of-principle experiment is explained. Next, the experimental results are presented and discussed. Finally, a summary of the experimental findings is given, together with an outlook of the application of EDPA to high-power ultrafast lasers. Electro-optically controlled divided-pulse amplification In classic DPA approaches, a single pulse is temporally split up, by a set of N delay lines, into a burst of 2 N pulses, which leads to a lower pulse peak intensity in the laser amplifiers. For the subsequent temporal recombination with delay lines, and, therefore, an increased peak power in the end, the pulse burst has to have a distinctive polarization and phase pattern in order to send the individual pulse replicas into their corresponding delay path. The necessary pattern can be found by propagating a single pulse backwards through the combining stage, as depicted exemplary for four pulses in Fig. 1 (from right to left). Here, the single pulse on the right is initially linearly polarized in a 45° angle, i.e. it comprises equal amounts of p-and spolarization. Propagating the pulse to the left, the polarizing beam splitter (PBS) 2 transmits the p-polarized component while the s-polarized component is reflected into the delay line (DL) 2. Therefore, the s-polarized part is delayed and, finally, reflected a second time by PBS 2, lining up the s-polarized component behind the previously transmitted p-polarized component. Since both pulse replicas are orthogonally polarized, a rotation by 45° with a half-wave plate (HWP) allows separating their p-and s-polarized components again by means of PBS 1, sending the spolarized parts into DL 1. In this case, DL 1 has half the length of the previously passed DL 2. Consequently, a burst of four pulses with alternating orthogonal polarizations and a distinctive phase pattern is generated. In reverse, a pulse burst with this pattern going through the combining stage (from left to right in Fig. 1) will be combined into a single pulse. There are several approaches to generate this pattern. For instance, in passive DPA [13], the polarization pattern is generated by the very same delay line arrangement which is subsequently used for combining. In ADPA [14], on the other hand, the division and combining stages are separated, requiring a second set of delay lines. EDPA employs an identical setup for the pulse combination as conventional DPA. However, the pulse burst with the specific polarization pattern is generated without any delay lines in a compact fiber-integrated front-end. The principle of the pattern generation is illustrated in Fig. 2 (from left to right) once again for four pulses. It starts with a p-polarized burst of equal pulses, which can be cut out directly from the pulse train emitted by an oscillator by means of an amplitude modulator. The burst is then spatially split in two equal parts that are sent to two amplifier channels, each containing a phase controlling element. Here, distinctive patterns of the relative phases are imprinted on both bursts. These phase patterns eventually will produce the polarization pattern required for the temporal combination and consist, in this idealized case, of relative phases of 0 and π. Since both channels contain amplifiers, this allows for a temporally and spatially separated amplification. Thereafter, the polarization of one channel is rotated by 90°, so that one channel is p-polarized and the other one is s-polarized. The interferometric superposition of both channels leads to a burst of pulses with, in this case, an alternating orthogonal polarization pattern. Next, the polarization is rotated by 45° degrees, so that the individual pulses of the generated burst are either p-or s-polarized. Due to the chosen phase patterns, this polarization pattern is exactly the one required for the combining stage that has been described above. Analogous to the case of the polarization pattern, the phase patterns that have to be imprinted on the pulses are found by propagating the polarization pattern in reverse (i.e. from right to left) through the setup shown in Fig. 2. It is important to note that this intuitive technique of generating the necessary polarization pattern and, consequently, the required phase patterns by propagating a pulse backwards through the system can be applied to any arbitrary number of optical delay lines. In conclusion, EDPA uses an interferometric superposition of two orthogonally polarized pulse bursts with distinctive phase patterns to generate the polarization pattern which is required for the temporal combination with a set of delay lines. The corresponding phase patterns, which need to be imprinted, can be found by propagating a single pulse backwards through the whole system to the phase modulators. As soon as gain saturation occurs in the amplifiers, some amplitude and nonlinear phase mismatches between the individual pulses appear and they reduce the performance of the combining process. This is where EDPA shows its full potential, since it allows for compensating the amplitude mismatch for any arbitrary number of pulses by pre-shaping the burst with the amplitude modulator. Furthermore, EDPA has the potential to significantly reduce the impact of the nonlinear phase mismatch, since it provides access to the phase of every individual pulse in the burst. To do this, in theory, the phase modulators simply have to add extra phases, individually customized to the demands of every single pulse, to the phases 0 and π of the phase patterns used in the idealized case without saturation. This pre-compensates variations of the accumulated nonlinear phase between the pulses in the subsequent amplification and substantially increases the performance of the combination. As another key feature, the number of pulses to be combined can be easily and arbitrarily scaled (but the final number of pulses always has to be a power of two) by changing the duration of the transmission window of the amplitude modulator and adjusting the phase patterns imprinted by the phase modulators (provided that the required delay lines are included in the combining stage). Furthermore, the extensive amplitude and phase corrections can be freely added to any desired number of pulses, posing an essential enhancement of former DPA techniques in the generation of high-power ultrafast laser systems. In this proof-of-principle experiment, however, only the general functional capability and performance of this technique are investigated, leaving the examination of phase and amplitude pre-compensation to future high-power experiments. Experimental setup The setup of the EDPA experiments used for the combination of four temporally separated pulses is schematically depicted in Fig. 3. It mainly consists of two parts: a fiber-coupled frontend (using PM980 fibers) and a free-space combining stage. At the beginning of the master oscillator power amplifier (MOPA) architecture, a homebuilt mode-locked fiber oscillator with a repetition rate of frep=108 MHz emits pulses with a duration of 190 ps at a center wavelength of 1030 nm. A synchronous countdown counts the generated laser pulses with the help of a photo diode (PD). This is used to set the temporal distance between two subsequent pulse bursts, which is the inverse of the burst repetition rate fburst. For this, the synchronous countdown emits an electric signal every NSC laser pulses, which is referred to as the clock signal, since it determines the timing of all electronic devices to be used. An arbitrary waveform generator (AWG) is used to drive an acousto-optic modulator (AOM) that applies a 37 ns transmission window to the pulse train coming from the oscillator. Thus, bursts of four pulses are created with the burst repetition rate fburst = frep / NSC. Next, the resulting laser signal is spatially divided into two channels. Each of them contains a fiber-coupled EOM, both driven by a two-channel AWG with a sampling rate of 1.2 GS/s. The EOMs add the phase patterns to the pulse bursts in each channel. In the simple case of a negligible nonlinear phase mismatch between the pulses, the phase pattern is set to consist of 0 and π. However, as soon as the nonlinear phase mismatch shows a noticeable impact on the combining, phase adjustments can be freely added to the phase patterns by means of the AWG. One ytterbium-doped polarization-maintaining 6 µm-fiber amplifier per channel amplifies the signals to up to 300 mW each. The signals are coupled out of the fibers, being collimated by aspheric lenses with a focal length of 8 mm. Next, both beams, which are initially p-polarized, pass a PBS with orthogonal propagation directions. On the one side, a 0° mirror with a quarterwave plate (QWP) in front of it rotates the signal from the one channel to s-polarization and reflects it back to the PBS. The lengths of both channels are interferometrically matched, such that the spatial combination of the two beams results in a pulse burst pattern with alternating orthogonal polarizations. A telescope is applied to increase the beam diameter to approximately 5 mm, reducing the impact of divergence in the long delay lines used for pulse stacking by increasing the Rayleigh length to about 19 m. With a HWP the polarization of the generated pulse burst is rotated by 45°, finally resulting in the same polarization and phase pattern as in conventional DPA. It consists of alternating s-and p-polarized pulses which allows for a subsequent temporal combination. The combining stage begins with a PBS, which reflects all s-polarized pulses into a delay line, while the p-polarized ones simply go through it. This first delay line has a length of l1 = c / frep = 2.76 m, determined by the fundamental repetition rate frep of the master oscillator and the speed of light in air c. With this delay line, the delayed pulses (the first and the third one) are coherently combined with their particular subsequent neighbors (the second and the fourth pulse, respectively). A 4f-arrangement is set up in the delay line to further reduce beam divergence and, therefore, maximize the spatial overlap in the combination. Due to the phase patterns imprinted with the EOMs, the polarization pattern after this first temporal combining step again consists of orthogonally polarized pulses. Thus, after a polarization rotation by 45° with a HWP, the pulses are again alternatingly s-polarized and p-polarized. The second delay line has twice the length of the first one, that is l2 = 2•l1 = 5.52 m, allowing for a temporal combination into a single pulse. After another rotation with a HWP, the combined pulse is transmitted by another PBS, which, for the most part, separates the successfully combined from the uncombined radiation. Since the overall setup is an interferometer with multiple arms, an active phase stabilization is required. In this setup, the technique of locking of optical coherence by single-detector electronic-frequency tagging (LOCSET) [17] was used in conjunction with piezo-mounted mirrors in every combining step, modulating distinctive frequencies onto the combined signal. A fraction of the combined signal is detected by a photodiode (PD) as required for the closedloop control. A demodulation at the respective frequencies delivers the required information to calculate the phase corrections that have to be applied with the piezo-mounted mirrors. However, since in spatio-temporal combining multiple stable states can be phase locked [18], a fast electronic switch is implemented. This introduces a short temporal window in the acquisition of the error signal around the temporal position of the correctly combined pulse. This way, the phase locking of unintended secondary stable states is avoided. Experimental results The quality of the spatial and temporal combination processes is crucial in order to obtain the maximum achievable performance of the corresponding system. Therefore, as a figure of merit, the combining efficiency (1) is introduced. It depends on the energy of the successfully combined pulse Ecomb and the uncombined energy Eloss. Another characteristic value, which additionally includes information about the loss inside the combining system due to losses at the optical elements ηOE < 1, is the system efficiency (2) This parameter is defined as the ratio between the energy of the combined pulse Ecomb and the total energy of the amplified pulse burst before the temporal combining stage, represented by comb comb loss comb E E E h = + comb sys comb OE tot . E E h h h = × = Etot. The temporal efficiency ηtemp of the combining is the energy contained in the temporal slot that corresponds to the successfully combined pulse divided by the total energy in the combined signal, which, therefore, also includes pre-and post-pulses. As confirmed later in detail by investigating the temporal contrast, which is in direct relation to the temporal efficiency, the temporal efficiency in EDPA is found to be ηtemp ≈ 1. Therefore, the measurements of the system efficiency and combining efficiency allow using the corresponding average powers Pcomb, Ploss and Ptot instead of the energies. During the experiments, the pulse burst repetition rate is varied while the average power of the signal seeding the main amplifiers is held constant by readjusting the pump power of the pre-amplifier. This results in a change of the total energy and, therefore, of the energy of the combined pulse. However, due to the limited operating range of the active stabilization and the driving EOM electronics, only repetition rates from 135 kHz to 1529 kHz can be applied. The achieved total power is in the range between 455 mW and 562 mW. Consequently, when combining four temporally separated pulses, total energies between 0.35 µJ and 3.5 µJ are generated. The efficiency measurements show consistently high values for all repetition rates applied. Although the theoretically estimated saturation energy of 14 µJ per channel was not reached, considerable signs of saturation were observed which can be seen in the progressively decreasing amplitudes of the individual pulses within the burst as illustrated in Fig. 4(a). In spite of these signs of saturation, there were no noticeable impairing effects on the combining apart from the need to readjust the HWPs in the combining stage. The detailed results of the combining efficiency measurements are shown in blue as a function of the total energy in Fig. 4(b). The values range from 92.5 % to 95.4 %. The system efficiency, colored red, is also included in Fig. 4(b). It shows a similar behavior and it settles between 78.1 % and 82.7 %. The discrepancy between both efficiencies originates from losses at the optical elements. This assumption is confirmed in a separate measurement, showing power losses of the laser beam of up to 13% during a simple propagation through the delay lines. Such power losses are reasonable, since many standard HR mirrors are used in the folded delay lines and the contrast of the PBSs and the accuracy of the wave-plates are limited. is introduced. Ecomb and Epre denote the energy of the combined pulse and the energy of the strongest pre-pulse, respectively. Both can be seen in the PD voltage trace in Fig. 5(a). For this measurement, only the most powerful pre-pulse is of concern. This is, because, on the one hand, comb 10 pre 10 log E E C × = for most applications only the pre-pulse contrast is of interest, since an early impact might disturb the targeted object before the main pulse arrives. On the other hand, the signal is almost symmetric with respect to the combined pulse when the combining efficiency is optimized. Therefore, a measurement of the post-pulse contrast would provide similar values. The contrast is evaluated from the photo diode (PD) voltage traces of the combined signal, as depicted in Fig. 5(a). As the contrast is high, the peak voltage of the main pulse is measured with neutral density (ND) filters, while the pre-pulse is recorded without any attenuation. Therefore, similar PD voltages can be ensured, eliminating the possibility of falsified measurement data due to PD saturation. Subsequently, by determining the precise attenuation of every single ND filter, the actual contrast is calculated. In Fig. 5(b), the result is depicted as a function of the total energy, showing a temporal contrast in the range of 23.2 dB to 27.3 dB. Since other pre-pulses are negligibly small, the achieved temporal contrast corresponds to a temporal efficiency of ηtemp > 0.99, even though a post-pulse that is similar to the pre-pulse is included. This confirms the earlier assumption of ηtemp ≈ 1. For a theoretical estimation of the achievable contrast, Eq. (3) is used. The highest possible Epre originates from the last combining step. Before this step, two pulses remain, but only the first one, having an energy of almost exactly ½•Ecomb, contributes to the pre-pulse. Since the specification of the transmissivity of the PBSs of s-polarized light is given by Ts < 0.005, a worst-case value Ts = 0.005 is assumed, leaving a pre-pulse with an energy of 0.005•½•Ecomb. This pre-pulse is polarization-rotated by the following HWP and its energy is cut in half by the last PBS, dumping one half of the energy on the loss port. Consequently, the resulting pre-pulse energy is Epre = ½•0.005•½•Ecomb, which can be applied to Eq. (3), resulting in (4) The measured contrast is high but still slightly lower than this theoretical maximum, indicating that small combining losses due to, for instance, beam size mismatches or wave front distortions exist. Furthermore, the phase jitter introduced by LOCSET also leads to a small decrease of the temporal contrast on average. However, by optimizing the control parameters of LOCSET, its influence on the temporal contrast can be minimized. Finally, the stability of the system is investigated. For this, the power of the combined signal resulting from the individual combining steps is temporally resolved with a PD. The signal is then filtered by a 500 kHz low-pass filter, recorded with an oscilloscope and Fourier max 10 2 2 10 log 29dB . 0.005 C × = × » transformed using the Hann window. The resulting power spectral densities (PSD) are depicted in Fig. 6(a) along with the corresponding integrated power spectral densities in Fig. 6(b). For the temporal combining of four pulses a low relative intensity noise (RIN) of 0.54 % is achieved, while the RIN of a single channel and using exclusively spatial combining is lower at approximately 0.2 %, as expected. The disturbances in the upper kHz regime of the four-pulse combining, which manifest as distinct steps in the integrated PSD, can be assigned to the LOCSET jitter frequencies, which are located at 5 kHz, 6 kHz and 9 kHz. Therefore, in the integrated PSD of the spatial combining, only a single step occurs at 5 kHz, since 6 kHz and 9 kHz were the jitter frequencies for the temporal combining. The edge at 80 Hz is most probably a mechanical resonance caused by the cooling fans of the electric control devices. Fig. 6. Power spectral density (a) and integrated power spectral density (b) measured for different combining steps (spatial combining in red and temporal combining of four pulses from two channels each in yellow), a single channel before combining (colored blue) and the background (green) in EDPA at fburst=1529 kHz. The combining of four temporally separated pulses from two channels was analyzed thoroughly and proves that EDPA is performing well. Therefore, the scalability of EDPA is investigated by doubling the amount of pulses to be combined. For the combining of eight pulses different adjustments and extension need to be made. To generate the pulse burst, the duration of the transmission window of the AOM is doubled to 74 ns. In accordance with Fig. 3, a third delay line is added to the end of the combining stage, between the former last delay line and the last PBS which separates the combined output from the loss port. The new delay line has a length of 11.04 m, which is twice the length of the longest delay line from the four-pulse combining. It is folded by 9 mirrors in 16 reflections and has three subsequent 4f-arrangements set up inside to counteract beam divergence. The phase patterns applied by the two EOMs and, consequently, the generated polarization pattern required for the combining, are determined again by propagating a single pulse backwards through the setup, as explained in Sec. 2 and depicted in Fig. 1 and Fig. 2. Doubling the number of pulses barely shows any influence on the handling of EDPA. Therefore, with the mentioned adjustments, the scaling of the pulse number in EDPA is relatively straight-forward. Already at the first try, the eight pulses were combined successfully at a fixed pulse burst repetition rate of fburst=1075 kHz. Few optimizations of the temporal and spatial alignment in the optical combining setup directly lead to excellent results. A combining efficiency of 89.7 % is reached, which is just barely lower than the result from the four-pulse combining. However, this small decrease of the combining efficiency indicates that there is still potential for improvement, especially in the last delay line. For instance, aberrations and beam parameter mismatches due to imperfect adjustments of the imaging systems and a limited choice of lenses decrease the constructive beam overlap, finally reducing the performance of the combining process. Furthermore, the large number of optical elements in the delay line creates many possibilities for phase front errors. The measurement of the corresponding system efficiency shows a high value of 76.8 %, which is consistent with the slight reduction of the combining efficiency. As before, the power losses are expected to originate from the large amount of imperfect optical elements. In a last measurement of the eight-pulse combining, the photo diode voltage trace of the combined signal is investigated. Again, the recording shows a temporal contrast with similar results as those seen in the experiments with the combining of four temporally separated pulses. Conclusion and prospect In this contribution, a novel technique for temporal pulse division, subsequent amplification and combination is presented. In a proof-of-principle experiment, four temporally separated pulses from two channels each are combined into a single pulse. The corresponding investigations show high combining efficiencies of up to 95.4 % and good system efficiencies of up to 82.7 %. A high interferometric stability with a measured RIN of 0.54 % together with a high temporal contrast of up to 27.3 dB is achieved, being close to the estimated theoretical optimum. The scalability of this technique is investigated by combining eight temporally separated pulses. A combining efficiency of 89.7 % and a system efficiency of 76.8 % are measured. The complexity of the optical setup as well as the required free-space propagation distance is reduced drastically in comparison with similarly performing ADPA setups. In addition, without any changes to the optical setup, EDPA already provides the possibility for amplitude preshaping of pulse bursts with arbitrary lengths to compensate for gain saturation. Furthermore, the EOMs offer access to the phase of every individual pulse. In conventional DPA nonlinear phase mismatches from gain saturation remain despite amplitude pre-shaping. Therefore, phase control poses a powerful tool in all types of DPA, since it potentially allows for an active compensation of the accumulated nonlinear phases and, therefore, a matching of the phases between the individual pulses. In theory, the EOMs also allow adding the phase jitter to the individual pulses, which is required for the active stabilization with LOCSET. This will again simplify the optical setup and probably further improve the performance of EDPA, since the jitter will not need to be applied mechanically with piezo mounted mirrors anymore. Few issues remain, however, such as, for instance, the considerable losses at the optical elements or the aberrations caused by imperfect imaging in the delay lines. A solution to this will be given by the application of multipass cells, e.g. Herriott type ones [19], with customized high-efficiency concave mirrors. Additionally, highly stable mirror mounts and tailored housings are expected to further improve the temporal stability. These upgrades will also have a positive influence on the temporal contrast, which will be increased even more in future experiments by replacing the PBSs with high-contrast thin film polarizers. In conclusion, EDPA shows an excellent efficiency, contrast and stability. In conjunction with the significantly reduced size and complexity of the optical setup and the potential for extensive amplitude and phase shaping, EDPA is a promising technique for the generation and further scaling of high peak power radiation with ultrafast laser systems. Funding Free State of Thuringia (2015FE9158) "PARALLAS", co-funded by the European Union within the framework of the European Regional Development Fund (ERDF); European Research Council (ERC) (617173, 670557). Seise, A. Klenke, J. Limpert, and A. Tünnermann, "Coherent addition of fiber-amplified ultrashort laser pulses," Opt. Express 18(26), 27827-27835 (2010). [7] R. Jones and J. Ye, "Femtosecond pulse amplification by coherent addition in a passive optical cavity," Opt. Lett. 27(20), 1848-1850 (2002). [8] S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza and J. Limpert, "A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities," Light Sci. Appl. 3(10), e211 (2014). [9] Fig. 1 . 1Simplified representation of the combination of four pulses (when going from left to right). The same setup can be used for pulse division when going from right to left. In this latter case, the rotation by the half-wave plate (HWP) and the propagation in the delay lines (DL) are reversed. (PBS: polarizing beam splitter.) Fig. 2 . 2Generation of the polarization and phase pattern (from left to right). The individual steps are: splitting of the burst and imprinting the phase patterns; rotating one channel by 90°; amplifying and superposing both channels; rotating the polarization by 45°. In case of pattern decomposition (from right to left) the polarization rotations are reversed. Fig. 3 . 3Schematic illustration of the EDPA setup used for the combination of four temporally separated pulses from two channels. (PD: photo diode, AOM: acousto-optic modulator, AWG: arbitrary waveform generator, EOM: electro-optic modulator, QWP: quarter-wave plate, HWP: half-wave plate, PBS: polarizing beam splitter.) Fig. 4 . 4(a) PD trace of a pulse burst at a total energy of Etot = 3.4 µJ at a burst repetition rate of fburst = 135 kHz before temporal combining. (b) Experimental result of the combining efficiency and system efficiency measurements for the temporal combining of four pulses.As another figure of merit, the temporal contrast (3) Fig. 5 . 5(a) PD voltage trace of the combined signal from EDPA with four pulses, Etot = 3.5 µJ and fburst = 135 kHz. The blue arrow highlights the strongest pre-pulse. The visible distortion at about 60 ns has been proven to be a measurement artifact originating from a reflection of the electric signal of the main pulse (ringing). (b) Experimental result of the temporal contrast measurements for the temporal combining of four pulses using EDPA.

The probabilistic threshold query is one of the most common queries in uncertain databases, where a result satisfying the query must be also with probability meeting the threshold requirement. In this paper, we investigate probabilistic threshold keyword queries (PrTKQ) over XML data, which is not studied before. We first introduce the notion of quasi-SLCA and use it to represent results for a PrTKQ with the consideration of possible world semantics. Then we design a probabilistic inverted (PI) index that can be used to quickly return the qualified answers and filter out the unqualified ones based on our proposed lower/upper bounds. After that, we propose two efficient and comparable algorithms: Baseline Algorithm and PI index-based Algorithm. To accelerate the performance of algorithms, we also utilize probability density function. An empirical study using real and synthetic data sets has verified the effectiveness and the efficiency of our approaches.

10.1109/tkde.2013.67

1301.2362

7e991c4e172653c728622144825e86f85925a451

Quasi-SLCA based Keyword Query Processing over Probabilistic XML Data 11 Jan 2013 Jianxin Li Chengfei Liu Rui Zhou Member, IEEEJeffrey Xu Yu Quasi-SLCA based Keyword Query Processing over Probabilistic XML Data 11 Jan 20131Index Terms-Probabilistic XMLThreshold Keyword QueryProbabilistic Index The probabilistic threshold query is one of the most common queries in uncertain databases, where a result satisfying the query must be also with probability meeting the threshold requirement. In this paper, we investigate probabilistic threshold keyword queries (PrTKQ) over XML data, which is not studied before. We first introduce the notion of quasi-SLCA and use it to represent results for a PrTKQ with the consideration of possible world semantics. Then we design a probabilistic inverted (PI) index that can be used to quickly return the qualified answers and filter out the unqualified ones based on our proposed lower/upper bounds. After that, we propose two efficient and comparable algorithms: Baseline Algorithm and PI index-based Algorithm. To accelerate the performance of algorithms, we also utilize probability density function. An empirical study using real and synthetic data sets has verified the effectiveness and the efficiency of our approaches. INTRODUCTION Uncertainty is widespread in many web applications, such as information extraction, information integration, web data mining, etc. In uncertain database, probabilistic threshold queries have been studied extensively where all results satisfying the queries with probabilities equal to or larger than the given threshold values are returned [1], [2], [3], [4], [5]. However, all of these works were studied based on uncertain relational data model. Because the flexibility of XML data model allows a natural representation of uncertain data, uncertain XML data management has become an important issue and lots of works have been done recently. For example, many probabilistic XML data models were designed and analyzed [6], [7], [8], [9], [10]. Based on different data models, query evaluation [7], [10], [11], [12], [13], algebraic manipulation [8] and updates [6], [10] were studied. However, most of these works concentrated on structured query processing, e.g., twig queries. In this paper, we propose and address a new interesting and challenging problem of Probabilistic Threshold Keyword Query (PrTKQ) over uncertain XML databases based on quasi-SLCA semantics, which is not studied before as far as we know. In general, an XML document could be viewed as a rooted tree, where each node represents an element or contents. XIRQL [14] supports keyword search in XML based on structured queries. However, users may not have the knowledge of the structure of XML data or the query language. As such, supporting pure keyword search in XML has attracted extensive research. The LCA-based approaches will identify the LCA node first, which contains every keyword under its subtree at least once [15], [16], [17], [18], [19], [20], [21]. Since the LCA nodes sometimes are not very specific to users' query, Xu and Papakonstantinou [20] proposed the concept of SLCA (smallest lowest common ancestor), where a node v is regarded as an SLCA if (a) the subtree rooted at the node v, denoted as T sub (v), contains all the keywords, and (b) there does not exist a descendant node v ′ of v such that T sub (v ′ ) contains all the keywords. In other words, if a node is an SLCA, then its ancestors will be definitely excluded from being SLCAs. The SLCA semantics of model keyword search result on a deterministic XML tree are also applied [22], [16], [19]. Based on the SLCA semantics, [23] discussed top-k keyword search over a probabilistic XML document. Given a keyword query q and a probabilistic XML document (PrXML), [23] returned the top k most relevant SLCA results (PrSLCAs) based on their probabilities. Different from the SLCA semantics over deterministic XML documents, a node v being a PrSLCA can only exclude its ancestors from being PrSLCAs by a probability. This probability can be calculated by aggregating the probabilities of the deterministic documents (called possible worlds) W implied in the PrXML where v is an SLCA in each deterministic document ∈ W . However, it is not suitable to directly utilize the PrSLCA semantics for evaluating PrTKQs because the PrSLCA semantics are too strong. In some applica-tions, users tend to be confident with the results to be searched, so relatively high probability threshold values may be given. Consequently, it is very likely that no qualified PrSLCA results will be returned. To solve this problem, we propose and utilize a so-called quasi-SLCA semantics to define the results of a PrTKQ by relaxing the semantics of PrSLCA with regards to a given threshold value, i.e., besides the probability of v being a PrSLCA in PrXML, the probability of a node v being a quasi-SLCA in PrXML may also count the probability of v ′ s descendants being PrSLCAs in PrXML if their probabilities are below the specified threshold value. In other words, a node v being a quasi-SLCA will exclude its ancestors from being quasi-SLCAs by a probability only when this probability is no less than the given threshold; otherwise, this probability will be included for contributing to its ancestors. This is different from the PrSLCA semantics that excludes the probability contribution from child nodes. Example 1: Consider an aircraft-monitored battlefield application, where the useful information will be taken as Aerial photographies. Through analysing the photographies, we can extract the possible objects (e.g., road, factory, airport, etc.) and attach some text description to them with probabilities, which can be stored in the format of PrXML. Figure 1 is a snapshot of an aircraft-monitored battlefield XML data. By issuing a keyword query {hazard, building}, a military department would find the potential areas containing hazard buildings above a probability threshold. Based on the semantics of PrSLCA, any of the nodes library (probability = 0.3), area 1 ( = 0.14), sub region 1 ( = 0.168), heliport( = 0.24), sub region 2 ( = 0.32) and region( = 0.088) can become an PrSLCA result. The detailed procedure of calculating the probabilities of results will be shown later. As we know, the users generally specify a threshold value σ as the confidence score with their issued query, e.g., σ = 0.40 representing that the users prefer to see the answers with their probabilities up to 0.40. In this condition, no results can be returned to the users. However, from Figure 1, we can see that if the probabilities of library and area 2 could contribute to their parent nodes, area 1 and sub region 2 would become quasi-SLCA results. Unfortunately, the PrSLCA semantics exclude them from being results. This motivates us to relax the PrSLCA semantics to the quasi-SLCA semantics. According to the quasi-SLCA semantics, the probabilities of area 1 and sub region 2 being the quasi-SLCA results are 0.44 and 0.56 with the contributions of their child nodes library and area 2 , respectively. As such, area 1 and sub region 2 are deemed as the interesting places to be returned. Given a PrTKQ, our problem is to quickly compute all the quasi-SLCA nodes with their probabilities meeting the threshold requirement. For users issuing PrTKQs, they generally expect to see the complete quasi-SLCA answer set as early as possible and do not need to know the accurate probability of each answer, which motivates us to design a Probabilistic Inverted (PI) index and PI-based efficient algorithm for quickly identifying quasi-SLCA result candidates. We summarize the contributions of this paper as follows: • Based on our proposed quasi-SLCA result definition, we study probabilistic threshold keyword query over uncertain XML data, which satisfies the possible world semantics. To the best of our knowledge, this problem has not been studied before. • We design a probabilistic inverted (PI) index that can quickly compute the lower bound and upper bound for a threshold keyword query, by which lots of unqualified nodes can be pruned and qualified nodes can be returned as early as possible. To keep the effectiveness of pruning, the probability density function is employed based on the assumption of Gaussian distribution. • We propose two algorithms, a comparable baseline algorithm and a PI-based Algorithm, to efficiently find all the quasi-SLCA results meeting the threshold requirement. • Experimental evaluation has demonstrated the efficiency and effectiveness of the proposed approaches. The rest of this paper is organized as follows. In Section 2, we introduce the probabilistic XML model and the problem definition of probabilistic threshold keyword query. Section 3 shows the procedure of efficiently finding quasi-SLCA results using an example. Section 4 first presents the data structure of PI index, discusses the basic building operations and pruning techniques of PI index, and provides the building algorithm of PI index. In Section 5, we propose a comparable baseline algorithm and a PI-based algorithm to find the qualified quasi-SLCA results. We report the experimental results in Section 6. Section 7 discusses related works and Section 8 concludes the paper. PROBABILISTIC DATA MODEL AND PROBLEM DEFINITION Probabilistic Data Model: A PrXML document defines a probability distribution over a space of deterministic XML documents. Each deterministic document belonging to this space is called a possible world. A PrXML document represented as a labelled tree has ordinary and distributional nodes. Ordinary nodes are regular XML nodes and they may appear in deterministic documents, while distributional nodes are only used for defining the probabilistic process of generating deterministic documents and they do not occur in those documents. In this paper, we adopt a popular probabilistic XML model, PrXML {ind,mux} [12], [23], which was first discussed in [7]. In this model, a PrXML document is considered as a labelled tree where distributional nodes have two types, IND and MUX. An IND node has children that are independent of each other, while the children of a MUX node are mutually-exclusive, that is, at most one child can exist in a random instance document (called a possible world). A real number from (0,1] is attached on each edge in the XML tree, indicating the conditional probability that the child node will appear under the parent node given the existence of the parent node. An example of a PrXML document is given in Fig. 1. Unweighted edges have 1 as the default conditional probability. The Semantics of PrSLCA in PrXML: According to the semantics of possible worlds, the global probability of a node v being a PrSLCA with regard to a given query q in the possible worlds is defined as follows: (1) where w 1 , . . . , w m denotes the possible worlds implied by slca(q, v, w i ) = true indicates that v is an SLCA in the possible world w i for the query q. P r(w i ) is the existence probability of the possible world w i . The symbol G means P r G slca (q, v) is the global probability of a node v being an SLCA w.r.t. q in all possible worlds. Figure 2.{b,c,d,e,f,g,h,i} where the solid line represents the existence of the edge while the dashed line represents the absence of the edge. Given a possible world, we can compute its global probability based on the existence/absence of the edges in the possible world, e.g., P r(w d ) = (1−0.5) * 0.3 * 0.4 = 0.06. P r G slca (q, v) = m i=1 {P r(w i )|slca(q, v, w i ) = true} Given a keyword query q = {k 1 , k 2 }, we can compute the global probability of c 2 being a PrSLCA w.r.t. q by using P r G slca (q, c 2 ) = P r(w b )+ P r(w d )+ P r(w f )+ P r(w h ) = 0.06 + 0.06 + 0.09 + 0.09 = 0.30. Similarly, we have the global probability of a 4 being a PrSLCA w.r.t. q by using P r G slca (q, a 4 ) = P r(w e ) = 0.14. The Semantics of quasi-SLCA in PrXML: Definition 1: Quasi-SLCA: Given a keyword query q and a threshold value σ, a node v is called a quasi-SLCA if and only if (1) v or its descendants are SLCAs in a set W of possible worlds; (2) the aggregated probability of v and its descendants to be SLCAs in W is no less than σ; (3) no descendant nodes of v satisfy both of the above conditions in any set of possible worlds that overlaps with W . In other words, if a descendant node v d of v is a quasi-SLCA, then the probability of v d has to be excluded from the probability of v being a quasi-SLCA. It means that the set of possible worlds that v d appears does not overlap with the set of possible worlds that v or its other descendants appear. Given a query q, we can compute P r L slca (q, v) in a bottom-up manner, where P r L slca (q, v) stands for the local probability for v being an SLCA in the probabilistic subtree rooted at v. For example, a 4 in Figure 2(a) is a subtree of Figure 1. P r L slca (q, a 4 ) can be used to compute the PrSLCA probability of a 2 and a 1 . From P r L slca (q, v), we can easily get P r G slca (q, v) by P r G slca (q, v) = P r(path r→v ) × P r L slca (q, v) where P r(path r→v ) indicates the existence probability of v in the possible worlds. It can be computed by multiplying the conditional probabilities along the path from the root r to v. Now, we define quasi-SLCA based on PrSLCA and the parent-child relationship. For an IND node v, we have: P r G quasi−slca (q, v) = P r G slca (q, v) + P r(path r→v )× (1 − v ′ ∈child(v)∧v ′ / ∈Vquasi (1 − P r L slca (q, v ′ ))) (2) where the child node v ′ of v is an SLCA node, but not a quasi-SLCA node. For MUX node v, we have: P r G quasi−slca (q, v) = P r G slca (q, v) + P r(path r→v )× {P r L slca (q, v ′ )|v ′ ∈ child(v) ∧ v ′ / ∈ V quasi }(3) Note, IND or MUX nodes are normally not allowed to be SLCA result nodes because they are only distributional nodes. As such, for the above IND or MUX node v, we may use its parent node v p (with v as a sole child) to represent the SLCA result node. Example 3: Let's consider Example 2 again. First assume the specified threshold value is 0.40, then the global probability of a 4 being a quasi-SLCA result can be calculated by using P r G quasi−slca (q, a 4 ) = P r G slca (q, a 4 ) + P r(path r→a4 ) * (1 − (1 − P r L slca (q, c 2 ))) = 0.14 + 0.30 = 0.44 because child c 2 is an SLCA node but not a quasi-SLCA node w.r.t. the given threshold. So c 2 's SLCA probability contributes to its parent node a 4 . If the threshold is decreased to 0.30, then c 2 will be taken as a qualified quasi-SLCA result and will not contribute to a 4 . In this case, a 4 cannot become a quasi-SLCA result because P r G quasi−slca (q, a 4 ) = P r G slca (q, a 4 ) = 0.14 < 0.30. If the threshold is further decreased to 0.14, both c 2 and a 4 are qualified quasi-SLCA results. Definition 2: Probabilistic Threshold Keyword Query: (PrTKQ) Given a keyword query q and a threshold σ, the results of q over a probabilistic XML data T is a set R of quasi-SLCA nodes with their probabilities equal to or larger than σ, i.e., P r G quasi−slca (q, v) ≥ σ for ∀v ∈ R. In this work, we are interested in how to efficiently compute the quasi-SLCA answer set for a PrTKQ over a probabilistic XML data. OVERVIEW OF THIS WORK A naive method to answer a PrTKQ is to enumerate all possible worlds and apply the query to each possible world. Then, we can compute the overall probability of each quasi-SLCA result and return the results meeting the probability threshold. However, the naive method is inefficient due to the huge number of possible worlds over a probabilistic XML data. Another method is to extend the work in [23] to compute the probabilities of quasi-SLCA candidates. Although it is much more efficient than the naive method, it needs to scan the keyword node lists and calculate the keyword distributions for all relevant nodes. Therefore, that motivates our development of efficient algorithms which not only avoids generating possible worlds, but also prunes more unqualified nodes. To accelerate query evaluation, in this paper we propose a prune-based probabilistic threshold keyword query algorithm, which determines the qualified results and filters the unqualified candidates by using off-line computed probability information. To do this, we need to first calculate the probability of each possible query term within a node, which is stored as an off-line computed probabilistic index. Within a node, any two of its contained terms may appear in the IND or MUX ways. To precisely differentiate IND and MUX, we utilize different parts to represent the probabilities of possible query terms appearing in MUX way, while the terms in each part hold IND relationships. In other words, the different parts of terms in a node are mutual-exclusive (MUX), e.g., a 1 and a 5 in Figure 3 consists of three parts. Given a keyword query and a threshold value, we first load the corresponding off-line computed probabilistic index w.r.t. the keyword query and then on-the-fly calculate the range of probabilities of a node being a result of the keyword query using the precomputed probabilistic index in a bottom-up strategy. Here, the range of probabilities can be represented by two boundary values: lower bound and upper bound. By comparing the lower/upper bounds of candidates, the qualified results can be efficiently identified. The followed two examples briefly demonstrate how we calculate the lower/upper bounds based on a given keyword query and the off-line computed probabilistic index, and how we apply the on-line computed lower/upper bounds to prune the unqualified candidates and determine the qualified ones. Figure 1 where the probability of each individual term is calculated offline while the lower/upper bounds are computed on-the-fly based on the given query keywords. Let's first introduce the related concepts briefly: the probability of a term in a node represents the total local probability of the term appearing in all possible worlds to be generated for the probabilistic subtree rooted at the node, e.g., P r(k 1 , a 2 ) = 0.65 and P r(k 2 , a 2 ) = 0.916; the lower bound value represents the minimal total local probability of the given query keywords appearing in all the possible worlds w.r.t. the probabilistic subtree, e.g., LB(k 1 k 2 , a 2 )=0.65*0.916=0.595; the upper bound value represents the maximal total local probability of the given query keywords appearing in all the possible worlds w.r.t. the probabilistic subtree because the keywords may be independent or co-occur, e.g., UB(k 1 k 2 , a 2 ) = min{0.65, 0.916} = 0.65 no matter whether they are independent. By multiplying the path probability, the local probability can be transformed into the global probability. For the nodes containing MUX semantics, we group the probabilities of its terms into different parts, any two of which are mutuallyexclusive as shown in a 1 , a 3 and a 5 in Figure 3. The details of computing the lower/upper bounds for the IND and MUX semantics in the following section. Example 4: Consider a PrTKQ {k 1 , k 2 } with σ=0.40 again. a 5 , c 2 and c 7 can be pruned directly without calculation because their upper bounds are all lower than 0.40. We need to check the rest nodes a 1 , a 2 , a 3 and a 4 . For a 4 , after computation, the probability of a 4 being a quasi-SLCA result is 0.44, which is larger than the specified threshold value 0.40, so a 4 will be taken as a result. After that, the result of a 4 can be used to update the lower bound and upper bound of a 2 , (LB=0.595, UB=0.65) → (LB=0.155, UB=0.21). As a consequence, a 2 should be filtered due to U B(a 2 ) = 0.21 < σ = 0.40. Similarly, a 3 can be computed and selected as a result because its probability is 0.56. Since a 3 and a 4 having been the quasi-SLCA results, the bounds of a 1 can be updated as (LB=0.890, UB=0.950) → (LB=0.136, UB=0.196). As such, a 1 can be pruned because its upper bound is lower than 0.40. From this example, we can find that many answers can be pruned or returned without the need to know their accurate probabilities, and the effectiveness of pruning would be accelerated greatly with the increase of users' search confidence. As an acute reader, you may find that we have to compute the probability of a 4 being a quasi-SLCA because it cannot determine whether or not a 4 is a qualified result to be output only based on its lower/upper bound values. To exactly calculate the probability of a 4 being a quasi-SLCA, we have to access its child/descendant nodes, e.g., c 1 , c 2 , c 3 , although c 2 has been recognized as a pruned node before we start to process a 4 . If an internal node depends on a larger number of pruned nodes, the effectiveness of pruning will be degraded to some extent. To fix this challenging problem, we will introduce Probability Density Function PDF that can be used to approximately compute the probability of a node, the result of which can be used to update the lower bound and upper bound of its ancestor nodes further. The details are provided and discussed with algorithms later. PROBABILISTIC INVERTED INDEX In this section, we describe our Probabilistic Inverted (PI) index structure for efficiently evaluating PrTKQ queries over probabilistic XML data. In keyword search on certain XML data, inverted indexes are popular structures, e.g., [16], [20]. The basic technique is to maintain a list of lists, where each element in the outer list corresponds to a domain element (i.e., a keyword). Each inner list stores the ids of XML nodes in which the given keyword occurs, and for each node, the frequencies or the weight at which the keyword appears or takes. In this work, we introduce a probabilistic version of this structure, in which we store for each keyword a list of node-ids. Along with each node-id, we store the probability values that the subtree rooted at the node may contain the given keyword. The probability values in inner lists can be used to compute lower bound and upper bound onthe-fly during PrTKQ evaluation. Figure 4 shows an example of a probabilistic inverted index of the data in Figure 1. At the base of the structure is a list of keywords storing pointers to lists, corresponding to each term in the XML data T . This is an inverted array storing, for each term in T , a pointer to a list of triple tuples. In the list k i .list corresponding k i ∈ T , the triple (v id, Pr(path r→v ), {p 1 , ...}) records the node v 1 , the conditional probability from the root to v, and the probability set that may contain single probability value or multiple probability value. Single probability value represents that all the keyword instances in the subtree can be considered as independent in probability, e.g., the confidence of a 2 containing k 1 is {0.65}, while multple probability value means that the keyword instances belonging to different sets occur mutually, e.g., the confidence of a 3 containing k 1 is a set {0.8, 0.86, 0.82}, that represents the different possibilities of k 1 occurring in a 3 . Basic Operations of Building PI Index To build PI index, we need to traverse the given XML data tree once in a bottom-up method. During the data traversal, we will apply the following operations that may be used solely or in their combinations. The binary operation X ⊲⊳ / Y promotes the probability of Y to its parent node X. The binary operation X ⊲⊳ sibling Y promotes the probabilities of two sibling nodes X and Y to their parent node. The n-ary case can be processed by calling for the corresponding binary cases one by one. Assume v 1 contains the keywords {k 1 , k 2 , ..., k i , ..., k m } and the conditional probability P r(path vp−>v1 ) is λ 1 ; and v 2 contains the keywords {k i , k i+1 , ..., k mi } and the conditional probability P r (path vp−>v2 ) is λ 2 . Operator1-v 1 ✶ sibling,IN D v 2 : If v 1 and v 2 are independent sibling nodes, we can directly promote their probabilities to their parent v p , then we have, P r(k j , v p ) =        λ 1 * P r(k j , v 1 ) j < i; 1 − (1 − λ 1 * P r(k j , v 1 )) (1 − λ 2 * P r(k j , v 2 )) i ≤ j ≤ m ≤ m i ; λ 2 * P r(k j , v 2 ) m ≤ j ≤ m i ;(4) Fig. 4. A probabilistic Inverted Index Operator2-v 1 ✶ /,IN D v 2 : If v 2 is an independent child of v 1 , we can directly promote the probability of v 2 to v 1 , then we have, Figure 2 using Operator1 and Operator2. Firstly, we compute c 1 ✶ sibling,IN D c 2 and promote the probability of keywords to their parent a 4 by Operator1, i.e., P r(k 1 , a 4 ) = 1 -(1 -0.5*1.0)*(1 -0.3*1.0) = 0.65 and P r(k 2 , a 4 ) = 0.3. And then, we compute a 4 ✶ /,IN D c 3 using operator2, i.e., P r(k 2 , a 4 ) = 1 -(1 -0.3)*(1 -0.4) = 0.58 while P r(k 1 , a 4 ) do not change because c 3 only contains k 2 here. And the conditional probability from the root to a 4 is 1.0. Therefore, k 1 → (a 4 , 1.0, 0.65) and k 2 → (a 4 , 1.0, 0.58) will be inserted in PI index, respectively. P r(k j , v 1 ) =        P r(k j , v 1 ) j < i; 1 − (1 − P r(k j , v 1 )) (1 − λ 2 * P r(k j , v 2 )) i ≤ j ≤ m ≤ m i ; λ 2 * P r(k j , v 2 ) m ≤ j ≤ m i ; (5) Example 5: Let's show the procedure of comput- ing c 1 ✶ sibling,IN D c 2 ✶ sibling,IN D c 3 in Operator3-v 1 ✶ sibling,MUX v 2 : If v 1 and v 2 are two mutually-exclusive sibling nodes and v p is their parent, then we generate two parts in v p by v p ✶ /,IN D v 1 and v p ✶ /,IN D v 2 , respectively. Operator4-v 1 ✶ /,MUX v 2 : If v 2 is a mutuallyexclusive child node of v 1 , then we can get the aggregated probability by v 1 ✶ /,IN D v 2 . In the above four basic operators, we assume the terms independently appear in v 1 and v 2 . When the nodes v 1 and v 2 contain mutually-exclusive parts, we need to deal with each part using the four basic operators. Given two independent sibling nodes v 1 (λ 1 ) and v 2 (λ 2 ) where only v 2 contains a set of mutually-exclusive parts {p m1 , p m2 , ...} with conditional probability λ mi . In this case, we can apply the operation v 1 ✶ sibling,IN D p mi for each part p mi . The computed results are maintained in different parts in their parent v p . Example 6: Consider an independent node c 5 and a node a 5 consisting of c 6 , c 7 and c 8 in Figure 1. We first promote c 6 , c 7 and c 8 to a 5 that consists of three parts: 1 2 3 k2 k1 k2 k1 0.5 0.3 0.3 0.1 , as shown in Figure 3 a 5 . Because c 5 and a 5 are independent sibling nodes, the operation c 5 ✶ sibling,IN D a 5 can be called to compute the probability with regards to their parent a 3 . To do this, we apply c 5 ✶ sibling,IN D part i for each part i ∈ a 5 using Operator1. The results are shown in Figure 3 a 3 . After that, we can insert k 1 → (a 3 , 0.8, 0.8, 0.86, 0.82) and k 2 → (a 3 , 0.8, 0.5, 0.3, 0) into PI index. If both v 1 and v 2 contain a set of mutually-exclusive parts, respectively, then we can do pairwise aggregations across the two sets of parts. Building PI index needs to scan the given probabilistic XML data only once. Assume that the probabilistic XML has been encoded using probabilistic Dewey codes. The basic idea of building PI index is to progressively process the document nodes sorted by Dewey codes in ascending order, i.e., the data can be loaded and processed in a streaming strategy. When a leaf node v l is coming, we will compute the probability of each term in the leaf node v l . After that, the terms with their probabilities in v l will be written into PI index. Next, we need promote the terms and their probabilities of v l to the parent v p of v l based on the operation types in Section 4.1. After the node stream is scanned completely, the building algorithm of PI index will be terminated. We don't provide the detailed building algorithm in this paper. Pruning Techniques using PI Index In this subsection, we first show how to prune the unqualified nodes using the proposed lower/upper bounds. And then, we explain how to compute lower/upper bounds, and how to update the upper/lower bounds based on intermediate results during the query evaluation. By default, the node lists in PI index are sorted in the document order. P r(k i , v) represents the overall probability of a keyword k i in a node v. It is obvious that the overall probability of a keyword appearing in a node is larger than or equal to that of the keyword appearing in its descendant nodes. And the overall probability value for each keyword in a node can be computed and stored in PI index offline. Consider a node v and a PrTKQ q containing a set of keywords {k 1 , k 2 , ..., k t }. If all the terms in v are independent, then we have, LB(q, v) = t i=1 P r(k i , v) (6) U B(q, v) = min{P r(k i , v)|1 ≤ i ≤ t}(7) Most of the time, v consists of a set of parts {vp 1 , vp 2 , ..., vp m } that are mutually-exclusive. In this case, the lower bound of v would be generated from a part vp j that gives the highest lower bound value while the upper bound of v would be generated from another part vp i that gives the highest upper bound value, in which j may be equal to or not equal to i. LB(q, v) = max 1≤j≤m { t i=1 P r(k i , vp j )} (8) U B(q, v) = max 1≤j≤m {min{P r(k i , vp j )|1 ≤ i ≤ t}}(9) Where vp j must satisfy the criteria: (1) LB(q, v) > 0; (2) cannot find another part vp ′ ) and LB(q, v) will be set as zero. j having t i=1 P r(k i , vp ′ j ) > 0 and min{P r(k i , vp ′ j )|1 ≤ i ≤ t} > min{P r(k i , vp j )|1 ≤ i ≤ t}. Otherwise, UB(q, v Example 7: Let's consider a 3 in Figure 3 as an example. The first and second parts can generate lower and upper bounds: Part 1 → LB({k 1 , k 2 },a 3 )=0.32, UB({k 1 , k 2 },a 3 )=0.4; and Part 2 → LB({k 1 , k 2 },a 3 )=0.206, UB({k 1 , k 2 },a 3 )=0.24. Because Part 1 can produce a higher upper bound than Part 2, the lower and upper bounds of a 3 will come from Part 1, which guarantees that a 3 can be a quasi-SLCA candidate with a higher probability. Since Part 3 does not contain full keywords, i.e., missing k 2 , it cannot generate lower and upper bounds. Property 1: [Upper Bound Usage] A node v can be filtered if the overall probability P r(k i , v) of any keyword k i ( k i ∈ v and k i ∈ q) is lower than the given threshold value σ, i.e., ∃k i , P r(k i , v) < σ. Proof: Since P r G quasi−slca (q, v) ≤ min{P r(k i , v)|k i ∈ q} ≤ P r(∀k i ∈ q, v), we have min{P r(k i , v)|k i ∈ q} as the upper bound probability of v becoming a qualified quasi-SLCA node. Therefore, if a node v holds the inequation P r(k i , v) < σ, then P r G quasi−slca (q, v) must be lower than σ. As such, v can be filtered. Proof: U B(q, v d ) < σ means that all the keyword nodes in the subtree rooted at v will contribute their probabilities to node v. In other words, no decendant node of v could be a quasi-SLCA so the lower bound probability LB(q, v) will not be deducted. Therefore, if we have for LB(q, v) ≥ σ, then P r G quasi−slca (q, v) ≥ σ. As such, v can be returned as a quasi-SLCA result. Example 8: Let's continue Example 7. a 3 can be directly returned as a qualified answer for the given threshold σ( = 0.4). This is because c 2 , c 7 and a 5 are filtered due to their upper bound less than the threshold σ( = 0.4). To update the lower/upper bound values during query evaluation, one way is to treat the different types of nodes differently, by which the updated lower/upper bounds may obtain better precision. But the disadvantage of this way is to easily affect the efficiency of bound update. This is because, given a current node having multiple quasi-SLCA nodes as its descendant nodes, it is required to know the detailed relationships (IND or MUX) among the multiple quasi-SLCA nodes. To avoid the disadvantage, we do not separate the different types of distributional nodes, under which the multiple quasi-SLCA nodes appear. In other words, we unify them into a uniform formula based on the following two properties. Property 3: No matter node v is an IND or ordinary or MUX node, we can update their upper bound values as follows: U B ′ (q, v) = U B(q, v) − 1 + m i=1 (1 − P r G quasi−slca (q, v ci ))(10) Where P r G quasi−slca (q, v ci ) ≥ σ should be held. Proof: According to the definition of upper bound, UB(q, v) represents the maximal probability of v being a quasi-SLCA node, which comes from the overall probability of a specific keyword. Therefore, the problem of updating upper bound can be alternatively considered as the percentage of the probability of the keyword has been used for the v ′ descendant nodes becoming qualified quasi-SLCA nodes. If we know there are m qualified descendant nodes of v as returned answers, then we can compute their aggregated probabilities by 1− m i=1 (1−P r G quasi−slca (q, v ci )). Therefore, the upper bound can be updated as U B(q, v) − 1 + m i=1 (1 − P r G quasi−slca (q, v ci )). Does the above update equation hold for MUX node? To answer this question, we utilize the properties in [23], from which we can compute the aggregated probability by using m i=1 P r G quasi−slca (q, v ci ). Therefore, we have U B ′ (q, v) = U B(q, v) − m i=1 P r G quasi−slca (q, v ci ). The equation can be con- verted into U B(q, v)−1+[1− m i=1 P r G quasi−slca (q, v ci )]. Since m i=1 (1 − P r G quasi−slca (q, v ci )) can be expressed as 1 − m i=1 P r G quasi−slca (q, v ci ) + ∆ where ∆ is a positive value, i.e., ≥ 0, we can de- rive that 1 − m i=1 P r G quasi−slca (q, v ci ) ≤ m i=1 (1 − P r G quasi−slca (q, v ci )) . As a consequence, we can obtain that U B ′ (q, v) = U B(q, v) − m i=1 P r G quasi−slca (q, v ci ) = U B(q, v) − 1 + [1 − m i=1 P r G quasi−slca (q, v ci )] ≤ U B(q, v) − 1 + m i=1 (1 − P r G quasi−slca (q, v ci )). Therefore, U B ′ (q, v) = U B(q, v) -1 + m i=1 (1 -P r G quasi−slca (q, v ci )) holds for IND, ordinary and MUX nodes. Property 4: No matter node v is an IND or ordinary or MUX node, we can update their lower bound values as follows: LB ′ (q, v) = LB(q, v) − m i=1 P r G quasi−slca (q, v ci ) (11) Where P r G quasi−slca (q, v ci ) ≥ σ should be held. Proof: For the lower bound update, we need to deduct the confirmed probability [1 − m i=1 (1 − P r G quasi−slca (q, v ci ))] for IND nodes or m i=1 P r G quasi−slca (q, v ci ) for MUX nodes, from the original lower bound LB(q, v). According to the procedure of the above proof, we have m i=1 (1 − P r G quasi−slca (q, v ci )) ≥ 1 − m i=1 P r G quasi−slca (q, v ci ). Consequently, we have the inequation, 1 − m i=1 (1 − P r G quasi−slca (q, v ci )) ≤ 1 − (1 − m i=1 P r G quasi−slca (q, v ci )) = m i=1 P r G quasi−slca (q, v ci ) . Therefore, it is safe to use the right side to update the lower bound values. Since U B ′ ({k 1 , k 2 }, a 2 ) < σ, a 2 can be filtered out effectively without computation. Property 3 is used to filter the unqualified nodes by reducing the upper bound value while Property 4 is used to quickly find the qualified required results by comparing the reduced lower bound value (for the probability of the remaining quasi-SLCAs) with the threshold value. Sometimes, we need to calculate the probability distributions of keywords in a node if the given threshold σ is in the range (LB(q, v), U B(q, v)]. The basic computational procedure is similar to the PrStack algorithm in [23]. Different from the PrStack algorithm, we will introduce probability density function (PDF) to approximately calculate the probability for a node if the node depends on a large number of pruned descendent nodes. To decide when to invoke the PDF while avoiding the risk of reducing precision significantly, we would like to select and compute some descendant nodes that may contribute large probabilities to the node v. For the remaining descendant nodes, we may choose to invoke the PDF, by which we can reduce the time cost while still guarantee the precision to some extent. The detailed procedure will be introduced in the next section. PRUNE-BASED PROBABILISTIC THRESH-OLD KEYWORD QUERY ALGORITHM A key challenge of answering a PrTKQ is to identify the qualified result candidates and filter the unqualified ones as soon as possible. In this work, we address this challenge with the help of our proposed probabilistic inverted (PI) index. Two efficient algorithms are proposed, a comparable Baseline Algorithm and a PI-based Algorithm. Baseline Algorithm In keyword search on certain XML data, it is popular to use keyword inverted index retrieving the relevant keyword nodes, by which the keyword search results are generated based on different algorithms, e.g., [20], [16], [24], [25]. In probabilistic XML data, [23] proposed PrStack Algorithm to compute top-k SLCA nodes. In this section, we propose an effective Baseline Algorithm that is similar the idea of PrStack Algorithm. To answer PrTKQ, we need to scan all the keyword inverted lists once. Firstly, the keywordmatched nodes will be read one by one based on their document order. After one node is processed, we check if its probability can be up to the given threshold value σ. If it is true, the node can be output as a quasi-SLCA node and its remaining keyword distributions (i.e., containing partial query keywords) can be continuously promoted to its parent node. Otherwise, we promote its complete keyword distributions (i.e., containing both all keywords or partial keywords) to its parent node. After that, the node at the top of the stack will be popped. Similarly, the above procedures will be repeated until all nodes are processed. The basic algorithm can be terminated when all nodes are processed. The detailed procedure is shown in Algorithm 1. Because Baseline Algorithm only needs to scan the keyword node lists once, it is a fast and simple algorithm. However, its core computation -keyword distribution computation would consume lots of time, which motivates us to propose the PI-based Algorithm that can quickly identify the qualified or unqualified candidates using offline computed PI index and only compute keyword distributions for a few candidates. Here, Baseline Algorithm is taken as a comparable base to show the pruning performance of the PI-based Algorithm described below. PI-based Algorithm To efficiently answer PrTKQ, the basic idea of PIbased Algorithm is to read the nodes from keyword node lists one by one in a bottom-up strategy. For each node, we quickly compute its lower bound and upper bound by accessing PI index, which is far faster than computing the keyword distributions of the node directly. After comparing its lower/upper bounds Algorithm 1 Baseline Algorithm input: a query q = {k1, k2, ..., km} with threshold σ, keyword inverted (KI) index output: a set R of quasi-SLCA nodes 1: load keyword node lists L = {l1, l2, ..., lm} from KI index; 2: get the smallest Dewey v from L; 3: initiate a stack S1 using v; 4: while L = φ do 5: get the next smallest Dewey v from L; 6: while (S1.top() ≺ not v) do 7: x = S1.pop(); 8: if x contains full keywords and P r G quasi−slca (x) ≥ σ then 9: output x into R; 10: promote the rest keyword distributions of x to its parent xp using CombineProb(x, xp); 11: S1.push(v); 12: while S1 = φ do 13: a new node v ← S1.pop(); 14: if v contains full keywords and P r G quasi−slca (v) ≥ σ then 15: output v into R; 16: promote the rest keyword distributions of v to its parent vp using CombineProb(v, vp); 17: return R; with the given threshold value, we can decide if the node should be output as a qualified answer, skipped as an unqualified result, or cached as a potential result candidate. For example, if the current node's lower bound is larger than or equal to the threshold value, then the node can be output directly without further computation. This is because all its descendants have been checked according to the bottom-up strategy. If its upper bound is lower than the threshold value, then the node can be filtered out. Otherwise, it will be temporarily cached for further checking. Based on different cases, different operations would be applied. Only the nodes identified as potential result candidates need to be computed. Compared with Baseline Algorithm, PI-based algorithm can be accelerated significantly because Baseline Algorithm has to compute the keyword distributions for all nodes. The detailed procedure has been shown in Algorithm 2. Detailed Procedure of PI-based Algorithm In Algorithm 2, Line 1-Line 4 show that the procedures of initiating PI-based Algorithm. We first load the keyword node lists L from KI index and probability node lists P IL from PI index. And then we take the smallest node v from L to initiate a stack S 1 that is set using the dewey codes of v. Another stack S 2 is used to maintain the temporary filtered nodes. After that, the PI-based Algorithm is ready to start. Next, we need to check each node in L in document order. Different from Baseline Algorithm, we only compute the keyword distribution probabilities for a few nodes that are first identified using the lower bound and upper bound in P IL. Consider v be the Algorithm 2 PI-based Algorithm input: a query q = {k1, k2, ..., km} with threshold σ, keyword inverted (KI) index, PI index output: a set R of quasi-SLCA nodes 1: load keyword node lists L = {l1, l2, ..., lm} from KI index; 2: load probability node lists P IL = {P IL1, P IL2, ..., P ILm}; 3: get the smallest Dewey v from L; 4: initiate a stack S1 using v and an empty stack S2; 5: while L = φ do 6: get the next smallest Dewey v from L again; 7: while (S1.top() ≺ not v) do 8: x = S1.pop(); 9: UB(q,x) and LB(q,x) ← ComputeBound(x, {P ILi(x)}); 10: if LB(q,x)≥ σ then 11: output x into R; 12: UpdateBound({va ∈ S1|va ≺ x}, LB(q,x), UB(q,x)); 13: S2.pop(v d ∈ S2|v d ≻ x); 14: else if UB(q,x) ≥ σ > LB(q,x) then 15: P rob(x) ← ComputeProbDist(x, S2); 16: if P rob(x) ≥ σ then 17: output x into R; 18: UpdateBound({va ∈ S1|va ≺ x}, P rob(x)); 19: 21: S2.push(x); 22: S1.push(v); 23: while S1 = φ do 24: a new node v ← S1.pop(); 25: UB (q,v) and 26: process the node v using the same codes in Line 10 -Line 21; 27: return R; S2.pop(v d ∈ S2|v d ≻ x); 20: elseLB(q,v) ← ComputeBound(v, {P ILi(v)}); next smallest node to be processed. We compare it with the node x in stack S 1 . If v is the descendant node of x, then v will be pushed into S 1 and get the next smallest node from L. Otherwise, we pop out x from S 1 and check if it is a qualified quasi-SLCA answer. In Baseline Algorithm, it will compute the keyword distributions of x and combine its remaining distributions and the distribution of its parent based on promotion operations. Different from Baseline Algorithm, PI-based Algorithm will quickly compute the upper bound UB(q,x) and lower bound LB(q,x) using P IL, which is used to differentiate the nodes as qualified nodes -output, unqualified nodes -filter and uncertain nodes -to be further checked. By doing this, only a few nodes need to be computed. Since bound computation is far faster than computation of keyword distribution, lots of run time cost can be saved in PI-based Algorithm. Line 10-Line 21 show the detailed procedures. If the lower bound LB(q,x) is larger than or equal to the given threshold value σ, then x can be output as a qualified quasi-SLCA answer without computation. At this moment, the lower bound LB(q,x) can be taken as the temporary probability of x being a quasi-SLCA result because the exact probability of x is delayed until we need to calculate its exact probability value. Subsequently, the temporary probability value LB(q,x) and the probabilities of x ′ descendant quasi-SLCA results can be used to update the lower/upper bounds of the ancestors of x in stack S 1 based on Equation 11 and Equation 10, respectively. If the lower bound LB(q,x) is lower than σ while the upper bound UB(q,x) is larger than or equal to σ, then we need to compute the keyword distributions of x using the cached descendant nodes in S 2 . Based on the computed probability P rob(x) of x, it can be decided to be output as a qualified answer or filtered as an unqualifed candidate. If the upper bound UB(q,x) is lower than σ, then x will be pushed into S 2 for the possible computation of its ancestors. There are two main functions in PI-based Algorithm. The first one is ComputeProbDist(v, S 2 ) for computing the probability of full keyword distribution of v using the descendant nodes in S 2 . The second is UpdateBound({v a ≺ v|v a ∈ S 1 }, LB(q,v) or Prob(q,v)) for updating the bounds of the nodes to be processed. Function ComputeProbDist() The function ComputeProbDist(v, S 2 ) can be implemented in two ways, Exact Computation or Approximate Computation. Exact Computation is to actually calculate the probability of v being a quasi-SLCA node by scanning all the nodes in the stack S 2 that maintains the descendant nodes of v. The processing strategy is similar to Baseline Algorithm in Section 5.1. In other words, it needs to visit the nodes in S 2 one by one and compute the local keyword distribution of each node, and then promotes the intermediate results to its parent. After all nodes in S 2 are processed, the probability of v will be obtained because it aggregates all the probabilities from its descendant nodes. Approximate Computation is to approximately calculate the probability of v being a quasi-SLCA node based on a partial set of nodes in the stack S 2 that maintains the descendant nodes of v. The approximate computation can be made according to different distribution types, e.g., uniform distributions, piecewise polynomials, poisson distributions, etc. In this work, we consider normal or Gaussian distributions in more detail. As we know Gaussian distribution is considered the most prominent probability distribution in statistics. However, the PDF of Gaussian distribution cannot be applied to PrTKQ over probabilistic XML data directly due to two main challenges. The first challenge is to simulate the continuous distributions using discrete distributions based on the real conditions in order to reduce the approximate errors as much as possible, and the second is to embody the multiple keyword variables in the PDF. Generally, the probability density function of a Gaussian distribution N (µ, σ 2 ) of mean µ and variance σ 2 is: f (x) = 1 √ 2πσ 2 e −(x−µ) 2 /(2σ 2 )(12) Addressing Challenge 1: The density function has a shape of a bell centered in the mean value µ with variance σ 2 . Based on the definition of Gaussian distribution, the Gaussian distribution is often used to describe, at least approximately, measurements that tends to cluster around the mean. Therefore, consider the mean µ be the partial computed probability value of v be a quasi-SLCA node, which guarantees the real probability value will not be significantly different from the probability base that has already been calculated based on promising descendant nodes. The value of the variance σ 2 can be chosen from the range [1-#computed descendant nodes/#total descendant nodes, 1] based on the visited/unvisited descendant nodes in S 2 . This is because the more the descendant nodes are actually computed, the higher the percentage of the values would be drawn within one standard deviation σ away from the mean. Extremely, if all descendant nodes are computed actually, 100% of values can be drawn within one stardard deviation. Therefore, we select and compute a few descendant nodes of v from S 2 , which can contribute relatively higher probabilities to make v a quasi-SLCA node. In this work, we use heuristic method to select a few descendant nodes with the higher probabilities of single keywords in the descendant nodes of v. And then, we take the partially computed probability as the base of the probability density function of a Gaussian distribution. Consider v be a node to be evaluated and UB(q,v) 2 be its current upper bound value. We have, P r G,Gaussian quasi−slca (q, v) = UB(q,v) 0 f (x)dx(13) After substituting Equation 12 into Equation 13, we get, P r G,Gaussian quasi−slca (q, v) = UB(q,v) 0 1 √ 2πσ 2 e − (x−µ) 2 2σ 2 dx (14) Where µ is the partially computed probability, σ 2 is set as 1-#computed descendant nodes/#total descendant nodes. Addressing Challenge 2: To embody all the keyword variables in the PDF, we introduce the joint/conditional Gaussian distribution based on the work in [26]. Assume a PrTKQ contains two keywords k x and k y . We have the conditional PDF as follows. after substituting Equation 12 into Equation 15, we get f Y |X (y|x) = 1 2π(1 − ρ 2 )σ 2 Y e − [(y−µ Y )−(ρ( σ Y σ X )(x−µ X ))] 2 2σ 2 Y (15) Since f (x, y) = f X (x) * f Y |X (y|x),f (x, y) = 1 2πσX σY √ 1−ρ 2 e − (x−µ X ) 2 2σ 2 X − [(y−µ Y )−(ρ( σ Y σ X )(x−µ X ))] 2 2σ 2 Y(16) If we make an assumption that x and y are independent keyword variables i.e., ρ = 0, and assume µ X = µ Y = µ and σ X = σ Y = σ, then we have f (x, y) = 1 2πσ 2 e − (x−µ) 2 +(y−µ) 2 2σ 2(17) Therefore, Equation 17 can be easily extended to multiple keyword variables that are assumed as independent. We can compute the probability of v w.r.t. a PrTKQ {k 1 , k 2 , ..., k t }. P r G,Gaussian quasi−slca (q, v) = UB(q,v) 0 ... UB(q,v) 0 e − (x 1 −µ) 2 +...+(x t −µ) 2 2σ 2 (2π) t/2 σ t dx 1 ...dx t(18) Where µ is the partially computed probability, σ 2 is set as 1-#computed descendant nodes/#total descendant nodes. In the experiments, we call Matlab from Java to calculate Equation 18. The estimated results are used to show the comparison between the actual computation and approximation computation. The results verify the usability of Gaussian distribution to measure the probability. Function UpdateBound() For each ancestor node v a ∈ S 1 (v a ≺ v), we need to update the upper bounds and lower bounds using Function UpdateBound() based on Equation 10 and Equation 11, respectively. To guarantee the completeness of the answer set, the parameters of the function may be different based on conditions. For example, if LB(q,v) is larger than or equal to the threshold value σ as shown in Algorithm 2: Line 12, then the probability value LB(q,v) is used to update the upper bounds of v ′ ancestors while the probability value UB(q,v) is used to update the lower bounds of v ′ ancestors; if LB(q,v) is smaller than σ and UB(q,v) is larger than or equal to σ as shown in Algorithm 2: Line 18, the actual or approximate probability value P rob(v) computed by Function ComputeProbDist(v, S 2 ) will be utilized to update the upper/lower bounds of v ′ ancestors together. Here, we use two hashmaps to implement Function UpdateBound(). For a node, one hashmap is used to cache the dewey of the node as a key, and the lower/upper bounds as a value where the bounds are computed based on PI index. Another hashmap is used to record the probability that the descendants of the node having been identified as qualified quasi-SLCA answers. When a node is coming, we can quickly get the updated lower/upper bounds based on the two hashmaps. EXPERIMENTAL STUDIES We conduct extensive experiments to test the performance of our algorithms: Baseline Algorithm (BA); PI-based Exact-computation Algorithm (PIEA) that implements Function ComputeProbDist() by exactly computing the probability distributions of the keyword matched nodes; and PI-based Approximatecomputation Algorithm (PIAA) that makes approximated computation based on the Gaussian distribution of keywords while still exactly computing the probability distributions of the keyword matched nodes that have the higher probabilities. All these algorithms were implemented in Java and run on a 3.0GHz Intel Pentium 4 machine with 2GB RAM running Windows 7. Dataset and Queries We use two real datasets, DBLP [27] and Mondial [28], and a synthetic XML benchmark dataset XMark [29] for testing the proposed algorithms. For XMark, we also generate four datasets with different sizes. The three types of datasets are selected based on their features. DBLP is a relatively shallow dataset of large size; Modial is a deep and complex, but small dataset; XMark is a balanced dataset with varied depth, complex structure and varied size. Therefore, they are chosen as test datasets. For each XML dataset used, we generate the corresponding probabilistic XML tree, using the same method as used in [12]. We visit the nodes in the original XML tree in pre-order way. We first set the random ratio of IND:MUX:Ordinary as 3:3:4. For each node v visited, we randomly generate some distributional nodes with "IND" or "MUX" types as children of v. Then, for the original children of v, we choose some of them as the children of the new generated distributional nodes and assign random probability distributions to these children with the restriction that the sum of them for a MUX node is no greater than 1. The generated datasets are described in Table 1. And we select terms and construct a set of keyword queries to be tested for each dataset. Due to the limited space, we only show six of these queries for each dataset. For each different sets of queries, the terms in the first two queries have small size of keyword matched nodes; the terms of the middle two queries relate to a medium size of keyword matched nodes; the terms of the last queries are based on the computation of a larger number of keyword matched nodes. Figure 5 shows the experimental results when we run the 18 queries over the selected three datasets where X represents the queries over 20MB XMark dataset, M represents the queries over Mondial dataset, D represents the queries over DBLP dataset and I represents the queries over INEX dataset. And the required threshold value σ is set as 0.3. From the results, we can find that compared with the BA algorithm, most of time the PIEA algorithm can reduce the response time by nearly 40% using the pruning techniques based on the updated lower/upper bounds. The PIAA algorithm can further improve the time efficiency by about 20% with the assumption of probability distribution of keywords. For X1, X2 and M 1 , M 2 , the response time of BA algorithm is approaching to the time cost of the other two algorithms. Especially for query X2, PIEA algorithm is overwhelmed by BA algorithm. This is because both the number of keyword-matched nodes and the size of answer sets are smaller than the other queries. From the four figures on the left side of Figure 5, we find that the scalability of PIEA and PIAA algorithms is much better than that of BA algorithm by testing the queries with different sizes of answer sets. To measure the precision and recall of PIAA algorithm, we utilize the P&R equations in information retrieval area as follows. P recision = |RBA∩RP IAA | |RP IAA| ; Recall = |RBA∩RP IAA| |RBA| Because PIEA algorithm can find the same results with BA algorithm by exactly computing the required probability distributions, Figure 5 only demonstrates the precision and recall of PIAA algorithm for different queries over each dataset. From the experimental results, we find that the precision and recall can reach up to at least 0.7 for XMark, 0.6 for Mondial, 0.7 for DBLP, and 0.66 for INEX, respectively. Sometimes, it can be up to 0.9 at most, e.g., X1, X2, M1, M2, I1, I2, etc. Comparing all the tested queries, we can get a general conclusion that the precision and recall will be decreased with the increase of potential result size. However, from the experiments, they will not be lower than 0.6 because (1) the results with higher probabilities are exactly selected and computed, which does not need to depend on the Gaussian assumption; (2) the rest minor results are estimated by using Gaussian assumption over the keyword distributions that have been excluded by the results with higher probabilities. In other words, PIAA strategy can return the percentage (≥0.6) of significant results, but may underestimate the minor results. Varying Threshold Values To test the adaptability of the proposed algorithms to threshold query, we test the changes of response time and precision&recall with the increase of threshold value. Figure 6 shows the experimental results when the threshold value varies from 0.2 to 0.7 for queries X5, M5, D5 and I5. The left four figures in Figure 6 show that PIEA and PIAA algorithms can overwhelm BA algorithm greatly with the increase of threshold value. This is because BA algorithm has to scan and compute all the relevant nodes while PIEA and PIAA algorithms can skip more nodes when the threshold value becomes large. However, when the threshold value is up to 0.5, the change of the time cost will be smooth because once a quasi-SLCA node is identified, its ancestor nodes can be skipped definitely, which is true for the threshold values larger than 0.5. From the right four figures in Figure 6, we can find that the precision and recall will be affected by the change of threshold values. When the threshold value reaches up to 0.5, the precision and recall can be up to 0.8 at least. On the contrary, if the threshold value is lower than 0.2, the precision and recall would be decreased to 0.5 based on the selected datasets. Varying Probabilistic Document Size We firstly take XMark dataset as an example to test the performance of the three algorithms when we increase the document size. We test all the six queries of XMark dataset, but in this paper, we only show the results of the query X3 where the threshold value is specified as 0.3. From Figure 7(a), we can see that the response time of all the three algorithms will be increased when the document size increases from 10MB to 80MB. However, the increase of PIEA and PIAA algorithms is much slower. Particularly, PIAA just changes a bit. The comparison illustrates that PIEA and PIAA algorithms can obtain much better performance than BA algorithm. In addition, all algorithms show linear degradation, i.e., they have the similar scalability. Secondly, we evaluate the precision and recall of PIAA algorithm using a variant of F-measure that aggregates the precision and recall of all queries together. F − measure = 2 * P (qi) * R(qi) P (qi)+R(qi) Where P (q i ) = 6 1 (P (q i ))/6, and R(q i ) = 6 1 (R(q i ))/6. To evaluate the F-measure of PIAA algorithm, we test 24 queries with different threshold values: 0.3, 0.5 and 0.7. From the results in Figure 7(b), we can find that the F-measure can be over 0.75 for all the four datasets. RELATED WORK The topic of probabilistic XML has been studied recently. Many models have been proposed, together with structured query evaluations. Nierman et al. [7] first introduced a probabilistic XML model, ProTDB, with the probabilistic types INDindependent and MUXmutually-exclusive. Hung et al. [8] modeled the probabilistic XML as directed acyclic graphs, supporting arbitrary distributions over sets of children. Keulen et al. [9] used a probabilistic tree approach for data integration where its probability and possibility nodes are similar to MUX and IND, respectively. Cohen et al. [30] incorporated a set of constraints to express more complex dependencies among the probabilistic data. They also proposed efficient algorithms to solve the constraint-satisfaction, query evaluation, and sampling problem under a set of constraints. In [12], Kimelfeld et al. summarized and extended the probabilistic XML models previously proposed, the expressiveness and tractability of queries on different models are discussed with the consideration of IND and MUX. [11] studied the problem of evaluating twig queries over probabilistic XML that may return incomplete or partial answers with respect to a probability threshold to users. [13] proposed and addressed the problem of ranking top-k probabilities of answers of a twig query. All the above work focused on the discussions of probabilistic XML data model and/or structured XML query, e.g., twig query. The most closely related work is [23] that proposed two algorithms to answer top-k keyword queries over probabilistic XML data. However, compared with [23], in this work we propose a probabilistic inverted index that can be used to efficiently answer threshold keyword queries by reducing the computational cost of unqualified nodes. In addition, we also take into account the relaxation (i.e., quasi-SLCA) of results for keyword search w.r.t. a threshold value while [23] focused on the strict SLCA semantics of results. There are some other work to discuss probabilistic index for query evaluation and/or data management. Although [31] discussed probabilistic inverted index as ours, its data model is relational in which each tuple is associated with a probability value and all tuples are assumed independent. In our work, we built the probabilistic inverted index based on probabilistic XML data model with IND and MUX semantics. Another difference is that we answer keyword query while [31] processes equality query. Another work discussing probabilistic index is [32] that first generates possible worlds and then cluster them based on probability values with a limited distance. The problem is that generating all possible worlds is very timeconsuming in XML data. In our work, we avoided the generation of possible worlds. CONCLUSIONS In this work, we first proposed and investigated the problem of finding quasi-SLCA for PrTKQs over probabilistic XML data. And then we designed a PI index and analyzed the pruning features of PI index. Based on the lower and upper bounds computed from PI index, the proposed PI-based algorithm can quickly identify the qualified results and filter the unqualified ones. Our experimental results demonstrated the comparison of Baseline algorithm, PIbased Exact-computation Algorithm (PIEA) and PIbased Approximate-computation Algorithm (PIAA), which verified our motivation and approaches. Fig. 1 . 1A probabilistic XML data tree Fig. 2 . 2A small PrXML and its possible worlds Fig. 3 . 3PI index and Lower/Upper Bound for a query {k 1 , k 2 } over the given PrXML Figure 3 3shows the lower/upper bounds of each node in Property 2 : 2[Lower Bound Usage] The nodes v can be returned as required results if we have LB(q, v) ≥ σ and U B(q, v d ) < σ where v d is any child or descendant node of v. Example 9 : 9Consider a 4 that has been computed and its probability is 0.44. Given threshold σ (=0.4), a 4 is returned as a quasi-SLCA result. Consequently, we can update the lower/upper bound values of its ancestor a 2 , i.e., U B ′ ({k 1 , k 2 }, a 2 ) = 0.65 -1 + (1 -0.44) = 0.21 and LB ′ ({k 1 , k 2 }, a 2 ) = 0.595 -0.44 = 0.155. Fig. 5 . 5Evaluation of Keyword Queries over XMark20M, Mondial, DBLP, INEX where σ=0.3 Fig. 6 . 6Time, Precision and Recall vs. Varied Threshold Fig. 7 . 7Response Time and F-Measure for different datasets • JianxinLi, Chengfei Liu and Rui Zhou are with the Faculty of Information & Technology, Swinburne University of Technology, Australia. {jianxinli, cliu, rzhou}@swin.edu.au• Jeffrey Xu Yu is with the Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, China. [email protected] TABLE 1 1Properties of PrXML dataPIAA-Precision PIAA-RecallID name size #IND #MUX #Ordinary Doc1 XMark 10M 26k 26k 145k Doc2 20M 54k 52k 200k Doc3 40M 98k 100k 606k Doc4 80M 329k 368k 1M Doc5 Modial 1.2M 8k 9k 20k Doc6 DBLP 136M 759k 589k 3M Doc7 INEX 5,898M 13M 10M 52M 6.2 Varying Keyword Queries 0 50 100 150 200 250 300 X1 X2 X3 X4 X5 X6 Keyword Queries Response Time ms BA PIEA PIAA (a) XMark 0 0.2 0.4 0.6 0.8 1 X1 X2 X3 X4 X5 X6 Keyword Queries Precision & Recall . The symbol v is used to represent a node's name or a node's id without confusions in the following sections. Here, v is the id of the node v . Note that UB(q,v) has been updated if v has descendant nodes that are qualified answers, i.e., it minus the probability contributions of the qualified answers. Managing uncertainty in sensor database. R Cheng, S Prabhakar, SIGMOD Record. 324R. Cheng and S. Prabhakar, "Managing uncertainty in sensor database," SIGMOD Record, vol. 32, no. 4, pp. 41-46, 2003. Efficient indexing methods for probabilistic threshold queries over uncertain data. R Cheng, Y Xia, S Prabhakar, R Shah, J S Vitter, VLDB. R. Cheng, Y. Xia, S. Prabhakar, R. Shah, and J. S. Vitter, "Efficient indexing methods for probabilistic threshold queries over uncertain data," in VLDB, 2004, pp. 876-887. Indexing multi-dimensional uncertain data with arbitrary probability density functions. Y Tao, R Cheng, X Xiao, W K Ngai, B Kao, S Prabhakar, VLDB. Y. Tao, R. Cheng, X. Xiao, W. K. Ngai, B. Kao, and S. Prabhakar, "Indexing multi-dimensional uncertain data with arbitrary probability density functions," in VLDB, 2005, pp. 922-933. Ranking queries on uncertain data: a probabilistic threshold approach. M Hua, J Pei, W Zhang, X Lin, SIGMOD Conference. M. Hua, J. Pei, W. Zhang, and X. Lin, "Ranking queries on uncertain data: a probabilistic threshold approach," in SIGMOD Conference, 2008, pp. 673-686. Threshold query optimization for uncertain data. Y Qi, R Jain, S Singh, S Prabhakar, SIGMOD Conference. Y. Qi, R. Jain, S. Singh, and S. Prabhakar, "Threshold query optimization for uncertain data," in SIGMOD Conference, 2010, pp. 315-326. On the complexity of managing probabilistic xml data. P Senellart, S , PODS. P. Senellart and S. Abiteboul, "On the complexity of managing probabilistic xml data," in PODS, 2007, pp. 283-292. ProTDB: Probabilistic data in xml. A Nierman, H V Jagadish, VLDB. A. Nierman and H. V. Jagadish, "ProTDB: Probabilistic data in xml," in VLDB, 2002, pp. 646-657. Pxml: A probabilistic semistructured data model and algebra. E Hung, L Getoor, V S Subrahmanian, ICDE. 467E. Hung, L. Getoor, and V. S. Subrahmanian, "Pxml: A proba- bilistic semistructured data model and algebra," in ICDE, 2003, pp. 467-. A probabilistic xml approach to data integration. M Van Keulen, A De Keijzer, W Alink, ICDE. M. van Keulen, A. de Keijzer, and W. Alink, "A probabilistic xml approach to data integration," in ICDE, 2005, pp. 459-470. Querying and updating probabilistic information in xml. S Abiteboul, P Senellart, EDBT. S. Abiteboul and P. Senellart, "Querying and updating prob- abilistic information in xml," in EDBT, 2006, pp. 1059-1068. Matching twigs in probabilistic xml. B Kimelfeld, Y Sagiv, VLDB. B. Kimelfeld and Y. Sagiv, "Matching twigs in probabilistic xml," in VLDB, 2007, pp. 27-38. Query efficiency in probabilistic xml models. B Kimelfeld, Y Kosharovsky, Y Sagiv, SIGMOD Conference. B. Kimelfeld, Y. Kosharovsky, and Y. Sagiv, "Query efficiency in probabilistic xml models," in SIGMOD Conference, 2008, pp. 701-714. Query ranking in probabilistic xml data. L Chang, J X Yu, L Qin, EDBT. L. Chang, J. X. Yu, and L. Qin, "Query ranking in probabilistic xml data," in EDBT, 2009, pp. 156-167. Xirql: A query language for information retrieval in xml documents. N Fuhr, K Großjohann, SIGIR. N. Fuhr and K. Großjohann, "Xirql: A query language for information retrieval in xml documents," in SIGIR, 2001, pp. 172-180. XSEarch: A Semantic Search Engine for XML. S Cohen, J Mamou, Y Kanza, Y Sagiv, VLDB. S. Cohen, J. Mamou, Y. Kanza, and Y. Sagiv, "XSEarch: A Semantic Search Engine for XML," in VLDB, 2003, pp. 45-56. XRANK: Ranked Keyword Search over XML Documents. L Guo, F Shao, C Botev, J Shanmugasundaram, SIGMOD Conference. L. Guo, F. Shao, C. Botev, and J. Shanmugasundaram, "XRANK: Ranked Keyword Search over XML Documents," in SIGMOD Conference, 2003, pp. 16-27. Lca-based selection for xml document collections. G Koloniari, E Pitoura, WWWG. Koloniari and E. Pitoura, "Lca-based selection for xml document collections," in WWW, 2010, pp. 511-520. Effective keyword search for valuable lcas over xml documents. G Li, J Feng, J Wang, L Zhou, CIKM. G. Li, J. Feng, J. Wang, and L. Zhou, "Effective keyword search for valuable lcas over xml documents," in CIKM, 2007, pp. 31- 40. Schema-Free XQuery. Y Li, C Yu, H V Jagadish, VLDB. Y. Li, C. Yu, and H. V. Jagadish, "Schema-Free XQuery," in VLDB, 2004, pp. 72-83. Efficient Keyword Search for Smallest LCAs in XML Databases. Y Xu, Y Papakonstantinou, SIGMOD Conference. Y. Xu and Y. Papakonstantinou, "Efficient Keyword Search for Smallest LCAs in XML Databases," in SIGMOD Conference, 2005, pp. 537-538. Fast elca computation for keyword queries on xml data. R Zhou, C Liu, J Li, EDBT. R. Zhou, C. Liu, and J. Li, "Fast elca computation for keyword queries on xml data," in EDBT, 2010, pp. 549-560. Multiway slca-based keyword search in xml data. C Sun, C Y Chan, A K Goenka, WWWC. Sun, C. Y. Chan, and A. K. Goenka, "Multiway slca-based keyword search in xml data," in WWW, 2007, pp. 1043-1052. Top-k keyword search over probabilistic xml data. J Li, C Liu, R Zhou, W Wang, ICDE. J. Li, C. Liu, R. Zhou, and W. Wang, "Top-k keyword search over probabilistic xml data," in ICDE, 2011, pp. 673-684. Effective xml keyword search with relevance oriented ranking. Z Bao, T W Ling, B Chen, J Lu, ICDE. Z. Bao, T. W. Ling, B. Chen, and J. Lu, "Effective xml keyword search with relevance oriented ranking," in ICDE, 2009, pp. 517-528. Suggestion of promising result types for xml keyword search. J Li, C Liu, R Zhou, W Wang, EDBT. J. Li, C. Liu, R. Zhou, and W. Wang, "Suggestion of promising result types for xml keyword search," in EDBT, 2010, pp. 561- 572. Introduction to probability. D P Bertsekas, J N Tsitsiklis, Athena Scientific. 41st editionD. P. Bertsekas and J. N. Tsitsiklis, "Introduction to probability, 1st edition," Athena Scientific, vol. 4.7, 2002. [27] "http://dblp.uni-trier.de/xml/." [28] "http://www.dbis.informatik.uni-goettingen.de/mondial." [29] "http://monetdb.cwi.nl/xml/." Incorporating constraints in probabilistic xml. S Cohen, B Kimelfeld, Y Sagiv, ACM Trans. Database Syst. 343S. Cohen, B. Kimelfeld, and Y. Sagiv, "Incorporating con- straints in probabilistic xml," ACM Trans. Database Syst., vol. 34, no. 3, 2009. Indexing uncertain categorical data. S Singh, C Mayfield, S Prabhakar, R Shah, S E Hambrusch, ICDE. S. Singh, C. Mayfield, S. Prabhakar, R. Shah, and S. E. Ham- brusch, "Indexing uncertain categorical data," in ICDE, 2007, pp. 616-625. Clustering uncertain data with possible worlds. P B Volk, F Rosenthal, M Hahmann, D Habich, W Lehner, ICDE. P. B. Volk, F. Rosenthal, M. Hahmann, D. Habich, and W. Lehner, "Clustering uncertain data with possible worlds," in ICDE, 2009, pp. 1625-1632.

The paper presents a method to compute the Jacobi's elliptic function sn on the period parallelogram. For fixed m it requires first to compute the complete elliptic integrals K = K(m) and K = K(1 − m). The Newton method is used to compute sn(z, m), when z ∈ [0, K] ∪ [0, iK ). The computation in any other point does not require the usage of any numerical procedure, it is done only with the help of the properties of sn.2010 Mathematics Subject Classification: 65D20, 33F05.

1803.05017

3db972d9cb24c80afd2c475ea81f0d4a6f8867a9

On Computing Jacobi's Elliptic Function sn E Scheiber On Computing Jacobi's Elliptic Function sn elliptic functionselliptic integralsarithmetic-geometric mean The paper presents a method to compute the Jacobi's elliptic function sn on the period parallelogram. For fixed m it requires first to compute the complete elliptic integrals K = K(m) and K = K(1 − m). The Newton method is used to compute sn(z, m), when z ∈ [0, K] ∪ [0, iK ). The computation in any other point does not require the usage of any numerical procedure, it is done only with the help of the properties of sn.2010 Mathematics Subject Classification: 65D20, 33F05. Introduction The paper presents a method to compute the Jacobi's elliptic function sn on the period parallelogram. For fixed m ∈ (0, 1) it requires first to compute the complete elliptic integrals K = K(m) and K = K(1 − m). The function to compute the first complete elliptic integral uses the arithmetic-geometric mean, as a consequence of Gauss's theorem. The Newton method to solve a nonlinear algebraic equation is used to compute sn(z, m), when z ∈ [0, K] ∪ [0, iK ). The computation in any other point does not require the usage of any numerical procedure, it is done only with the help of the properties of sn and its values on some points from [0, K] ∪ i[0, K ). The validity of the method is exemplified with the help of a Scilab application. The obtained results are very good approximations of the values given by the corresponding functions from Scilab and Mathematica. The computation of the elliptic integrals and of the elliptic functions were studied in many papers, e.g. [2], [3], [5], as well as the included bibliography. Incomplete elliptic integral of first kind The following incomplete and complete elliptic integrals of first kind are defined respectively by, [9], F (φ, m) = sin φ 0 dt (1 − t 2 )(1 − mt 2 ) = φ 0 dθ 1 − m sin 2 θ and K(m) = 1 0 dt (1 − t 2 )(1 − mt 2 ) = F ( π 2 , m). In order to compute F (φ, m) we recall a result established by Carl Friedrich GAUSS (1777-1855) in 1799, [7], [1]: Theorem 2. 1 If a and b are positive reals and M (a, b) is their the arithmeticgeometric mean then 1 M (a, b) = 2 π π 2 0 dx a 2 cos 2 x + b 2 sin 2 x .(1) For a > b > 0 and 0 ≤ φ ≤ π 2 we shall take care of the integral I(a, b, φ) = φ 0 dx a 2 cos 2 x + b 2 sin 2 x = 1 a φ 0 dx 1 − 1 − b 2 a 2 sin 2 x = (2) = 1 a F φ, 1 − b 2 a 2 and I(a, b, π 2 ) = 1 a K 1 − b 2 a 2 . Thus, the equality (2) may be rewritten as 1 M (a,b) = 2 aπ K 1 − b 2 a 2 or K 1 − b 2 a 2 = π 2 1 1 a M (a, b) = π 2 1 M (1, b a ) . As in [7], for I(a, b, φ) the changing of variables sin x = 2a sin ϕ a + b + (a − b) sin 2 ϕ leads to the sequence I(a, b, φ) def = I 0 (a 0 , b 0 , φ 0 ) = I 1 (a 1 , b 1 , φ 1 ) = I 2 (a 2 , b 2 , φ 2 ) = . . .(3) where I k (a k , b k , φ k ) = φ k 0 dϕ a 2 k cos 2 ϕ + b 2 k sin 2 ϕ and the upper integration limits are generated by the sequence sin φ k−1 = 2a k−1 sin φ k a k−1 + b k−1 + (a k−1 − b k−1 ) sin 2 φ k . The sequence (sin φ k ) k∈N is decreasing and consequently the sequence (φ k ) k∈N is convergent. It results that sin φ k = a k−1 − a 2 k−1 cos 2 φ k−1 + b 2 k−1 sin 2 φ k−1 (a k−1 − b k−1 ) sin φ k−1 = y k (4) φ k = arcsin y k . From (3) it results I(a, b, φ) = lim k→∞ I k (a k , b k , φ k ) = φ ∞ M (a, b) , with φ ∞ = lim k→∞ φ k . Using (2) we get I(a, b, φ) = 1 a F φ, 1 − b 2 a 2 = φ ∞ M (a, b) and consequently F φ, 1 − b 2 a 2 = aφ ∞ M (a, b) = φ ∞ 1 a M (a, b) = φ ∞ M (1, b a ) . Denoting m = 1 − b 2 a 2 , (a > b > 0 ⇔ 0 < m < 1), the above equation becomes F (φ, m) = φ ∞ M (1, √ 1 − m) .(5) Therefore the computation of F (φ, m) returns to generate iteratively the sequences (a k ) k , (b k ) k , (φ k ) k until a stopping condition is fulfilled. The initial values are a 0 = 1, b 0 = √ 1 − m and φ 0 = φ. For a 0 = 1, instead of the sequences (a k ) k , (b k ) k we may compute the sequences, [8], s 0 = b 0 s k+1 = 2 √ s k 1+s k p 0 = 1 2 (1 + s 0 ) p k+1 = 1 2 (1 + s k )p k . Then lim k→∞ p k = M (1, b 0 ). If φ = π 2 then φ ∞ = π 2 and we retrieve K(m) = π 2M (1, √ 1 − m) .(6) From a practical point of view and as a drawback the method is not applicable when φ is small, e.g. 0 < φ < 10 −5 . The cause is the presence of the factor sin φ k−1 in the denominator in (4). In this case, from the Maclaurin series expansion of F (φ, m) we get F (φ, m) ≈ φ − m 6 φ 3 . The Jacobi elliptic function sn The Jacobi elliptic function sn(z, m) may be defined by the equation, [1], z = sn(z,m) 0 dt (1 − t 2 )(1 − mt 2 ) .(7) Throughout this paper the variable m is fixed and we use the shorter notation sn(z), omitting m. We shall use the following Jacobi elliptic functions, too cn 2 (z) = 1 − sn 2 (z), dn 2 (z, m) = 1 − m sn 2 (z). Again we shall use the shorter notation dn(z). If K = K(m) and K = K(1 − m) then the parallelogram period is the rectangle D = [0, 4K) + i[0, 2K ) and the points K i and 2K + iK are poles of first order, [8]. The following properties of the function sn will be used, [8], [6]: • sn(−z) = −sn(z)(8) • sn(x + y) = sn(x)cn(y)dn(y) + sn(y)cn(x)dn(x) 1 − m sn 2 (x)sn 2 (y) cn(x + y) = cn(x)cn(y) − sn(x)sn(y)dn(x)dn(y) 1 − m sn 2 (x)sn 2 (y) dn(x + y) = dn(x)dn(y) − m sn(x)sn(y)cn(x)cn(y) 1 − m sn 2 (x)sn 2 (y) • Because sn(K) = 1, cn(K) = 0, dn(K) = √ 1 − m from the above equalities it results sn(K ± z) = cn(z) dn(z) (12) cn(K + z) = − √ 1 − m sn(z) dn(z) (13) dn(K + z) = √ 1 − m dn(z)(14) • Knowing that sn(2K) = 0, cn(2K) = −1, dn(2K) = 1, from (9) it results sn(2K ± z) = ∓sn(z) • Knowing that sn(K + iK ) = 1 √ m , dn(K + iK ) = 0 from (9) it results sn(z + K + iK ) = 1 √ m dn(z) cn(z) .(16) The computation of sn(z) depends on the position of z in D and we suppose that we know K and K . • If z ∈ [0, K] or z ∈ i[0, K ) then sn(z) will be the solution u of the equation Φ(u) = u 0 dt (1 − t 2 )(1 − mt 2 ) − z = 0.(17) • Otherwise and excepting the poles the value of cn(z) will be computed using the properties of the function sn and its values on some points from [0, K] ∪ i[0, K ), without any other additional numerical procedure. Computing sn(z) in the segment [0, 4K) The following cases arise: 1. z ∈ [0, K]. Equation (17) may be solved using the Newton method with the iterations u k+1 = u k − Φ(u k ) Φ (u k ) = u k − (F (arcsin u k , m) − z) (1 − u 2 k )(1 − mu 2 k ) The linear interpolation between sn(z, 0) and sn(z, 1) gives the initial approximation u 0 = (1 − m) sin z + m tanh z. If z is small enough the method is rapidly converging and for z near K we set z = K − z and after computing sn(z ) = w as was described above, using (12) we have sn(z) = sn(K − z ) = cn(z ) dn(z ) = 1 − w 2 1 − m w 2 . 2. z ∈ (K, 4K). Let be z =    2K − z if z ∈ (K, 2K] z − 2K if z ∈ (2K, 3K] 4K − z if z ∈ (3K, 4K) . After computing sn(z ) = w , z ∈ [0, K], we have sn(z) = w if z ∈ (K, 2K] −w if z ∈ (2K, 4K) . Computing sn(z) for z ∈ i(0, 2K ) \ {iK }. The following cases arise: 1. z ∈ i[0, K ). Writing z = iy, y ∈ [0, K ), we are looking for the solution of the equation (17) in the form u = iv, v ∈ R. After the change of variable t = is there is obtained the equation Ψ(v) = v 0 ds (1 + s 2 )(1 + m s 2 ) − y = 0.(18) According to the Newton method, the iterations are v k+1 = v k − Ψ(v k ) Ψ (v k ) = = v k − v k 0 ds (1 + s 2 )(1 + m s 2 ) − y (1 + v 2 k )(1 + m v 2 k ) starting with v 0 = y. The above integral is computed using a quatrature procedure. 2. z ∈ i(K , 2K ]. Let be z = z − iK = iy with y ∈ (0, K ). From (16) we have sn(z) = sn(iK + iy ) = 1 √ m dn(iy − K) cn(iy − K) . After using (13) and (14) it results sn(z) = 1 √ m sn(z ) . Computing sn(z) in the rectangle period We describe here how to compute sn(z) when z belongs to the rectangle period excepting the poles and the lower and the left sides. Let z = x + iy such that x ∈ (0, 4K) and y ∈ (0, 2K ). The following cases arise: 1. y = K . Using (9) we have sn(z) = sn(x + iy) = sn(x)cn(iy)dn(iy) + sn(iy)cn(x)dn(x) 1 − m sn 2 (x)sn 2 (iy) . sn(x), sn(iy) are computed as was presented above and then compute cn(x), cn(iy), dn(x), dn(iy). It must be taken into account that if x ∈ (K, 3K) then cn(x) = − 1 − sn 2 (x). 2. y = K . We deduce through (14) sn(z) = sn(K + iK + x − K) = 1 √ m dn(x − K) cn(x − K) . As above, applying (13) and (14) we obtain sn(z) = 1 √ m sn(x) . On poles, z ∈ {iK , 2K + iK }, we set sn(z) = ∞. In his way we have computed the value of sn(z) for any z ∈ D. Numerical results Numeric computing softwares contains functions to compute elliptical integrals and elliptical functions. We recall some methods in Mathematica and Scilab in Table 1. We developed a Scilab program based on the method presented in this paper. The values K and K were computed using (6). Because the numbers have a floating point representation two numbers are considered to be equal if their distance is less than a tolerance. Meaning Function signature Mathematica F (φ, m) EllipticF[φ, m] K(m) EllipticK[m] sn(z, m) JacobiSN[z, m] cn(z, m) JacobiCN[z, m] dn(z, m) JacobiDN[z, m] Scilab F (φ, m) delip(φ, √ m) sn(z, m) %sn(z, m) Some results are given in Table 2: Results obtained using the presented method. The 3D image of the modules of the function sn(z, 0.81) computed on D is given in Figure 1. Finally we show the visualization of the complex function, using the method presented in [4]. In a point the value of the function is represented by a color obtained projecting that value into the colors cube. The procedure is based on the stereographic projection. The Figure 2 is given for calibration, representing the visualization of the identity function. The Figure 3 contains the visualization of sn(z, 0.81), z ∈ D. The zeros are colored in black while the poles are colored in white. Indeed, if z ∈ (K, 2K] then sn(z) = sn(2K − z ) = sn(z ) = w ; if z ∈ (2K, 3K] then sn(z) = sn(2K + z ) = −sn(z ) = −w and if z ∈ (3K, 4K) then sn(z) = sn(4K − z ) = sn(−z ) = −sn(z ) = −w . Figure 1 : 1For m = 0.81 the 3D image of the modules of sn. Figure 2 :Figure 3 : 23Function w → w, w ∈ [−5, 5] + i[−5, 5] Function w = sn(z, 0.81), z ∈ D. Table 1 : 1Elliptical function in Mathematica and Scilab. Table 2 . 2The function JacobiSN from Mathematica gives similar values (excepting the poles).z sn(z) computed %sn(z, m) Error |sn(z) − %sn(z, m)| m = 0.81 K=2.2805491 delip(1, 0.9) = 2.2805491 4.441D − 16 K'=1.6546167 delip(1, √ 0.19) = 1.6546167 2.220D − 16 0.5K 0.8345252 0.8345252 1.024D-09 1.4K 0.9038225 0.9038225 6.602D-10 2.7K -0.9501563 -0.9501563 7.363D-10 3.3K -0.9501563 -0.9501563 7.363D-10 i0.6K 1.4511449i 1.4511449i 1.554D-15 i1.3K -2.0696167i -2.0696167i 1.332D-15 0.8K + i0.3K 1.0085488 + 0.0420829i 1.0085488 + 0.0420829i 5.812D-10 0.5K + i1.7K 0.9048397 -0.1679796i 0.9048397 -0.1679796i 1.129D-09 1.3K + i1.7K 0.9892195 + 0.071665i 0.9892195 + 0.071665i 2.701D-10 2.5K + i0.4K -0.9592212 -0.2093038i -0.9592212 -0.2093038i 8.724D-10 3.6K + i0.4K; -0.8951883 + 0.3091877i -0.8951883 + 0.3091877i 1.656D-09 3.6K + i1.7K -0.8233279 -0.2419397i -0.8233279 -0.2419397i 1.519D-09 0.5K + iK 1.3314291 1.3314291 1.634D-09 2.5K + iK -1.3314291 -1.3314291 1.183D-09 K + iK 1.1111111 1.1111111 2.220D-16 K 1. 1. 2.220D-16 iK Nan + Infi 1.633D+16i Nan 2K + iK Nan + Infi -4.211D+15 + 1.170D+15i Nan . J M Borwein, B Borwein P, Pi, John Wiley & SonsNew YorkBORWEIN J.M., BORWEIN P.B., Pi and the AGM. John Wiley & Sons, New York, 1986. Numerical computation of inverse complete elliptic integrals of first and second kinds. Fukushima T, J. Computation and Applied Mathematics. 249FUKUSHIMA T., Numerical computation of inverse complete elliptic integrals of first and second kinds. J. Computation and Applied Math- ematics, 249 (2013), 37-50. Fast computation of complete elliptic integrals and Jacobian elliptic functions. Fukushima T, 10.1007/s10569-009-9228-zCelest Mech Dyn Astr. 105305FUKUSHIMA T. Fast computation of complete elliptic integrals and Jacobian elliptic functions. Celest Mech Dyn Astr (2009) 105: 305. https://doi.org/10.1007/s10569-009-9228-z. Visualizing Complex Functions. J L Richardson, RICHARDSON J.L., Visualizing Complex Functions. 2003, http://web.archive.org/web/20030802162645/http://physics. hallym.ac.kr/education/. The derivation of algorithms to compute elliptic integrals of the first and second kind by Landen transformation. Rösch N, 10.1590/S1982-21702011000100001Boletin de Ciências Geodésicas. 171RÖSCH N., The derivation of algorithms to compute elliptic integrals of the first and second kind by Landen transformation. Boletin de Ciências Geodésicas (Online), 17 (2011), no.1, http://dx.doi.org/10.1590/ S1982-21702011000100001. Applications of Elliptic Functions in Classical and Algebraic Geometry. J Snape, Collingwood College, University of DurhamMaster thesisSNAPE J., Applications of Elliptic Functions in Classical and Algebraic Geometry. Collingwood College, University of Durham, Master thesis, 2000. https://wwwx.cs.unc.edu/~snape/publications/mmath/. Elliptic functions: Introduction course. V G Tkachev, TKACHEV V.G., Elliptic functions: Introduction course. http:// users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf. A Course of Modern Analysis. Whittaker E T Watson G, N , Cambridge University PressWHITTAKER E.T., WATSON G.,N., A Course of Modern Analysis. Cambridge University Press, 1920. . * * *, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders17NIST Digital Library of Mathematical FunctionsRelease 1.0.* * *, NIST Digital Library of Mathematical Functions. http://dlmf. nist.gov/, Release 1.0.17 of 2017-12-22. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds.

We study an Unruh-DeWitt particle detector that is coupled to the proper time derivative of a real scalar field in 1+1 spacetime dimensions. Working within first-order perturbation theory, we cast the transition probability into a regulatorfree form, and we show that the transition rate remains well defined in the limit of sharp switching. The detector is insensitive to the infrared ambiguity when the field becomes massless, and we verify explicitly the regularity of the massless limit for a static detector in Minkowski half-space. We then consider a massless field for two scenarios of interest for the Hawking-Unruh effect: an inertial detector in Minkowski spacetime with an exponentially receding mirror, and an inertial detector in (1 + 1)-dimensional Schwarzschild spacetime, in the Hartle-Hawking-Israel and Unruh vacua. In the mirror spacetime the transition rate traces the onset of an energy flux from the mirror, with the expected Planckian late time asymptotics. In the Schwarzschild spacetime the transition rate of a detector that falls in from infinity gradually loses thermality, diverging near the singularity proportionally to r −3/2 .

10.1088/0264-9381/31/24/245007

1406.2574

fe50288c696d18f7d4f8041318c61f19e8d39a4d

Onset and decay of the 1+1 Hawking-Unruh effect: what the derivative-coupling detector saw Revised July 2014 arXiv:1406.2574v2 [gr-qc] 10 Jul 2014 Benito A Juárez-Aubry School of Mathematical Sciences University of Nottingham NG7 2RDNottinghamUK Jorma Louko School of Mathematical Sciences University of Nottingham NG7 2RDNottinghamUK Onset and decay of the 1+1 Hawking-Unruh effect: what the derivative-coupling detector saw Revised July 2014 arXiv:1406.2574v2 [gr-qc] 10 Jul 20141 We study an Unruh-DeWitt particle detector that is coupled to the proper time derivative of a real scalar field in 1+1 spacetime dimensions. Working within first-order perturbation theory, we cast the transition probability into a regulatorfree form, and we show that the transition rate remains well defined in the limit of sharp switching. The detector is insensitive to the infrared ambiguity when the field becomes massless, and we verify explicitly the regularity of the massless limit for a static detector in Minkowski half-space. We then consider a massless field for two scenarios of interest for the Hawking-Unruh effect: an inertial detector in Minkowski spacetime with an exponentially receding mirror, and an inertial detector in (1 + 1)-dimensional Schwarzschild spacetime, in the Hartle-Hawking-Israel and Unruh vacua. In the mirror spacetime the transition rate traces the onset of an energy flux from the mirror, with the expected Planckian late time asymptotics. In the Schwarzschild spacetime the transition rate of a detector that falls in from infinity gradually loses thermality, diverging near the singularity proportionally to r −3/2 . Introduction For quantum fields living on a pseudo-Riemannian manifold, the experiences of observers coupled to the field depend both on the quantum state of the field and on the worldline of the observer [1,2]. A celebrated example is the Unruh effect [3], in which uniformly accelerated observers in Minkowski spacetime experience a thermal bath in the quantum state that inertial observers perceive as void of particles. Other well-studied examples arise with black holes that emit Hawking radiation [4] and with observers in spacetimes of high symmetry [5]. A useful tool for analysing the experiences of an observer is a model particle detector that follows the observer's worldline and has internal states that couple to the quantum field. Such detectors are known as Unruh-DeWitt (UDW) detectors [3,6]. While much of the early literature on the UDW detectors focused on stationary situations, including the Unruh effect [7,8], the detectors remain well defined also in time-dependent situations, where they can be analysed within first-order perturbation theory [9,10,11,12,13,14,15,16,17,18,19,20,21] as well as by nonperturbative techniques [22,23,24,25,26]. A review with further references can be found in [27]. In this paper we consider a detector coupled to a quantised scalar field in 1+1 spacetime dimensions. A scalar field in 1+1 dimensions has local propagating degrees of freedom, and it exhibits the Hawking and Unruh effects just like a scalar field in higher dimensions. However, the dynamics of the field in 1+1 dimensions is significantly simpler than in higher dimensions, especially for a massless minimally coupled field, for which the field equation is conformally invariant and conformal techniques are available. In particular, the evolution of a massles minimally coupled field on a (1+1)dimensional collapsing star spacetime reduces essentially to that of a massless field on (1+1)-dimensional Minkowski spacetime in the presence of a receding mirror, and the system is explicitly solvable [28,29]. The simplifications in the dynamics come however at a cost: the Wightman function of a minimally coupled field in 1+1 dimensions is infrared divergent in the massless limit. While Hadamard states can still can be defined in terms of the short distance expansion of the Wightman function [30], the Hadamard expansion contains an additive constant that is not determined by the quantum state. While this undetermined additive constant does not contribute to stress-energy expectation values [1,31,32], it does contribute to the transition probability of an UDW detector that couples to the value at the field at the detector's location. In stationary situations the ambiguous contribution to transition probablities can be argued to vanish, under suitable assumptions about the switch-on and switch-off [7,11,12], but in nonstationary situations the ambiguity is more severe and has been found to lead to physically undesirable predictions in examples that include a receding mirror spacetime [19]. In this paper we analyse a detector that is insensitive to the infrared ambiguity of the (1 + 1)-dimensional Wightman function in the massless minimaly coupled limit: we couple the detector linearly to the derivative of the field with respect to the detector's proper time [9,22,33,34], rather than to the value of the field. Working in first-order perturbation theory, the detector's transition probability involves then the double derivative of the Wightman function, rather than the Wightman function itself. We show that the response of the (1+1)-dimensional derivative-coupling detector is closely similar to that of the (3 + 1)-dimensional detector with a non-derivative coupling [14,15]. In particular, the transition probablity can be written as an integral formula that involves no short-distance regulator at the coincidence limit but contains instead an additive term that depends only on the switching function that controls the switch-on and switch-off. A consequence is that in the limit of sharp switching the transition probability diverges but the transition rate remains finite. We carry out three checks on the physical reasonableness of the derivative-coupling detector in stationary situations in which the switch-on is pushed to asymptotically early times. First, we verify that the transition rate is continuous in the limit of vanishing mass for an inertial detector in Minkowski space, with the field in Minkowski vacuum, and we show that the same holds also for a static detector in Minkowski half-space with Dirichlet and Neumann boundary conditions. This is evidence that the derivative coupling manages to remove the infrared ambiguity of the massless field without producing unexpected discontinuities in the massless limit. Second, we show that the transition rate of a uniformly accelerated detector in Minkowski space, coupled to a massless field in Minkowski vacuum, coincides with that of an inertial detector at rest with a thermal bath, being in particular thermal in the sense of the Kubo-Martin-Schwinger (KMS) property [35,36]. This shows that the derivative-coupling detector sees the usual Unruh effect for a massless field. Third, we show that in a thermal bath of a massless field in Minkowski space, the transition rate of an inertial detector that is moving with respect to the bath is a sum of two terms that are individually thermal but at different temperatures, related to the temperature of the bath by Doppler shifts to the red and to the blue. As these terms stem respectively from the right-moving and left-moving components of the field, the temperature shifts are exactly as one would expect. After these checks, we focus on the massless minimally coupled field in two nonstationary situations, each of interest for the Hawking-Unruh effect. We first consider a Minkowski spacetime with a mirror whose exponentially receding motion makes the field mimic the late time behaviour of a field in a collapsing star spacetime [1,28,29]. We show that at late times the detector's transition rate is the sum of a Planckian contribution, corresponding to the field modes propagating away from the mirror, and a vacuum contribution, corresponding to the field modes propagating towards the mirror. While the detector couples to the sum of the two parts, the two parts can nevertheless be unambiguously identified by considering their dependence on the detector's velocity with respect to the rest frame in which the mirror was static in the asymptotic past. We also show numerical results on how the transition rate evolves from the asymptotic early time form to the asymptotic late time form. These properties of the transition rate are in full agreement with the energy flux emitted by the mirror [1,28,29]. We then consider a detector falling inertially into the (1 + 1)-dimensional Schwarz-schild black hole, with the field in the Hartle-Hawking-Israel (HHI) and Unruh vacua. Starting the infall at the asymptotic infinity, we verify that the early time transition rate in HHI vacuum is as in a thermal state, in the usual Hawking temperature, while in the Unruh vacuum it is as in a thermal state for the outgoing field modes and in the Boulware vacuum for the ingoing field modes. The outgoing and ingoing contributions can again be unambiguously identified by considering their dependence on the detector's velocity in the asymptotic past. The transition rate remains manifestly nonsingular on horizon-crossing, and we present numerical evidence of how the thermal properties are gradually lost during the infall. Near the black hole singularity the transition rate diverges proportionally to r −3/2 where r is the Schwarzschild radial coordinate. These results are in full agreement with the known properties of the HHI and Unruh vacua, including their thermality, their invariance under Schwarzschild time translations and their regularity across the future horizon. We anticipate that the derivative-coupling detector will be a useful tool for probing a quantum field in other situations where the infrared properties raise technical difficulties for the conventional UDW detector. One such instance is when the field has zero modes, which typically occur in spacetimes with compact spatial sections [37]. Other instances may arise in analogue spacetime systems [38] or in spacetimes where the back-reaction due to Hawking evaporation is strong (for a small selection of references see [39,40,41,42,43]). The structure of the paper is as follows. In Section 2 we introduce the derivativecoupling detector, write the transition probability in a regulator-free form and provide the formula for the transition rate in the sharp switching limit. Consistency checks in three stationary situations are carried out in Section 3. Sections 4 and 5 address respectively the receding mirror spacetime and the Schwarzschild spacetime. The results are summarised and discussed in Section 6. Details of technical calculations are deferred to four appendices. Our metric signature is (−+), so that the norm squared of a timelike vector is negative. We use units in which c = = k B = 1, so that frequencies, energies and temperatures have dimension inverse length. Spacetime points are denoted by Sans Serif characters (x) and spacetime indices are denoted by a, b, . . .. Complex conjugation is denoted by overline. O(x) denotes a quantity for which O(x)/x is bounded as x → 0, o(x) denotes a quantity for which o(x)/x → 0 as x → 0, O(1) denotes a quantity that is bounded in the limit under consideration, and o(1) denotes a quantity that goes to zero in the limit under consideration. Derivative-coupling detector In this section we introduce an UDW detector that couples linearly to the proper time derivative of a scalar field [9,22,33,34]. We show, within first-order perturbation theory, that in (1 + 1) spacetime dimensions the detector's transition probability and transition rate are closely similar to those of a (3 + 1)-dimensional UDW detector with a non-derivative coupling [14,15]. Derivative-coupling detector in d ≥ 2 dimensions Our detector is a spatially point-like quantum system with two distinct energy eigenstates. We denote the normalised energy eigenstates by |0 D and |ω D , with the respective energy eigenvalues 0 and ω, where ω = 0. The detector moves in a spacetime of dimension d ≥ 2 along the smooth timelike worldline x(τ ), where τ is the proper time, and it couples to a real scalar field φ via the interaction Hamiltonian H int = cχ(τ )µ(τ ) d dτ φ x(τ ) , (2.1) where c is a coupling constant, µ is the detector's monopole moment operator and the switching function χ specifies how the interaction is switched on an off. We assume that χ is real-valued, non-negative and smooth with compact support. Where H int (2.1) differs from the usual UDW detector [3,6] is that the interaction is linear in the proper time derivative of the field, rather than in the field itself. An alternative expression is H int = cχ(τ )µ(τ )ẋ a ∇ a φ x(τ ) , where the overdot denotes d/dτ . We take the detector to be initially in the state |0 D and the field to be in a state |ψ , which we assume to be regular in the sense of the Hadamard property [30]. After the interaction has been turned on and off, we are interested in the probability for the detector to have made a transition to the state |ω D , regardless the final state of the field. Working in first-order perturbation theory in c, we may adapt the analysis of the usual UDW detector [1,2] to show that this probability factorises as P (ω) = c 2 | D 0|µ(0)|ω D | 2 F(ω) ,(2.2) where | D 0|µ(0)|ω D | 2 depends only on the internal structure of the detector but neither on |ψ , the trajectory or the switching, while the dependence on |ψ , the trajectory and the switching is encoded in the response function F. With our H int (2.1), the response function is given by F(ω) = ∞ −∞ dτ ∞ −∞ dτ e −iω(τ −τ ) χ(τ )χ(τ ) ∂ τ ∂ τ W(τ , τ ) ,(2.3) where the correlation function W(τ , τ ) . = ψ|φ x(τ ) φ x(τ ) |ψ is the pull-back of the Wightman function ψ|φ(x )φ(x )|ψ to the detector's world line. From now on we drop the constant prefactors in (2.2) and refer to the response function as the transition probability. As |ψ is Hadamard and the detector's worldline is smooth and timelike, the correlation function W is a well-defined distribution on R × R [44,45,46,47]. As χ is smooth with compact support, it follows that F (2.3) is well defined: given a family of functions W that converges to the distribution W as → 0 + , F is evaluated by first making in (2.3) the replacement W → W , then performing the integrals, and finally taking the limit → 0 + . The limit → 0 + may however not necessarily be taken under the integrals. For the usual UDW detector, for which the response function is given as in (2.3) but without the derivatives, this issue is known to become subtle if one wishes to define an instantaneous transition rate by passing to the limit of sharp switching: the subtleties start in three spacetime dimensions and increase as the spacetime dimension increases and the correlation function becomes more singular at the coincidence limit [14,15,16,17]. We may expect similar subtleties for the derivative-coupling detector, and since the derivatives in (2.3) increase the singularity of the integrand at the coincidence limit, we may expect the subtleties to start in lower spacetime dimensions than for the usual UDW detector. We confirm these expectations in Subsections 2.2, 2.3 and 2.4 by analysing the response function (2.3) and the transition rate in (1 + 1) spacetime dimensions. (1 + 1) response function: isolating the coincidence limit We now specialise to (1 + 1) spacetime dimensions. In this subsection we write the response function (2.3) in a way where the contribution from the singularity of the integrand at the coincidence limit has been isolated. We start from (2.3) and write W = (W − W sing ) + W sing , where W sing is the locally integrable function W sing (τ , τ ) . =    − i sgn(τ − τ ) 4 − ln |τ − τ | 2π for τ = τ , 0 for τ = τ ,(2.4) and sgn denotes the signum function, sgn x . =      1 for x > 0 , 0 for x = 0 , −1 for x < 0 . (2.5) We obtain F(ω) = F reg (ω) + F sing (ω) , (2.6a) F reg (ω) = ∞ −∞ dτ ∞ −∞ dτ e −iω(τ −τ ) χ(τ )χ(τ ) ∂ τ ∂ τ W(τ , τ ) − W sing (τ , τ ) , (2.6b) F sing (ω) = ∞ −∞ dτ ∞ −∞ dτ e −iω(τ −τ ) χ(τ )χ(τ ) ∂ τ ∂ τ W sing (τ , τ ) , (2.6c) where the derivatives are understood in the distributional sense. Consider first F reg (2.6b). The Hadamard short distance form of the Wightman function ψ|φ(x )φ(x )|ψ [30] implies that both W(τ , τ ) and ∂ τ ∂ τ W(τ , τ ) − W sing (τ , τ ) are represented in a neighbourhood of τ = τ by locally integrable functions. It follows that the integral in (2.6b) receives no distributional contributions from τ = τ , and the integral can hence be decomposed into integrals over the subdomains τ > τ and τ < τ . In the subdomain τ > τ we write τ = u and τ = u − s, where u ∈ R and 0 < s < ∞, and in the subdomain τ < τ we write τ = u and τ = u − s, where again u ∈ R and 0 < s < ∞. Using the property W(τ , τ ) = W(τ , τ ) and the explicit form of W sing (τ , τ ) (2.4), we obtain F reg (ω) = 2 ∞ −∞ du ∞ 0 ds χ(u)χ(u − s) Re e −iωs A(u, u − s) + 1 2πs 2 , (2.7) where A(τ , τ ) . = ∂ τ ∂ τ W(τ , τ ) . (2.8) Note that the integrand in (2.7) is still a distribution, but it is represented by a locally integrable function in a neighbourhood of s = 0, and any distributional singularities must hence be isolated from s = 0. We evaluate F sing (2.6c) in Appendix A. Combining (2.7) and (A.6), we find F(ω) = −ωΘ(−ω) ∞ −∞ du [χ(u)] 2 + 1 π ∞ 0 ds cos(ωs) s 2 ∞ −∞ du χ(u)[χ(u) − χ(u − s)] + 2 ∞ −∞ du ∞ 0 ds χ(u)χ(u − s) Re e −iωs A(u, u − s) + 1 2πs 2 , (2.9) where we recall that A is given by (2.8). An equivalent form, using for F sing the alternative expression (A.7) given in Appendix A, is F(ω) = − ω 2 ∞ −∞ du [χ(u)] 2 + 1 π ∞ 0 ds s 2 ∞ −∞ du χ(u) [χ(u) − χ(u − s)] + 2 ∞ −∞ du ∞ 0 ds χ(u)χ(u − s) Re e −iωs A(u, u − s) + 1 2πs 2 . (2.10) The integral over s in the second term in (2.9) and (2.10) is convergent at small s since the integral over u produces an even function of s that vanishes at s = 0. The expression (2.10) for the response function is closely similar to that obtained in [14,15] for the usual, non-derivative UDW detector in (3 + 1) dimensions. This happens because of the similarity between the coincidence limit singularities of the twice differentiated (1 + 1)-dimensional correlation function, appearing in (2.3), and the undifferentiated (3 + 1)-dimensional correlation function that appears in the similar expression for non-derivative coupling. We re-emphasise that the last term in (2.9) and (2.10) may contain distributional contributions from s > 0. Similar distributional contributions were not considered for the (3 + 1)-dimensional non-derivative UDW detector in [15], but they can occur also there, and the analysis in [15] can be amended to include these contributions by proceeding as in the present paper. Similar distributional contributions can arise in any spacetime dimension: in (2 + 1) dimensions, examples on the Bañdados-Teitelboim-Zanelli black hole were encountered in [17]. (1 + 1) sharp switching limit: transition rate In this subsection we consider the sharp switching limit of the derivative-coupling detector in (1 + 1) dimensions. Following [14,15], we consider a family of switching functions given by χ(u) = h 1 u − τ 0 + δ δ × h 2 −u + τ + δ δ ,(2.11) where τ and τ 0 are pararameters satisfying τ > τ 0 , δ is a positive parameter, and h 1 and h 2 are smooth non-negative functions such that h 1 (x) = h 2 (x) = 0 for x ≤ 0 and h 1 (x) = h 2 (x) = 1 for x ≥ 1. In words, the detector is switched on over an interval of duration δ before time τ 0 , stays on at constant coupling from time τ 0 to time τ , and is finally switched off over an interval of duration δ after time τ . The profile of the switch-on is determined by h 1 and the profile of the switch-off is determined by h 2 . We are interested in the limit δ → 0. To begin with, suppose that A(τ , τ ) (2.8) is represented by a smooth function for τ = τ . Given the similarity between (2.10) and the (3 + 1)-dimensional non-derivative response function given by equation (3.16) in [15], we may follow the analysis that led to equations (4.4) and (4.5) in [15]. For the response function, we find F(ω, τ ) = − ω∆τ 2 + 2 τ τ 0 du u−τ 0 0 ds Re e −iωs A(u, u − s) + 1 2πs 2 + 1 π ln ∆τ δ + C + O(δ),(2.12) where ∆τ . = τ − τ 0 , C is a constant that depends only on h 1 and h 2 , and we have included in F(ω, τ ) the second argument τ to indicate explicitly the dependence on τ . The response function (2.12) hence diverges logarithmically as δ → 0, but the divergent contribution is a pure switching effect, independent of the quantum state and of the detector's trajectory. The transition rate, defined asḞ(ω, τ ) . = ∂ ∂τ F(ω, τ ), remains finite as δ → 0, and is in this limit given bẏ F(ω, τ ) = − ω 2 + 2 ∆τ 0 ds Re e −iωs A(τ, τ − s) + 1 2πs 2 + 1 π∆τ . (2.13) An equivalent expression, obtained by writing 1 = cos(ωs) + [1 − cos(ωs)] under the integral in (2.13), iṡ F(ω, τ ) = −ωΘ(−ω) + 1 π cos(ω∆τ ) ∆τ + |ω| si(|ω|∆τ ) + 2 ∆τ 0 ds Re e −iωs A(τ, τ − s) + 1 2πs 2 , (2.14) where si is the sine integral [48] and Θ is the Heaviside function, Θ(x) . = 1 for x > 0 , 0 for x ≤ 0 . (2.15) When the switch-on is in the asymptotic past and the fall-off of A(τ, τ − s) is sufficiently fast as large s, the ∆τ → ∞ limit of (2.14) giveṡ F(ω, τ ) = −ωΘ(−ω) + 2 ∞ 0 ds Re e −iωs A(τ, τ − s) + 1 2πs 2 . (2.16) The observational meaning of the transition rate relates to ensembles of ensembles of detectors (see Section 5.3.1 of [12] or Section 2 of [15]). When A(τ , τ ) (2.8) is not represented by a smooth function for τ = τ , the estimates leading to (2.12) and (2.13) need not hold, and the transition rate need not have a well-defined δ → 0 limit for all τ 0 and τ . In particular, if the detector is switched on at a finite time τ 0 and A(τ , τ ) has a distributional singularity at (τ , τ ) = (τ, τ 0 ), the integral expressions in (2.13) and (2.14) would not be well defined because the singularity occurs at an end-point of the integration. If the switch-on is in the asymptotic past, however, the transition rate formula (2.16) is well defined even when A(τ , τ ) has distributional singularities for τ = τ provided these singularities are sufficiently isolated. We shall encounter examples of such singularities in Subsection 3.1 and Section 4. Similar remarks about singularities of the correlation function at timelike-separated points apply also to the sharp switching limit of the non-derivative UDW detector in (3 + 1) dimensions. The transition rate results given in [15] for a stwich-on at a finite time hold when no such singularities are present. Stationary transition rate Suppose that the Wightman function is stationary with respect to the detector's trajectory, in the sense that W(τ , τ ) depends on τ and τ only through the difference τ − τ . When the detector is switched on in the asymptotic past, the transition rate (2.16) reduces tȯ F(ω) = −ωΘ(−ω) + 2 ∞ 0 ds Re e −iωs A(s, 0) + 1 2πs 2 = −ωΘ(−ω) + ∞ −∞ ds e −iωs A(s, 0) + 1 2πs 2 = −ωΘ(−ω) + C ds e −iωs A(s, 0) + 1 2πs 2 = ∞ −∞ ds e −iωs A(s, 0) , (2.17) where we have dropped the second argument τ fromḞ as the transition rate is now independent of τ , and A(s, 0) is understood as a distribution everywhere, including s = 0. In (2.17) we have first used the properties A(τ , τ ) = A(τ −τ , 0) and A(τ , τ ) = A(τ , τ ). Next, we have deformed the real s axis into a contour C in the complex s plane, such that C follows the real axis except that it dips into the lower half-plane near s = 0; this deformation is justified by the Hadamard short separation form of the Wightman function [30]. In the contour integral over C, we have then separated the two terms in the integrand, evaluated the integral of the second term by a standard contour technique, and noted that in the first term C can be deformed back to the real s axis provided the integrand is understood as a distribution for all s, including s = 0. The result (2.17) coincides with the transition rate that one obtains from the response function (2.3) by the usual procedure of setting χ = 1 and formally factoring out the infinite total detection time [1]. x > 0. We consider in M and M a scalar field of mass m ≥ 0, and in M we impose the Dirichlet or Neumann boundary condition that the field or its normal derivative vanish at x = 0. We set the field in M in the Minkowski vacuum |0 , and the field in M in the Minkowski-like vacuum |0 that is the no-particle state with respect to the timelike Killing vector ∂ t . Now, consider in M and M a detector on the static worldline x(τ ) = (τ, d) , (3.1) where d is a positive constant. In M the value of d has no geometric significance, but in M d is the distance of the detector from the mirror at x = 0. We take the detector to be switched on in the asymptotic past, so that the detector's transition rate is stationary and given by (2.17). We shall show that the transition rate is continuous in the limit m → 0. 3.1.1 m > 0 For m > 0, the Wightman function in M is [1] 0|φ(x)φ(x )|0 = 1 2π K 0 m (∆x) 2 − (∆t − i ) 2 , (3.2) where ∆x = x−x , ∆t = t−t , K 0 is the modified Bessel function of the second kind [48] , and the expression is understood as a distribution in the sense of → 0 + . The square root is positive when x and x are spacelike separated and → 0 + , and the continuation to general x and x is specified by the i prescription. By the method of images, the Wightman function in M is the sum of (3.2) and the image piece 0 |φ(x)φ(x )|0 − 0|φ(x)φ(x )|0 = η 2π K 0 m (x + x ) 2 − (∆t − i ) 2 ,(3.3) where η = −1 for Dirichlet and η = 1 for Neumann. We evaluate the transition rate (2.17) in Appendix B. We obtain M :Ḟ(ω) = ω 2 √ ω 2 − m 2 Θ(−ω − m) , (3.4a) M :Ḟ(ω) = ω 2 1 + η cos 2d √ ω 2 − m 2 √ ω 2 − m 2 Θ(−ω − m) . (3.4b) The transition rate is non-negative, and it is nonvanishing only for ω < −m, that is, for de-excitations exceeding the mass gap. These are properties that one would expect of a reasonable detector coupled to a massive field. m = 0 On M, the massive Wightman function (3.2) diverges as m → 0. However, the quantity 0|φ(x)φ(x )|0 + 1 2π ln[me γ /(2µ)], where γ is Euler's constant and µ is a positive constant of dimension inverse length, has at m → 0 a finite limit, given by [48] 0|φ (x)φ(x )|0 . = − 1 2π ln µ (∆x) 2 − (∆t − i ) 2 . (3.5) We take (3.5) as the definition of the Wightman function for m = 0. The constant µ is required for dimensional consistency, and its arbitrariness means that 0|φ(x)φ(x )|0 (3.5) is unique up to an additive constant. The massless Wightman function on M is the sum of (3.5) and the image piece 0 |φ(x)φ(x )|0 − 0|φ(x)φ(x )|0 = − η 2π ln µ (x + x ) 2 − (∆t − i ) 2 ,(3.6) where again η = −1 for Dirichlet and η = 1 for Neumann. Note that for η = −1, the massless Wightman function on M is independent of µ and can be obtained as the m → 0 limit of the massive Wightman function on M without introducing a subtraction by hand. We show in Appendix B that the transition rate is given by M :Ḟ(ω) = −ω Θ(−ω) , (3.7a) M :Ḟ(ω) = −ω [1 + η cos(2dω)] Θ(−ω) . (3.7b) The transition rate is non-negative, and it is nonvanishing only for de-excitations, as one would expect of a reasonable detector coupled to a massless field. We see from (3.4) and (3.7) that the massless transition rate is equal to the massless limit of the massive transition rate. This is the property that we wished to verify. Unruh effect Let again M be (1 + 1) Minkowski spacetime, and consider in M a massless field in the Minkowski vacuum. We consider a detector on the uniformly accelerated worldline x(τ ) = a −1 sinh(aτ ), a −1 cosh(aτ ) ,(3.8) where the positive constant a is the magnitude of the proper acceleration. The trajectory is stationary with respect to the boost Killing vector t∂ x +x∂ t , and |0 is invariant under this Killing vector. With the detector switch-on pushed to the asymptotic past, the transition rate is independent of time and given by (2.17). From (2.8), (3.5) and (3.8), we find A(τ , τ ) = − a 2 8π sinh 2 a(τ − τ − i )/2 . (3.9) Substituting (3.9) in (2.17), deforming the contour of s-integration to s = −iπ/a + r where r ∈ R, and using formula 3.985.1 in [49], we finḋ F(ω) = ω e 2πω/a − 1 . (3.10) The transition rate (3.10) satisfies the KMS relation [35,36], F(ω) = e −ω/TḞ (−ω) ,(3.11) with T = a/(2π), and is hence thermal at temperature a/(2π) in the KMS sense. We conclude that the detector does see the usual Unruh effect [3,7]. The Planckian form of the transition rate (3.10) is identical to that of a non-derivative detector coupled to a massless field on a uniformly accelerated trajectory in (3 + 1) dimensions [3,7]. Inertial detector in a thermal bath We consider again a massless field in (1 + 1) Minkowski spacetime M, but now in the thermal state |T of positive temperature T . Working in Minkowski coordinates (t, x) in which the thermal bath is at rest, the thermal Wightman function is obtained from the vacuum Wightman function by taking an image sum in t with period i/T [1]. With the vacuum Wightman function (3.5), the sum reads T |φ(x)φ(x )|T = − 1 4π ∞ n=−∞ ln µ 2 (∆x) 2 − (∆t − i + in/T ) 2 ,(3.12) and does not converge. However, differentiation of the sum in (3.12) term by term with respect to ∆x gives a new sum that converges and can be summed by residues into an elementary function. We integrate the elementary function with respect to ∆x and fix the ∆t-dependent integration constant by requiring that the massless Klein-Gordon equation is satisfied, and requiring evenness in ∆t for (∆x) 2 − (∆t) 2 > 0. The outcome is T |φ(x)φ(x )|T . = − 1 4π ln{sinh[πT (∆x + ∆t − i )]} − 1 4π ln{sinh[πT (∆x − ∆t + i )]} ,(3.13) uniquely up to an additive constant, and we take (3.13) as the definition of the thermal Wightman function. Note that (3.13) decomposes into the right-mover contribution that depends on ∆(x − t) and the left-mover contribution that depends on ∆(x + t). We consider the inertial detector worldline x(τ ) = (τ cosh λ, −τ sinh λ) ,(3.14) where λ ∈ R is the detector's rapidity parameter with respect to the rest frame of the bath. For later convenience, we have chosen the sign in (3.14) so that a detector with positive λ is moving towards decreasing x. From (2.8), (3.13) and (3.14) we find A(τ , τ ) = − 1 16π (2πT + ) 2 sinh 2 [πT + (τ − τ − i )] + (2πT − ) 2 sinh 2 [πT − (τ − τ − i )] , (3.15) where T ± . = e ±λ T . Taking the detector to be switched on in the asymptotic past, and proceeding as with (3.9), we find that the transition rate is given bẏ F(ω) = ω 2 1 e ω/T + − 1 + 1 e ω/T − − 1 , (3.16) simplifying in the special case λ = 0 tȯ F(ω) = ω e ω/T − 1 . (3.17) The λ = 0 transition rate (3.17) satisfies the KMS relation (3.11) in temperature T , and it coincides with the transition rate (3.10) of a uniformly accelerated detector when T = a/(2π). The λ = 0 transition rate (3.16) is a sum of the right-mover and left-mover contributions, each satisfying the KMS relation but in the respective Doppler-shifted temperatures T ± . These are properties that one would expect of a reasonable detector. Inertial detector with vacuum left-movers and thermal right-movers In preparation for the nonstationary situations that will be addressed in Sections 4 and 5, we consider here the inertial detector (3.14) in the state in which the left-movers are in the Minkowski vacuum but the right-movers are in a thermal bath with temperature T . As the left-movers and the right-movers decouple, the results can be read off from those given above in a straightforward way. Taking the switch-on to the asymptotic past, we findḞ (ω) = − ω 2 Θ(−ω) + ω 2 (e ω/T + − 1) . (3.18) The first term in (3.18) is the left-mover contribution, equal to half of the Minkowski transition rate. The second term is the right-mover contribution, which is Planckian in the Doppler-shifted temperature T + = e λ T . The receding mirror spacetime In this section we consider a massless field in (1 + 1)-dimensional Minkowski spacetime with a receding mirror that asymptotes at late times to a null line, in a fashion that mimics the late time redshift that occurs in a collapsing star spacetime [1,28,29]. Focusing on a specific mirror trajectory that is asymptotically inertial at early times, and choosing a vacuum with no incoming radiation from infinity, we compute the transition rate of an inertial, sharply-switched detector that is turned on in the asymptotic past. We show that the early time transition rate is Minkowskian and the late time transition rate has the expected form of Planckian radiation emitted from the mirror. Mirror spacetime and the in-vacuum Denoting a standard set of Minkowski coordinates by (t, x), we work in the double null coordinates u = t − x , (4.1a) v = t + x , (4.1b) in which ds = −du dv. We take the mirror trajectory to be v = − 1 κ ln(1 + e −κu ) , (4.2) where κ is a positive constant. When parametrised in terms of the proper time τ , the trajectory reads u = − 2 κ ln[sinh(−κτ /2)] , (4.3a) v = − 2 κ ln[cosh(−κτ /2)] , (4.3b) where −∞ < τ < 0. The velocity and acceleration are towards decreasing x, and the proper acceleration has the magnitude κ/ sinh(−κτ ). At early times the trajectory is asymptotically inertial, asymptoting to x = 0 from the left, with proper acceleration that vanishes exponentially in τ . At late times the trajectory asymptotes to the null line v = 0 from below, receding to infinity as τ → 0 − , and the proper acceleration diverges as −1/τ . A spacetime diagram is shown in Figure 1. We consider the spacetime that is to the right of the mirror. The mirror is hence receding, and the constant κ is analogous to the surface gravity in a collapsing star spacetime at late times [1,28,29]. We consider a massless scalar field φ with Dirichlet boundary conditions at the mirror. As the positive frequency mode functions, we choose [1,28,29] u k = i(4πk) −1/2 e −ikv − e −ikp(u) ,(4.4) where k > 0 and p(u) = − 1 κ ln(1 + e −κu ) . These modes satisfy the massless Klein-Gordon equation, they satisfy the Dirichlet boundary condition at the mirror, and they are Dirac orthormal, (u k , u k ) = −(u k , u k ) = δ(k − k ) and (u k , u k ) = (u k , u k ) = 0, where ( · , · ) is the Klein-Gordon inner product on a hypersurface of constant t. In the distant past, the modes reduce to the usual Dirichlet boundary condition modes in the static half-space x > 0. We note that to verify the orthonormality, it suffices to consider the static half-space limit on a constant t hypersurface in the distant past: the inner product is constant in t due to the Klein-Gordon equation and the Dirichlet boundary condition. We denote by |0, in the no-particle state with respect to the modes (4.4). In the distant past, |0, in coincides with the usual no-particle state in the half-space x > 0, and we call it the in-vacuum. Computing the Wightman as a mode sum from (4.4) gives [1] 0, in|φ(x)φ(x )|0, in = − 1 4π ln p(u) − p(u ) − i (v − v − i ) v − p(u ) − i p(u) − v − i ,(4.6) where i arises from the conditional ultraviolet convergence as usual. The mode sum is infrared convergent because of the Dirichlet boundary condition. Inertial detector: static in the distant past We consider a detector on the inertial worldline (3.1), where d is again a positive constant. In the asymptotic past, the detector is hence at distance d from a static mirror. We take the detector to be switched on in the asymptotic past, and we take the field to be in the in-vacuum |0, in . Using (3.1), (4.1) and (4.6), we find A(τ , τ ) = − 1 4π p (u )p (u ) [p(u ) − p(u ) − i ] 2 + 1 (v − v − i ) 2 − p (u ) [v − p(u ) − i ] 2 − p (u ) [p(u ) − v − i ] 2 ,(4.7) where u = τ − d, v = τ + d, u = τ − d and v = τ + d. The prime on p denotes derivative with respect to the argument. From (2.16) we then havė F(ω, τ ) =Ḟ 0 (ω, τ ) +Ḟ 1 (ω, τ ) +Ḟ 2 (ω, τ ) , (4.8a) F 0 (ω, τ ) = −ωΘ(−ω) + 1 2π ∞ 0 ds cos(ωs) − p (τ − d)p (τ − d − s) [p(τ − d) − p(τ − d − s)] 2 + 1 s 2 , (4.8b) F 1 (ω, τ ) = 1 2π ∞ 0 ds cos(ωs) p (τ − d − s) [τ + d − p(τ − d − s)] 2 , (4.8c) F 2 (ω, τ ) = 1 2π ∞ 0 ds Re e −iωs p (τ − d) [p(τ − d) − τ − d + s − i ] 2 . (4.8d) In (4.8b) and (4.8c) we have set = 0 as the integrand has no singularities. The in (4.8d) needs to be kep as the integrand has a singularity, arising because the points τ − s and τ on the detector's trajectory are connected by a null ray that is reflected from the mirror, as shown in Figure 1. The integral is well defined despite this singularity since the switch-on is in the asymptotic past so that the range of s cannot end at the singularity. We show in Appendix C that the early and late time forms of the transition rate (4.8) areḞ (ω, τ ) = −ω [1 − cos(2dω)] Θ(−ω) + O(e κτ ) as τ → −∞ , (4.9a) F(ω, τ ) = − ω 2 Θ(−ω) + ω 2 (e 2πω/κ − 1) + o(1) as τ → ∞ . (4.9b) 4.3 Inertial detector: travelling towards the mirror in the distant past We next consider a detector on the inertial worldline x(τ ) = (τ cosh λ, −τ sinh λ) ,(4.10) where λ > 0. In the asymptotic past, where the mirror is static, the detector is moving towards the mirror with speed tanh λ. Proceeding as above, we finḋ F(ω, τ ) =Ḟ 0 (ω, τ ) +Ḟ 1 (ω, τ ) +Ḟ 2 (ω, τ ) , (4.11a) F 0 (ω, τ ) = −ωΘ(−ω) + 1 2π ∞ 0 ds cos(ωs) − p (e λ τ )p e λ (τ − s) e 2λ p(e λ τ ) − p e λ (τ − s) 2 + 1 s 2 , (4.11b) F 1 (ω, τ ) = 1 2π ∞ 0 ds cos(ωs) p e λ (τ − s) e −λ τ − p e λ (τ − s) 2 , (4.11c) F 2 (ω, τ ) = 1 2π ∞ 0 ds Re e −iωs p (e λ τ ) [p(e λ τ ) − e −λ (τ − s) − i ] 2 ,(4.11d) and we show in Appendix C that the early and late time forms arė F(ω, τ ) = −ω 1 − e 2λ cos(2τ sinh λ e λ ω) Θ(−ω) + O(τ −1 ) as τ → −∞ , (4.12a) F(ω, τ ) = − ω 2 Θ(−ω) + ω 2 (e 2πe −λ ω/κ − 1) + o(1) as τ → ∞ . (4.12b) Onset of thermality We are now ready to discuss the sense in which the transition rate exhibits the onset of thermality as the mirror continues to recede. Consider first the distant past. For the detector (3.1), static with respect to the mirror, the transition rate (4.9a) agrees with that (3.7b) of the same detector in the static half-space M. For the detector (4.10), drifting towards the mirror, the transition rate (4.12a) can be verified to agree with that of the same detector in M. Compared with (4.9a), the non-Minkowski part of (4.12a) has the static distance d replaced by the time-dependent distance −τ sinh λ, ω replaced by the blueshifted frequency e λ ω, and an additional blueshift factor e λ . Consider then the distant future. The distant future transition rates (4.9b) and (4.12b) agree with the transition rate (3.18) of an inertial detector in Minkowski space when the left-movers are in the Minkowski vacuum and the right-movers are thermal in temperature κ/(2π). Note that the detector's velocity shows up by a Doppler blueshift in the right-mover contribution. The late time transition rates (4.9b) and (4.12b) can hence be interpreted to consist of a contribution from the left-moving part of the field, undisturbed by the mirror, and and a contribution from the right-moving part of the field, excited by the mirror to induce a Planckian response. This interpretation is consistent with the fact that the stress-energy tensor of the field contains at late times an energy flux to the right [1,28,29,50,51]. This late time result is consistent with that quoted in [1] for a non-derivative UDW detector with a mirror trajectory with similar late time asymptotics, in the sense that the left-mover contribution was not explicitly written out in [1]. Figures 2 and 3 show numerical plots for the evolution of the transition rate from early to late times, for the detector (3.1) that is static with respect to the mirror in the distant past. The asymptotic late time value is reached via a ring-down of oscillations whose period equals 2π/κ within the range of the numerical experiments. We have not attempted to examine this oscillation analytically. (1 + 1) Schwarzschild spacetime In this section we consider a detector in the (1+1)-dimensional Schwarzschild spacetime, obtained by dropping the angular dimensions from the (3+1)-dimensional Schwarzschild metric [52]. We first establish the notation, recall the definitions of the Boulware, HHI and Unruh vacua [1], and discuss briefly the case of a static detector in the exterior. The main objective is to study a geodesically infalling detector. Spacetime and vacua We write the metric of the (1+1)-dimensional maximally extended Schwarzschild spacetime in the notation of [1] as ds 2 = − 2M e −r/(2M ) r dū dv ,(5.1) where M > 0 is the Schwarzschild mass parameter, the Kruskal null coordinatesū and v increase towards the future and satisfyūv < (4M ) 2 , and r ∈ R + is the unique solution to The metric has the Killing vector ξ = (4M ) −1 (−ū∂ū +v∂v), which is timelike forūv < 0 (r > 2M ), spacelike forūv > 0 (r < 2M ) and null at the Killing horizonūv = 0 (r = 2M ). The right-going (respectively left-going) branch of the Killing horizon is u = 0 (v = 0). The Killing horizon divides the spacetime into four quadrants as summarised in Table 1. We denote by u and v the tortoise null coordinates defined by u = −4M ln[−ū/(4M )] forū < 0 , (5.3a) v = 4M ln[v/(4M )] forv > 0 . (5.3b) In Quadrant I (right-hand exterior), where r > 2M , we can hence introduce the usual exterior Schwarzschild coordinates (t, r) by u = t − r − 2M ln[r/(2M ) − 1] , (5.4a) v = t + r + 2M ln[r/(2M ) − 1] , (5.4b) so that t = 2M ln(−v/ū) ,(5.5) the metric reads ds 2 = −(1 − 2M/r) dt 2 + dr 2 (1 − 2M/r) ,(5.6) and ξ = ∂ t . In Quadrant II, where r < 2M , we can similarly introduce the Schwarzschildlike coordinates (t, r) by (5.2) andt = 2M ln(v/ū) ,(5.7) so that the metric takes the form ds 2 = − dr 2 [(2M/r) − 1] + [(2M/r) − 1] dt 2 (5.8) Quadrantūv ξ a ξ a r I: right-hand exterior − + − 2M < r < ∞ II: black hole interior + + + 0 < r < 2M III: left-hand exterior + − − 2M < r < ∞ IV: white hole interior − − + 0 < r < 2M and ξ = ∂t. A pair of coordinates that covers Quadrants I and II and the black hole horizon that separates them is (ū, v). We shall not need the explicit form of the metric in these coordinates. We consider a massless minimally coupled scalar field in three distinguished states. First, in Quadrant I we consider the Boulware vacuum |0 B , defined by the positive and negative frequency decomposition with respect to ∂ t in (5.6) [53]. At the asymptotically flat infinity of Quadrant I, |0 B reduces to the Minkowski vacuum. Second, on the whole spacetime we consider the HHI vacuum |0 H , defined by the positive and negative frequency decomposition with respect to ∂ū and ∂v on the Killing horizon [54,55]. In Quadrant I, |0 H is a thermal equilibrium state with respect to ∂ t , at the local Hawking temperature T loc = 1 8πM 1 − 2M/r . (5.9) Third, in Quadrants I and II and on the black hole horizon that separates them, we consider the Unruh vacuum |0 U , defined by the positive and negative frequency decomposition with respect to ∂ū and ∂ v in the coordinates (ū, v) [3]. |0 U mimics a state that results from the collapse of a star at late times when there is initially no incoming radiation from infinity, and it has the left-moving part of the field in a Boulware-like state and the right-moving part of the field in a HHI-like state. The Wightman functions for the three vacua are [1] [31,32]. The non-invariance of the Unruh vacuum Wightman function is due to the infrared properties of the (1+1)-dimensional conformal field and has no counterpart in higher dimensions [56]. 0 B |φ(x)φ(x )|0 B = − 1 4π ln[( + i∆u)( + i∆v)] , (5.10a) 0 H |φ(x)φ(x )|0 H = − 1 4π ln[( + i∆ū)( + i∆v)] , (5.10b) 0 U |φ(x)φ(x )|0 U = − 1 4π ln[( + i∆ū)( + i∆v)] ,(5. Static detector We consider first a detector in Quadrant I on the static, noninertial trajectory r = R, where R > 2M is a constant. Using (2.17) and (5.10), the calculations are closely similar to those in Section 3, and we omit the detail. We finḋ F B (ω) = −ωΘ(−ω) , (5.11a) F H (ω) = ω e ω/T loc − 1 , (5.11b) F U (ω) = − ω 2 Θ(−ω) + ω 2 (e ω/T loc − 1) , (5.11c) for respectively the Boulware, HHI and Unruh vacua, where T loc is the local Hawking temperature (5.9) evaluated at r = R. These results conform fully to expectations. The Boulware vacuum transition rate is that of an inertial detector in Minkowski space in Minkowski vacuum, while the HHI vacuum transition rate is thermal in the local Hawking temperature (5.9). The Unruh vacuum transition rate is the average of the two, with the two pieces arising respectively from the left-moving and right-moving parts of the field. These results are also consistent with what was reported for the non-derivative UDW detector in [1], in the sense that the left-mover contribution in (5.11c) was not explicitly written out in [1]. Finally, the similarity between (5.11c) and our receding mirror spacetime results (4.9b) and (4.12b) is an additional confirmation that the Unruh vacuum mimics the late time properties of a state created in a collapsing star spacetime [1,3]. Interlude: geodesics We next turn to inertial detectors. In this subsection we recall a convenient parametrisation for the geodesics. We give the full expressions in a form that applies only to Quadrant I, where the equations of a timelike geodesic in the Schwarzschild coordinates (5.6) take the formṫ = E 1 − 2M/r , (5.12a) r 2 = E 2 − 1 + 2M/r , (5.12b) where E is a positive constant and the overdot denotes derivative with respect to the proper time τ . The continuation beyond Quadrant I can be done by passing to the Kruskal coordinates (ū,v). When E > 1, the geodesic has at infinity the nonvanishing speed √ 1 − E −2 with respect to the Killing vector ξ. We consider a geodesic that is sent in from the infinity, so thatṙ < 0. The geodesic can be parametrised as τ = M (E 2 − 1) 3/2 (sinh χ − χ) , (5.13a) r = M (E 2 − 1) (cosh χ − 1) , (5.13b) t = M E (E 2 − 1) 3/2 sinh χ + (2E 2 − 3)χ + 2M ln − tanh(χ/2) + √ 1 − E −2 − tanh(χ/2) − √ 1 − E −2 , (5.13c) where the parameter χ takes values in (∞, 0), so that the trajectory starts at the infinity in the asymptotic past at χ → −∞ and hits the singularity at χ → 0. The additive constant in (5.13a) is chosen so that −∞ < τ < 0. The horizon-crossing occurs at χ = χ h . = −2 arctanh √ 1 − E −2 . Equation (5.13c) applies only in Quadrant I, where −∞ < χ < χ h . When E = 1, the geodesic has at infinity a vanishing speed with respect to ξ. We consider again a geodesic that is sent in from the infinity. The geodesic takes the form r = 2M [−3τ /(4M )] 2/3 , (5.14a) t = τ − 4M [−3τ /(4M )] 1/3 + 2M ln [−3τ /(4M )] 1/3 + 1 [−3τ /(4M )] 1/3 − 1 , (5.14b) where −∞ < τ < 0. The horizon-crossing occurs at τ = τ h . = −4M/3, and the singularity is reached as τ → 0. Equation (5.14b) applies only in Quadrant I, where −∞ < τ < τ h . When 0 < E < 1, the geodesic has a maximum value of r. The geodesic can be parametrised as τ = M (1 − E 2 ) 3/2 (ϕ + sin ϕ) , (5.15a) r = M (1 − E 2 ) (1 + cos ϕ) , (5.15b) t = M E (1 − E 2 ) 3/2 sin ϕ + (3 − 2E 2 )ϕ + 2M ln 1 + √ E −2 − 1 tan(ϕ/2) 1 − √ E −2 − 1 tan(ϕ/2) , (5.15c) where the parameter ϕ takes values in (−π, π), so that the trajectory starts at the white hole singularity at ϕ → −π and ends at the black hole singularity at ϕ → π. The additive constant in (5.15a) is chosen so that τ = 0 at the moment when r reaches its maximum value, 2M/(1 − E 2 ). The total proper time elapsed between the singularities is 2πM (1 − E 2 ) −3/2 . The horizon-crossings occur at ϕ = ∓ϕ h where ϕ h . = 2 arctan √ E −2 − 1 . Equation (5.15c) applies only in Quadrant I, where −ϕ h < ϕ < ϕ h . Finally, there exist also timelike geodesics that pass from the white hole to the black hole through the horizon bifurcation pointū =v = 0, without entering Quadrant I (or Quadrant III). These geodesics take the form u =v = 4M sin(ϕ/2) exp 1 2 cos 2 (ϕ/2) , (5.16) where the parameter ϕ takes values in (−π, π), and τ and r are given by (5.15a) and (5.15b) with E = 0. The isometry generated by ξ has been used in (5.16) to setū =v without loss of generality. Inertial detector The transition rate of the inertial detector is obtained by inserting the Wightman functions (5.10) and the geodesic trajectories of subsection 5.3 into the integral formulas of subsection 2.3. The transition rate is expressible as the integral of an elementary function for all values of E; for E > 1 (respectively E < 1) this is accomplished by writing the differentiations and the integration in terms of χ (ϕ). We address the near-infinity and near-singularity limits analytically and the intermediate regime numerically. Near the infinity We consider the E > 1 trajectories (5.13) and the E = 1 trajectory (5.14), all of which fall in from the infinity, and we push the switch-on to the infinite past. It is shown in Appendix D that at early times, τ → −∞, we havė F B (ω, τ ) = −ωΘ(−ω) + o(1) , (5.17a) F H (ω, τ ) = ω 2 (e ω/T − − 1) + ω 2 (e ω/T + − 1) + o(1) , (5.17b) F U (ω, τ ) = − ω 2 Θ(−ω) + ω 2 (e ω/T + − 1) + o(1) , (5.17c) where T ± . = e ±λ /(8πM ) and λ . = arctanh √ 1 − E −2 . For E = 1, we have T + = T − = 1/(8πM ), so that the two terms in (5.17b) are equal and combine to the Planckian response. The asymptotic past results (5.17) conform fully to physical expectations. The Boulware vacuum transition rate is that in Minkowski vacuum (3.7a), consistently with the interpretation of the Bouware vacuum as the no-particle state with respect to ξ. The HHI vacuum transition rate is that of an inertial detector in a thermal bath in Minkowski space (3.16), with the temperature given by the Hawking temperature at the infinity, 1/(8πM ), and with each of the two Planckian terms containing a Doppler shift factor that accounts for the detector's velocity at the infinity. The Unruh vacuum transition rate sees a Planckian term only in the outgoing part of the field, as confirmed by the Doppler shift to the blue in this term, while the term that corresponds to the ingoing part of the field is Minkowski-like. Near the singularity We consider the transition rate in the HHI and Unruh vacua in the limit where the detector approaches the black black hole singularity. We allow all values of the nonnegative constant E. We also allow the switch-on moment to remain arbitrary, subject only to the condition that for 0 ≤ E < 1 the switch-on in the HHI vacuum takes place after the trajectory emerges from the white hole singularity, and the switch-on in the Unruh vacuum takes place after the trajectory crosses the past horizon. It is shown in Appendix D that in this near-singularity limit we havė F(ω, τ ) = 1 8πM 2M r(τ ) 3/2 + 1 + E 2 2 2M r(τ ) 1/2 + O(1) ,(5. Intermediate regime: loss of thermality For a trajectory falling in from the infinity in the HHI and Unruh vacua, it is seen from the limits (5.17) and (5.18) that the thermal character of the transition rate is lost during the infall. Numerical evidence of how this loss takes place for the E = 1 trajectory is shown in Figures 4 and 5. The numerical evidence shows that the Planckian form of the transition rate is lost before the trajectory crosses the horizon. Numerical evidence also corroborates that the overall magnitude of the transition rate increases during the infall and grows without bound near the singularity. A sample plot for the E = 1 trajectory in the HHI vacuum is shown in Figure 6. Finally, suppose the field is in the HHI vacuum, and consider the trajectory that passes from the white hole to the black hole through the horizon bifurcation point. We use the parametrisation (5.16), so that (5.15a) and (5.15b) hold with E = 0. We switch the detector on at ϕ = −9π/10, close to but well separated from the white hole singularity at ϕ = −π. Figure 7 shows a perspective plot of the transition rate as a function of ω and τ . The plot shows clearly both the divergence when τ approaches the switch-on time, arising from the last term in (2.13), and the divergence when the trajectory approaches the black hole singularity. To examine the thermal character of the transition rate, we have evaluated numerically the quantity If T as,KMS were (approximately) independent of ω for fixed τ , the transition rate would satisfy (approximately) the KMS condition (3.11) and T as,KMS would be equal to (approximate) KMS temperature, which could possibly be τ -dependent. Within the parameter range that our numerical experiments have been able to probe, we have however found no regimes in which T as,KMS would be (approximately) independent of ω. Conclusions In this paper we have analysed an UDW detector that is coupled to the proper time derivative of a scalar field in a (1 + 1)-dimensional spacetime. Working within first-order perturbation theory, we showed that although the derivative makes the interaction between the detector and the field more singular, the singularity is no worse than that of the non-derivative UDW detector in (3 + 1) spacetime dimensions, and issues of switching can be handled by the same techniques. In particular, even though the transition probability diverges in the sharp switching limit, the transition rate remains well defined and allows the detector to address strongly time-dependent situations. Our main aim was to show that the derivative-coupling detector provides a viable tool for probing a (1 + 1)-dimensional massless field, whose infrared properties create ambiguities for the conventional UDW detector in time-dependent situations. We presented strong evidence that the derivative-coupling detector does remain well-behaved for the massless field, with and without time-dependence. As specific time-dependent examples, we analysed an inertial detector in a Minkowski spacetime with an exponentially receding mirror and a detector falling inertially into the (1 + 1)-dimensional Schwarzschild black hole. In both cases we recovered the expected thermal results due to the Hawking-Unruh effect in the appropriate limits. In the receding mirror spacetime we saw the thermality gradually set in as the mirror's acceleration approaches the asymptotic late time behaviour, tailored to model the late time effects of a gravitational collapse. In the (1 + 1)-dimensional Schwarzschild spacetime we saw thermality gradually lost as the detector falls, and we saw the transition rate diverge as the detector approaches the black hole singularity, for both the HHI and Unruh vacua. Our results about the time-dependence of the Hawking-Unruh effect complement those obtained via Bogoliubov coefficient techniques or via a quasi-temperature approximation to the Wightman function [57,58,59,60]. A key input in our analysis was to characterise the time-dependence of the response in terms of the instantaneous transition rate, defined by taking the sharp switching limit, and mathematically well defined in our spacetimes even when the time-dependence is strong. A conceptual disadvantage of the instantanous transition rate is however that it cannot be measured by a single detector, or even by an ensemble of detectors, but the measurement requires an ensemble of ensembles of detectors [12,15]. A technical disadvantage is that the instantanous transition rate becomes singular when the Wightman function has singularities that typically occur with spatial periodicity [37]. Further, the limit of sharp switching and the limit of large energy gap need not commute [61], which becomes an issue when one attempts to identify characteristics of thermal behavour in the response of a detector that operates for a genuinely finite interval of time. While we hence do not advocate the instantaneous transition rate as a definitive quantifier of time-dependence in the detector's response, our results strongly suggest that the instantaneous transition rate conveys a physically expected picture about the onset and decay of the Hawking-Unruh effect. A Evaluation of F sing (2.6c) In this appendix we show that F sing (2.6c) can be written as (A.6) or (A.7). Starting from (2.6c) and integrating the distributional derivatives by parts, we have F sing (ω) = ∞ −∞ dτ ∞ −∞ dτ Q ω (τ )Q ω (τ ) W sing (τ , τ ) , (A.1) where Q ω (τ ) . = e −iωτ χ(τ ) and the prime denotes derivative with respect to the argument. Note that the integrand in (A.1) is a locally integrable function, containing no distributional parts. Using the explicit form of W sing (2.4), we obtain F sing (ω) = F sing,1 (ω) + F sing,2 (ω) , (A.2a) F sing,1 (ω) = − i 4 ∞ −∞ dτ ∞ −∞ dτ Q ω (τ )Q ω (τ ) sgn(τ − τ ) , (A.2b) F sing,2 (ω) = − 1 2π ∞ −∞ dτ ∞ −∞ dτ Q ω (τ )Q ω (τ ) ln |τ − τ | . (A.2c) For F sing,1 , integrating over τ in (A.2b) gives F sing,1 (ω) = − i 2 ∞ −∞ du Q ω (u)Q ω (u) = − i 2 ∞ −∞ du [χ (u) − iωχ(u)] χ(u) = − ω 2 ∞ −∞ du [χ(u)] 2 , (A.3) where we have renamed τ as u, used the definition of Q ω , and finally noted that ∞ −∞ du χ (u)χ(u) = 1 2 ∞ −∞ du d du [χ(u)] 2 = 0. For F sing,2 , we break the integral in (A.2c) into the subdomains τ > τ and τ < τ . In the subdomain τ > τ we write τ = u and τ = u − s, where u ∈ R and 0 < s < ∞, and in the subdomain τ < τ we write τ = u and τ = u − s, where again u ∈ R and 0 < s < ∞. This gives In the last expression in (A.4), writing F sing,2 (ω) = − 1 π ∞ 0 ds ln s ∞ −∞ du Re Q ω (u)Q ω (u − s) = 1 π ∞ 0 ds ln s ∞ −∞ du Re Q ω (u)Q ω (u − s) = 1 π ∞ 0 ds ln s d 2 ds 2 ∞ −∞ du Re Q ω (u)Q ω (u − s) = 1 πχ(u)χ(u − s) = [χ(u)] 2 − χ(u)[χ(u) − χ(u − s)] gives F sing,2 (ω) = ω π ∞ −∞ du [χ(u)] 2 ∞ 0 ds sin(ωs) s + 1 π ∞ 0 ds s d ds cos(ωs) ∞ −∞ du χ(u)[χ(u) − χ(u − s)] = |ω| 2 ∞ −∞ du [χ(u)] 2 + 1 π ∞ 0 ds cos(ωs) s 2 ∞ −∞ du χ(u)[χ(u) − χ(u − s)] , (A.5) where in the first term we have used the identity ∞ 0 ds s −1 sin(ωs) = 1 2 π sgn ω, and in the second term we have integrated by parts. The integral over s in the second term is convergent at small s because F sing (ω) = −ωΘ(−ω) ∞ −∞ du [χ(u)] 2 + 1 π ∞ 0 ds cos(ωs) s 2 ∞ −∞ du χ(u)[χ(u) − χ(u − s)] . (A.6) An alternative expression is F sing (ω) = − ω 2 ∞ −∞ du [χ(u)] 2 + 1 π ∞ 0 ds s 2 ∞ −∞ du χ(u) [χ(u) − χ(u − s)] + 1 π ∞ −∞ du ∞ 0 ds χ(u)χ(u − s) [1 − cos(ωs)] s 2 , (A.7) which may be obtained from (A.6) by writing cos(ωs) = 1 − [1 − cos(ωs)] and using the identity B.1 m = 0 We consider first the massless field, with the Wightman function given by (3.5) and (3.6). In M, we find from (2.8), (3.1) and (3.5) that A(τ , τ ) = −1/ 2π(τ − τ − i ) 2 . Evaluating (2.17) as a contour integral gives (3.7a). In M, we find from (3.6) that the integrand in (2.17) contains the additional piece ∆A(τ , τ ) = − η 2π (τ − τ − i ) 2 + 4d 2 [(τ − τ − i ) 2 − 4d 2 ] 2 . (B.1) Evaluating the contribution to (2.17) as a contour integral leads to (3.7b). We note that ∆A(τ , τ ) (B.1) has distributional singularities at τ − τ = ±2d. The geometric reason for these singularities is that the points τ and τ on the detector's trajectory are connected by a null ray that is reflected from the mirror. As we have seen, the stationary transition rate is well defined despite these singularities. Were we however the consider a detector that operates for a finite duration, the singularities would interfere with the sharp switching limit manipulations that led to (2.13) when ∆τ = 2d. B.2 m > 0 For the massive field, the Wightman function is given by (3.2) and (3.3). We consider M and M in turn. B.2.1 M In M, we find from (2.8), (3.1) and (3.2) that A(τ , τ ) = m 2 2π K 0 m + i(τ − τ ) , (B.2) where the prime denotes derivative with respect to the argument. From (2.17) we then obtainḞ (ω) = m 2 2π C ds e −iωs K 0 (ims) , (B.3) where the contour C in the complex s plane follows the real axis from −∞ to +∞ except that it drops in the lower half-plane near s = 0, and K 0 has its principal branch when s is negative imaginary. We now assume ω = −m: it follows then from the asymptotics of (ω) = ω 2 2π C ds e −iωs K 0 (ims) = ω 2 2 Im ∞ 0 ds e −iωs H(2)0 (ms) , (B.4) where we have first integrated by parts twice, as allowed by the large s behaviour of the integrand, then deformed C to the real s axis, as allowed by the merely logarithmic singularity of the integrand at s = 0, and finally used the Bessel function analytic continuation formulas [48]. The integral in (B.4) was encountered in [19] in the context of a non-derivative detector, and from equations (5.11) and (5.14) therein we havė F(ω) = ω 2 √ ω 2 − m 2 Θ(−ω − m) , (B.5) which is the result (3.4a) used in the main text. B.2.2 M In M, we find from (3.3) that the integrand in (2.17) contains the additional piece ∆A(τ, τ ) = − η 2π d 2 dτ 2 K 0 m 4d 2 − (τ − τ − i ) 2 , (B.6) where the branch of the square root is as explained in the main text. The additional piece in the transition rate (2.17) is hence ∆Ḟ(ω) = − η 2π ∞ −∞ ds e −iωs d 2 ds 2 K 0 m 4d 2 − (s − i ) 2 = ηω 2 2π ∞ −∞ ds e −iωs K 0 m 4d 2 − (s − i ) 2 = ηω 2 π Re ∞ 0 ds e −iωs K 0 m 4d 2 − (s − i ) 2 , (B.7) again assuming ω = −m and integrating by parts twice. The integral in (B.7) was encountered in [19], and from equations (5.15) and (5.25) therein we have ∆Ḟ(ω) = ηω 2 cos 2d √ ω 2 − m 2 √ ω 2 − m 2 Θ(−ω − m) , (B.8) which leads to the result (3.4b) in the main text. C Asymptotic past and future transition rate in the receding mirror spacetime In this appendix we find the asymptotic past and future forms (4.9) and (4.12) of the transition rate of an inertial detector in the receding mirror spacetime of Section 4. C.1 Static in the distant past We wish to extract the asymptotic behaviour of (4.8) as τ → −∞ and as τ → ∞. C.1.1 τ → −∞ Consider (4.8b). Using (4.5) and letting h . = 1 + e κ(d−τ ) −1 , we havė F 0 (ω, τ ) = −ωΘ(−ω) + 1 2π ∞ 0 ds cos(ωs) 1 X + 1 s 2 , (C.1) where X = − [1 − h(1 − e −κs )] κs + ln[1 − h(1 − e −κs )] 2 κ 2 (1 − h) 2 . (C.2) The limit τ → −∞ is now the limit h → 0 + . Following the technique used in Subsection 5.3 of [13], we make in the integrand of (C.1) the re-arrangement 1 X + 1 s 2 = −X − s 2 s 4 1 + −X − s 2 s 2 −1 . (C.3) Taylor expanding the numerator of (C.2) to quartic order in h(1 − e −κs ) shows that the second factor in (C.3) is of the form 1 + O(h), uniformly in s, and yields for the first factor in (C.3) an estimate that can be applied under the integral over s and whose leading term is proportional to h. We hence havė F 0 (ω, τ ) = −ωΘ(−ω) + O(h) . (C.4) Consider then (4.8c). Proceeding similarly, we finḋ F 1 (ω, τ ) = 1 4πd + |ω| 2π cos(2dω) si(2d|ω|) − sin(2d|ω|) Ci(2d|ω|) + O(h) , (C.5) where si and Ci are the sine and cosine integrals [48]. Consider finally (4.8d). Integrating by parts once reduces the integral to a form that can be evaluated exactly in terms of the sine and cosine integrals [48], with the resulṫ F 2 (ω, τ ) = 1 − h 2π − 1 B + |ω| sin(B|ω|) Ci(B|ω|) − cos(Bω) si(B|ω|) + 2πω cos(Bω)Θ(−ω) , (C.6) where B . = 2d − κ −1 ln(1 − h). A small h expansion in (C.6) giveṡ F 2 (ω, τ ) = − 1 4πd + |ω| 2π sin(2d|ω|) Ci(2d|ω|) − cos(2dω) si(2d|ω|) + ω cos(Bω)Θ(−ω) + O(h) . (C.7) Combining (C.4), (C.5) and (C.7), we havė F(ω, τ ) = −ω [1 − cos(2dω)] Θ(−ω) + O(e κτ ) as τ → −∞ . (C.8) C.1.2 τ → ∞ Consider (4.8b ). Letting f . = 1/(1 + e κ(τ −d) ), and adding and subtracting κ 2 cos(ωs)[8π sinh 2 (κs/2)] −1 in the integrand, we obtaiṅ F 0 (ω, τ ) = −ωΘ(−ω) + 1 2π ∞ 0 ds cos(ωs) 1 s 2 − κ 2 4 sinh 2 (κs/2) + κ 2 2π ∞ 0 ds cos(ωs) 1 4 sinh 2 (κs/2) − f 2 e κs [1 + f (e κs − 1)] ln[1 + f (e κs − 1)] 2 . (C.9) In the last term in (C.9), the integrand goes to zero pointwise as f → 0, and a monotone convergence argument shows that the integral vanishes as f → 0. The second term plus half of the first term is equal to half of the transition rate encountered in Subsection (3.2) (with a → κ) and evaluated to (3.10). We hence havė F 0 (ω, τ ) = − ω 2 Θ(−ω) + ω 2 (e 2πω/κ − 1) + o(1) as f → 0 . (C.10) In (4.8c), a straightforward monotone convergence argument givesḞ 1 (ω, τ ) = o(1). In (4.8d), (C.6) givesḞ 2 (ω, τ ) = O(f ). Combining, we havė F(ω, τ ) = − ω 2 Θ(−ω) + ω 2 (e 2πω/κ − 1) + o(1) as τ → ∞ . (C.11) C.2 Travelling towards the mirror in the distant past We wish to extract the asmptotic behaviour of (4.11) as τ → −∞ and as τ → ∞. For (4.11c), we havė F 1 (ω, τ ) = 1 2π ∞ 0 cos(ωs) ds 1 + ge −κse λ se λ − 2τ sinh λ + κ −1 ln 1 + ge −κse λ 2 , (C.13) where g = e κτ e λ . When τ < 0, we may bound the absolute value ofḞ 1 (ω, τ ) by the replacements cos(ωs) → 1 and g → 0 in (C.13), and evaluating the integral that ensues givesḞ 1 (ω, τ ) = O(τ −1 ). For (4.11d), we proceed as with (C.6), obtaining the exact resulṫ F 2 (ω, τ ) = (1 − h) e 2λ 2π − 1 C + |ω| sin(C|ω|) Ci(C|ω|) − cos(Cω) si(C|ω|) + 2πω cos(Cω)Θ(−ω) , (C.14) where h = g/(1 + g) and C . = −(e 2λ − 1)τ − κ −1 e λ ln(1 − h). As τ → −∞, we have C → ∞, and using formulas (6.2.17) and (6.12 .3) in [48] giveṡ F 2 (ω, τ ) = e 2λ ω cos(2τ sinh λ e λ ω)Θ(−ω) + O(τ −3 ) . (C.15) Combining, we havė F(ω, τ ) = −ω 1 − e 2λ cos(2τ sinh λ e λ ω) Θ(−ω) + O(τ −1 ) . (C.16) C.2.2 τ → ∞ For (4.11b), proceeding as in (C.9) giveṡ F 0 (ω, τ ) = − ω 2 Θ(−ω) + ω 2 (e 2πe −λ ω/κ − 1) + o(1) as τ → ∞ . (C.17) For (4.11c), using (C.13) and substituting s = τ + r giveṡ F 1 (ω, τ ) = κ 2 2π ∞ −τ cos[ω(τ + r)] ds 1 + e −κre λ κτ e −λ + ln 1 + e κre λ 2 . (C.18) We may assume τ > 0. To bound the absolute value of (C.18), we make in the integrand the replacement cos[ω(τ + r)] → 1 and extend the integration to be over the full real axis in r. Elementary estimates then show that the contribution from −∞ < r < 0 is O(τ −2 ) and the contribution from from 0 < r < ∞ is O(τ −1 ). HenceḞ 1 (ω, τ ) = O(τ −1 ). For (4.11d), (C.14) givesḞ 2 (ω, τ ) = O e −e λ κτ . Combining, we havė F(ω, τ ) = − ω 2 Θ(−ω) + ω 2 (e 2πe −λ ω/κ − 1) + o(1) as τ → ∞ . (C.19) D Near-infinity and near-singularity transition rates in the (1 + 1)-dimensional Schwarzschild spacetime In this appendix we verify the near-infinity and near-singularity transition rate formulas (5.17) and (5.18) for the inertial detector in the (1 + 1)-dimensional Schwarzschild spacetime. D.1 Near-infinity transition rate We consider the E ≥ 1 trajectories (5.13) and (5.14) in Quadrant I. We wish to find the transition rate in the early time limit, assuming that the detector is switched on in the asymptotic past. Using (2.16) and (5.10), we finḋ A v (τ, τ ) = −v (τ )v(τ ) 4π[v(τ ) − v(τ )] 2 , (D.2b) Aū(τ, τ ) = −u (τ )u(τ ) 4π[ū(τ ) −ū(τ )] 2 , (D.2c) Av(τ, τ ) = −v (τ )v(τ ) 4π[v(τ ) −v(τ )] 2 . (D.2d) Using (5.13) and (5.14), it is straightforward to verify that as τ → −∞ with fixed positive s, we have A u (τ, τ − s) → − 1 4πs 2 , (D.3a) A v (τ, τ − s) → − where λ = arctanh √ 1 − E −2 . Taking the τ → −∞ limit under the integrals in (D.1), justified by the monotone convergence argument given below, and proceeding as in subsection (3.2), leads to the formulas (5.17) in the main text. What remains is to provide the monotone convergence argument. Let q stand for eitheru orv, and note from (5.4) has a fixed sign for all s > 0 when τ is sufficiently large and negative. Introducing in (D.6) a new integration variable by p = E 2 − 1 + 2M/r(τ ) , we see that it suffices to show that each of the functions f 1 (p) = 1 2 p f p dp (E + ηp )[p 2 − E 2 + 1] 2 − p f − p (E + ηp)[p 2 f − E 2 + 1] 2 + (E + ηp f )[p 2 − E 2 + 1] 2 , (D.7a) f 2 (p) = tanh 1 2 p f p dp (E + ηp )[p 2 − E 2 + 1] 2 − p f − p (E + ηp)[p 2 f − E 2 + 1] 2 + (E + ηp f )[p 2 − E 2 + 1] 2 , (D.7b) defined on the domain √ E 2 − 1 < p < p f , where p f ∈ √ E 2 − 1 , E is a parameter, has a fixed sign when p f is sufficiently close to √ E 2 − 1. Consider f 1 . f 1 is a rational function whose sign can be analysed by elementary methods, with the outcome that f 1 is negative when p f is sufficiently close to √ E 2 − 1. Hence f 1 is positive when p f is sufficiently close to √ E 2 − 1. Consider then f 2 . When p f is sufficiently close to √ E 2 − 1, an elementary analysis shows that the second term in (D.7b) is negative and strictly increasing, and there is a p 1 ∈ √ E 2 − 1 , p f such that this term takes the value −1 at p = p 1 . With p f this close to √ E 2 − 1, it follows that f 2 is negative for p ≤ p 1 , whereas for p 1 < p < p f f 2 has the same sign as f 3 can be analysed by the same methods as f 1 , with the outcome that f 3 is negative when p f is sufficiently close to √ E 2 − 1. Collecting, we see that f 2 is negative when p f is sufficiently close to √ E 2 − 1. This completes the monotone convergence argument. D.2 Near-singularity transition rate We consider the trajectories (5.13), (5.14), (5.15) and (5.16), with E ≥ 0, and with the field in the HHI and Unruh vacua. The switch-off moment τ is assumed to be in Quadrant II. The switch-on-moment τ 0 either is finite and in a region of the spacetime where the vacuum is regular, or for E ≥ 1 may alternatively be pushed to the asymptotic past. Let τ sing be the value of τ at the black hole singularity, and let τ 1 be a constant such that the detector is somewhere in Quadrant II at proper time τ 1 . In the limit τ → τ sing with everything else fixed, we havė cos[ω(τ − τ )] −v (τ ) v(τ ) − v(τ ) + 1 τ − τ + O(1) = 1 16πM 2M r(τ ) 3/2 − E 2M r(τ ) + 1 + E 2 2 2M r(τ ) 1/2 + O(1) , (D.13) where we have first integrated by parts, observing that the new integral term is O(1) by virtue of near-singularity estimates that ensue from (D.11) and (D.12), and then evaluated the limit using (D.11) and (D.12). For Gv(ω, τ, τ 1 ) we may proceed similarly, usingv = 4M exp[v/(4M )]. The differences from G v (ω, τ, τ 1 ) turn out to be O(1), so that Gv(ω, τ, τ 1 ) = 1 16πM 2M r(τ ) For Gū(ω, τ, τ 1 ) the analysis is as for Gv(ω, τ, τ 1 ) but with E → −E, with the result Gū(ω, τ, τ 1 ) = 1 16πM 2M r(τ ) Figure 1 : 1Minkowski spacetime with the receding mirror (4.2) and an inertial detector (3.1) that is static with respect to the mirror in the asymptotic past. Dashed lines show a selection of null geodesics that bounce off the mirror. Figure 2 : 2−ūv = (4M ) 2 r/(2M ) − 1 e r/(2M ) .(5.2) The figure shows a perspective plot of the transition rateḞ(ω, τ ) (4.8) for the detector (3.1) that is asymptotically static with respect to the mirror in the distant past, with d = 1/κ. Figure 3 : 3Cross-sections of the plot in Figure 2 at (a) ω = −κ and (b) ω = κ. The dashed horizontal lines show the past and future asymptotic values (4.9). 18) for both the HHI vacuum and the Unruh vacuum: the differences between the two vacua show up only in the O(1) part. In terms of τ , the leading term in (5.18) is 1/[6π(τ sing − τ )], where τ sing is the value of τ at the black hole singularity. TFigure 4 :Figure 5 : 45as,KMS (ω, τ ) . = ω ln Ḟ (−ω, τ )/Ḟ(ω, τ ). The solid (red) curve shows MḞ as a function of M ω for the E = 1 trajectory in the HHI vacuum, at the times (a) τ = −10M , (b) τ = −3.5M and (c) τ = −1.5M , all of them before the horizon-crossing, which occurs at τ = τ h = −(4/3)M . The dashed (blue) curve shows M times the Minkowski thermal bath response (3.16) at the local Hawking temperature T loc (5.9) and with the Doppler shift factor λ = λ loc = arctanh 2M/r , as a function of M ω. The discrepancy between the two curves shows that the Planckian character of the transition rate is lost as τ approaches τ h , where the solid curve remains finite but the dashed curve disappears to +∞. The solid (red) curve is as inFigure 4but for the Unruh vacuum. The dashed (blue) curve shows M times the Minkowski response (3.18) in a state with vacuum leftmovers and thermal right-movers, at the local Hawking temperature T loc (5.9) and with the Doppler shift factor λ = λ loc = arctanh 2M/r . The discrepancy between the two curves again shows loss of the Planckian character as τ approaches τ h , where the solid curve remains finite but the dashed curve disappears to +∞. Note the discontinuous slope of the dashed curve at ω = 0. Figure 6 : 6(a) The solid (blue) curve shows MḞ 1/(4πM ) as a function of τ /M for the E = 1 trajectory in the HHI vacuum. The dashed (red) line shows the value 1/[4π(e 2 −1)] to which the solid curve asymptotes at τ /M → −∞. (b) The solid (blue) curve shows a close-up of (a) near the horizon-crossing, τ /M = −4/3. The dash-dotted (green) curve shows the τ -dependent terms included in the asymptotic τ → 0 expression (5.18). Figure 7 : 7The transition rate for a detector on the E = 0 trajectory (5.16) in the HHI vacuum, with the switch-on at ϕ = −9π/10, where τ ≈ −3.136M . The white hole singularity is at ϕ = −π, where τ = −πM . The divergence of the transition rate in the limit of short detection time and in the limit of approaching the black hole singularity is evident in the plot. have first integrated by parts in u, then written the derivatives in Q ω (u − s) as s-derivatives outside the u-integral, then used the definition of Q ω , and finally integrated by parts in s. The substitution term from s = 0 in the integration by parts vanishes because cos(ωs) ∞ −∞ du χ(u)χ(u − s) is even in s, and the integral over s in the last expression in (A.4) is convergent at small s for the same reason. χ(u)[χ(u) − χ(u − s)] vanishes at s = 0 and is even in s. Combining (A.3) and (A.5), we obtain B Evaluation of the static detector's transition rate in Minkowski (half-)spaceIn this appendix we verify formulas (3.4) and (3.7) for the transition rate of a static detector in Minkowski space and Minkowski half-space. We use the Wightman functions found in Subsection 3.1 and evaluate the transition rate from (2.17). F 0 0(ω, τ ) = −ωΘ(−ω) + O e e λ κτ . (C.12) FF B (ω, τ ) = −ωΘ(−ω) + 2 ∞ 0 ds cos(ωs) A u (τ, τ − s) + A v (τ, τ − s) H (ω, τ ) = −ωΘ(−ω) + 2 ∞ 0 ds cos(ωs) Aū(τ, τ − s) + Av(τ, τ − s) + 1 2πs 2 , (D.1b) F U (ω, τ ) = −ωΘ(−ω) + 2 ∞ 0 ds cos(ωs) Aū(τ, τ − s) + A v (τ, τ − s) + 1 2πs 2 , (D.1c) where A u (τ, τ ) = −u (τ )u(τ ) 4π[u(τ ) − u(τ )] 2 ,(D.2a) Aū(τ, τ − s) → − e 2λ 4π(8M ) 2 sinh 2 e λ s/(8M ) , (D.3c) Av(τ, τ − s) → − e −2λ 4π(8M ) 2 sinh 2 e −λ s/(8M ) , (D.3d) = 1 for q =v and η = −1 for q =u. well defined for all s > 0 when τ is sufficiently large and negative. For monotone convergence, it suffices to show that each of the expressions in (D.5) is monotone in τ for all s > 0 when τ is sufficiently large and negative. Differentiating (D.5) with respect to τ , it suffices to show that each of the expressions ηp)[p 2 f − E 2 + 1] 2 + (E + ηp f )[p 2 − E 2 + 1] 2 . (D.8) F H (ω, τ ) = Gū(ω, τ, τ 1 ) + Gv(ω, τ, τ 1 ) + O(1) , (D.9a) F U (ω, τ, τ 1 ) = Gū(ω, τ, τ 1 ) + G v (ω, τ, τ 1 ) + O(cos[ω(τ − τ )] Aū(τ, τ ) + 1 4π(τ − τ ) 2 , (D.10c)and A v , Av and Aū are given in (D.2). Consider first G v (ω, τ, τ 1 ), and assume τ 1 < τ < τ sing . Working in the coordinates (v, r), well defined in Quadrant II, the equations for the trajectory reaḋ r = − E 2 − 1 + 2M/r , D.9), (D.13), (D.14) and (D.15) yields (5.18). Table 1 : 1The four quadrants of the extended Schwarzschild spacetime. The columns show the signs of the Kruskal coordinatesū andv, the norm squared of the Killing vector ξ, and the range of the function r. In the exteriors, where ξ is timelike, it is future-pointing in Quadrant I and past-pointing in Quadrant III. 10c ) 10cwhere ∆u = u − u and similarly for the other coordinates. Each of the Wightman functions is unique up to a real-valued additive constant. The Boulware and HHI vacuum Wightman functions are invariant under the isometries generated by the Killing vector ξ. The Unruh vacuum Wightman function changes under these isometries by an additive constant; however, the Unruh vacuum may nevertheless be considered invariant under the isometries since the stress-energy tensor and other quantities built from derivatives of the Wightman function are invariant Stationary checks: massless limit, the Unruh effect, and inertial response in a thermal bath In this section we perform reasonableness checks on the derivative-coupling detector in three stationary situations. 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We study extreme ultraviolet (EUV) radiation induced defects in single-layer graphene. Two mechanisms for inducing defects in graphene were separately investigated: photon induced chemical reactions between graphene and background residual gases, and breaking sp 2 bonds, due to photon and/or photoelectrons induced bond cleaving. Raman spectroscopy shows that D peak intensities grow after EUV irradiation with increasing water partial pressure in the exposure chamber. Temperature-programmed desorption (TPD) experiments prove that EUV radiation results in water dissociation on the graphene surface. The oxidation of graphene, caused by water dissociation, is triggered by photon and/or photoelectron induced dissociation of water. Our studies show that the EUV photons cleave the sp 2 bonds, forming sp 3 bonds, leading to defects in graphene.

1401.2352

bef6c0ad313ae6c99ad9528d062204ad9f777f78

Extreme ultraviolet radiation induced defects in single-layer graphene 3 Jan 2014 A Gao FOM-Dutch Institute for Fundamental Energy Research Edisonbaan 143439 MNNieuwegeinthe Netherlands ) XUV group MESA+ Institute for Nanotechnology University of Twente PO Box 2177500 AEEnschedethe Netherlands E Zoethout FOM-Dutch Institute for Fundamental Energy Research Edisonbaan 143439 MNNieuwegeinthe Netherlands J M Sturm FOM-Dutch Institute for Fundamental Energy Research Edisonbaan 143439 MNNieuwegeinthe Netherlands ) XUV group MESA+ Institute for Nanotechnology University of Twente PO Box 2177500 AEEnschedethe Netherlands Materials innovation institute M2i Mekelweg 22628 CDDelftthe Netherlands C J Lee FOM-Dutch Institute for Fundamental Energy Research Edisonbaan 143439 MNNieuwegeinthe Netherlands ) XUV group MESA+ Institute for Nanotechnology University of Twente PO Box 2177500 AEEnschedethe Netherlands F Bijkerk FOM-Dutch Institute for Fundamental Energy Research Edisonbaan 143439 MNNieuwegeinthe Netherlands ) XUV group MESA+ Institute for Nanotechnology University of Twente PO Box 2177500 AEEnschedethe Netherlands Extreme ultraviolet radiation induced defects in single-layer graphene 3 Jan 2014(Dated: 13 January 2014)PACS numbers: 6148De We study extreme ultraviolet (EUV) radiation induced defects in single-layer graphene. Two mechanisms for inducing defects in graphene were separately investigated: photon induced chemical reactions between graphene and background residual gases, and breaking sp 2 bonds, due to photon and/or photoelectrons induced bond cleaving. Raman spectroscopy shows that D peak intensities grow after EUV irradiation with increasing water partial pressure in the exposure chamber. Temperature-programmed desorption (TPD) experiments prove that EUV radiation results in water dissociation on the graphene surface. The oxidation of graphene, caused by water dissociation, is triggered by photon and/or photoelectron induced dissociation of water. Our studies show that the EUV photons cleave the sp 2 bonds, forming sp 3 bonds, leading to defects in graphene. I. INTRODUCTION Graphene, a two-dimensional hexagonal packed sheet of carbon atoms, has attracted a lot of attention from different research fields due to its unique physical and chemical properties 1-8 . Graphene can act as a diffusion barrier by providing physical separation between an underlying substrate and reactant gases. Indeed, studies show that graphene is highly impermeable to gases 9 . Furthermore, single-layer graphene is also highly transparent 10 , which makes it a promising protection layer for optical devices, such as mirrors, lenses and screens. Additionally, graphene can be grown on large scales by chemical vapor deposition (CVD) and transferred to arbitrary substrate 11 , making the range of potential applications very wide. Graphene devices may be used in harsh environments e.g., space applications, soft x-ray systems and extreme ultraviolet (EUV) lithography systems. Since the light source of current EUV lithography systems operates at a wavelength of 13.5 nm, the optical system is in vacuum, but with a significant partial pressure of water. The physical and chemical stability of graphene in such an environment is of critical importance if it is to be used as a part of an optical component. Early experimental work showed that defects were generated in multilayer graphene after EUV exposure. And, even in a reducing environment, a small partial pressure of oxidizing agents (water) may cause oxidation 12 . The damage to graphene observed in our previous study was attributed to two possible effects: photo-induced chemistry between graphene and background residual gases, and breaking sp 2 bonds, due to EUV photon and/or photoelectron ina) Electronic mail: [email protected]. duced bond cleaving. In this work, we investigate both of these possible EUV-induced damage mechanisms on CVD grown single layer graphene. In previous work 12 , the oxidative damage to graphene was tentatively attributed to the presence of a reactive low density water plasma. But in the ref 12 , the amount of water (which was the main source of oxidative damage to graphene) adsorbed to the graphene surface was not controlled, nor was it possible to perform experiments without a substantial plasma-surface interaction. In this work, the influence of the EUV-induced water plasma is controlled in two different ways. In the first set of experiments, the partial pressure of water in the vacuum chamber was set by introducing water vapor into the interaction chamber. In the second set of experiments, water layers were deliberately deposited on a cold graphene surface (83 K). In the first set of experiments, the graphene is exposed to an EUV-induced plasma, which contains different concentrations of water plasma. In the second set of experiments, the solid water layer can be ionized into much denser plasma under EUV irradiation. In addition, since this water layer is physisorbed onto the graphene surface, the reaction probability is much higher compared with that in the first set of experiments. By comparing the nature and density of defects induced in these two experiments, it is possible to determine the relative contribution of the EUV-induced plasma to the damage observed in graphene after exposure. Finally, in a third set of experiments, naturally accumulated hydrocarbon contamination (0.7 nm) on a graphene surface was used as a barrier layer between the residual water and graphene surface. In this way, it was possible to study the damage to graphene, while minimizing the reaction rate between graphene and water plasma. In our experiments, the graphene was character-ized using Raman spectroscopy, X-Ray photoelectron spectroscopy (XPS), and scanning electron microscopy (SEM), before and after exposure. The ratio of the D and G Raman spectral features is used as a measure of how well-ordered the graphene crystalline structure is, while XPS is used to determine the amount of oxidation. SEM was used to qualitatively compare the graphene layer completeness. The adsorbed water was examined using temperature-programmed desorption (TPD) spectroscopy to understand the water layer morphology, water-water, and water-graphene interactions II. EXPERIMENTS For the first and second sets of experiments, single layer graphene samples, obtained from Graphene Supermarket Inc., were grown on copper substrates by CVD and transferred to SiO 2 /Si substrates. The substrate size was 10 mm x 10 mm with a 285 nm thick layer of SiO 2 . For the third set of experiments, single layer graphene samples on Cu substrates were purchased from Graphenea, which were stored in ambient condition for several months, leading to a layer of hydrocarbons about 0.7 nm thick on top of the graphene layer. For the first and third sets of experiments, the graphene samples were exposed to EUV from a Xe plasma discharge source (Philips EUV Alpha Source 2) with a repetition rate of 500 Hz. The EUV beam profile has a Gaussian distribution with FWHM= 3 mm. The EUV intensity at the sample surface was estimated to be 5 W/cm 2 with a dose of 10 mJ/cm 2 per pulse. The base pressure of the exposure chamber was 2x10 −9 mbar. For the second set of experiments, the graphene samples were exposed with EUV intensity of 0.05 W/cm 2 , and the base pressure of the exposure chamber was 1x10 −9 mbar. All exposed samples were characterized using Raman spectroscopy, XPS, and SEM after exposure. Raman spectra were collected with a home-built system, based on a 532 nm solid state laser system and a Solar M266 spectrometer with a resolution of 1 cm −1 . The illumination intensity was set at 200 W/cm 2 . The collection optics and pixel size of the detector result in a spatial resolution of 100 × 100 um 2 . The collection efficiency of the detector system was calibrated using the HG-1 Mercury Argon Calibration Light Source and AvaLight-D(H)-S Deuterium-Halogen Light Source. XPS was measured by using a monochromatic Al-K, Thermo Fisher Theta probe with a beam footprint of 1 mm diameter. III. RESULTS AND DISCUSSION A. Photo-induced plasma of the residual water Four graphene samples (single layer graphene on SiO 2 /Si substrate) were exposed to EUV for 30 min with various water partial pressures. One of the graphene samples was exposed without adding water into the chamber (water partial pressure less than 10 −9 mbar), while another three samples were exposed with water partial pressures of 1x10 −7 mbar, 1x10 −6 mbar, and 1x10 −5 mbar, respectively. These four samples were kept at 289 K by backside cooling during the exposure. After exposure, all the samples were examined by Raman spectroscopy and XPS. A typical Raman spectrum of graphene has three prominent features i.e., D, G and 2D peaks, located at approximately 1350 cm −1 , 1580 cm −1 , and 2680 cm −1 . The G peak is a first order Raman scattering process, corresponding to an in plane stretching of sp 2 bonds. The D band is due to the breathing modes of six-atom rings, and requires a defect for activation. The ratio of I(D)/I(G) is commonly used to quantify the defect density 13 . Fig. 1a shows the Raman spectra of the reference graphene sample (unexposed) and the exposed graphene samples. It clearly shows that the D peak height grows after exposure for all samples, indicating defect generation. Furthermore, the dependence of the I(D)/I(G) ratio on water partial pressure is clearly indicated in Fig. 1b. The Raman spectra, however, do not clearly show the source of the increased defect density. Fig. 2a shows the curve fit for the XPS spectrum of the C1s peak of the pristine sample. The four components of the C1s spectrum, corresponding to sp 2 bonds in graphitic like carbon (284.3 eV), sp 3 hybridization (285.2 eV), hydroxyl (C-O) groups (286.1 eV), and carboxyl (C=O) groups (288 eV), are plotted separately for the exposed sample. The appearance of sp 3 , C-O, and C=O bonds can be attributed to photo-induced bond cleaving and photo-induced oxidation of graphene, due to presence of water. However, Fig. 2b also shows that the total carbon thickness of the exposed samples increases, due to the well-known effect of hydrocarbon deposition during EUV exposure 14 . The contribution of the deposited hydrocarbons to the XPS spectra makes a single interpretation of the data impossible, since the sp 3 , C-O, and C=O contributions can also come from the hydrocarbon contamination. Note that the EUV transmission will decrease by less than 1 % due to 0.3 nm carbon contamination 15 , thus, the difference of EUV dose on graphene samples is negligible, and cannot explain the differences between Raman spectra for the five samples. B. Photo-induced plasma of the adsorbed water In order to prevent hydrocarbon contamination, a layer of water was deliberately deposited on the graphene surface by dosing the surface with water while it was held at a temperature of 83 K. Directly before water dosing, the graphene sample was heated to 600 K to remove surface contamination, after which a calibrated surface coverage of water was adsorbed onto the sample. The water dose is expressed in Langmuir, with 1 L = 1.33x10 −6 mbar. Temperature-programmed desorption (TPD) spec- The curve fitting results of the XPS spectrum for the graphene samples on SiO2/Si substrate exposed to EUV with water partial pressure of 1x10 −5 mbar. The solid curve indicates the C1s peak while the four dot-dash curves are fitted curves. (b) The total carbon thickness of the exposed samples with various water partial pressures. Note that the unexposed sample has slightly hydrocarbon contamination on the surface due to transport and storage in atmospheric conditions. The thickness is calculated based on the angle resolved XPS measurements assuming the carbon density of 2.1 g/cm 3 . The x axis is in log scale. troscopy was performed to investigate how water adsorbs on to the graphene surface. The graphene sample was heated at a rate of 0.5 K/s and water desorption was detected by a quadruple mass spectrometer. To ensure that the signal is dominated by sample surface desorption, the entrance to the mass spectrometer consists of a cone that faces the sample surface. The cone has an entrance aperture of 4 mm, located approximately 2 mm from the graphene surface. The temperature was measured with a thermocouple pressed onto the sample with a molybdenum clamp. Since this clamp has a relatively good thermal contact with the cold finger of the manipulator, the measured temperature will underestimate the actual sample temperature. In our experimental procedure, the graphene sample was dosed with water, and TPD spectroscopy was performed prior to EUV exposure. The sample was then re-dosed with water and exposed to EUV. A second TPD spectrum was obtained from the exposed water-graphene sample. Finally, the sample was dosed with water a third time and a third TPD was performed on the exposed graphene. This procedure was repeated three times on the same sample after EUV exposure of 10 min, 30 min, and 180 min respectively. The EUV intensity in this exposure was estimated to be 100 µ J/cm 2 , about 100 times lower than used in the first set of experiments. The TPD results are shown in Fig. 3a. The spectra at low water dosages and high water dosages are markedly different, indicating that the structure of the adsorbed water is different, or evolves differently as function of both temperature and initial dose. As graphene (with certain hydrocarbon contamination) has a hydrophobic surface, water films tend to dewet upon heating. At low coverage, the water molecules form two-dimensional (2D) FIG. 4: (color online) (a) Raman spectra of the graphene sample before (pristine) and after EUV radiation with water adsorbed onto the sample (exposed).(b) Raman spectra of the samples exposed with or without water adsorbed onto graphene surface. clusters. Water molecules at edges of these clusters have low coordination to other water molecules, such that their desorption energy is low, resulting in the desorption peak around 130 K [16][17][18] . However, as the coverage increases, water molecules can form 3D clusters of crystalline-ice like structure, resulting in a second peak at 134 K. The position (temperature) of this peak increases with coverage, while the leading edge remains similar. This is characteristics for zero-order desorption of water molecules from ice 17,18 . The rearrangement of the water suppresses the peak at 130 K for water doses above 3 L. At a water dose of 5 L the 130 K peak disappears, which indicates that all water molecules form multilayers or 3D ice clusters. The TPD data from Fig. 3a also indicates that water does not dissociate on adsorption to the graphene surface, thus, any water dissociation and resulting changes to the graphene are driven by EUV-induced processes as discussed in the following. In Fig. 3b, the TPD spectra of water on graphene before and after exposure are shown for a water dose of 1 L. It is shown in Fig. 3b that the spectra of the graphene sample after exposure exhibits lower peak intensities than that of before exposure. This indicates that the amount of water on the surface decreases due to EUV-induced desorption of water. Note that, aside from the decrease in the total amount of water on the graphene surface, a second TPD peak appears at higher temperature after 30 min of exposure. This second peak is likely due to the water layer rearranging itself during EUV exposure. EUV photons with an energy of 92 eV can also cause water dissociation by direct photon excitation or by an indirect process involving secondary electrons emitted from the substrate. Many studies have shown that the indirect process, induced by secondary electrons, dominates over the direct photon excitation in the surface pho-tochemistry 14,19,20 . Therefore, direct EUV-dissociation can be neglected and the water molecules can be dissociated according to the following reactions 14 : e − + H 2 O → H + + OH − (1) e − + OH − → H + + O −(2) Upon dissociation, the reactive species can either desorb from the surface or react (mainly atomic oxygen) with the graphene. Fig. 4a shows the Raman spectra of the graphene sample before and after EUV exposure with adsorbed water layer. Comparing these two spectra in Fig. 4a, there is a clear D peak intensity increase at around 1350 cm −1 , which is attributed to the presence of defects in graphene. Previous work has shown that defects in graphene can be induced by EUV radiation 12 , however, the shift of the G peak from 1585 cm −1 to 1595 cm −1 and of the 2D peak 2681 cm −1 to 2691 cm −1 , are usually interpreted as evidence of oxide doping 21 . Fig. 4b shows the Raman spectra of samples exposed with and without water adsorbed on to the graphene surface. The two spectra differ in both the I(D)/I(G) ratio and the spectrum fluorescence background. Clearly, the I(D)/I(G) ratio of the sample exposed with water is much higher than that of the sample exposed without water, indicating more defects were generated in the former sample. The water layer adsorbed to the graphene surface will result in a very dense water plasma, and such a plasma should have a faster reaction rate than that of water plasma generated in the background residual gas. The large fluorescence signal in the sample exposed without water is attributed to hydrocarbon contamination. The XPS spectra of the graphene sample exposed to EUV with adsorbed water onto graphene. The solid curve indicates the C1s peak while the four dot-dash curves are fitted curves. Fig. 5 shows the curve fit for the C1s peak of the sample exposed to EUV with adsorbed water. The components of the C1s spectrum: sp 2 , sp 3 , C-O, and C=O, are plotted. The atomic concentration of each component and full width half maximum of the spectral components for the pristine sample, and samples exposed to EUV with and without adsorbed water are summarized in Tab. I. The broadening of the sp 2 peak from 0.88 eV to 0.98 eV is usually evidence of a transition from a highly ordered graphite-like carbon to a less ordered carbon state, supporting the conclusion from the Raman results that the exposed graphene has more defects. The changes in C-O and C=O concentrations are not significant, considering that the uncertainty in the atomic concentration is ±2.5 at.%. The C-O and C=O bonds are most likely to be due to residual poly (methyl meth acrylate), used during the transfer process, and clearly observed on all samples using SEM (see Fig. 6). From the SEM images in Fig. 6, it is apparent that there is less PMMA on the graphene after exposure. Cracks and holes are also observed in the SEM images for the exposed samples, indicating an oxidative etching effect. We can see that the exposed graphene samples show more cracks and holes than the pristine sample. And these holes and cracks are predominantly along graphene grain boundaries, a phenomenon also observed for thermal oxidation of graphene 22,23 . The formation of C-O or C=O groups as a result of oxidation is due to fully dissociated water (e.g., atomic oxygen), or partially dissociated water (e.g., OH groups). The small holes in the pristine sample may come from the transfer process. It is also noted that the total thickness of carbon in the exposed sample decreases, due to both the etching of graphene and the removal of PMMA. This can be seen in the XPS data in Tab. I, where the concentration of the sp 2 and sp 3 components decrease, while both the C-O and C=O contributions remain almost unchanged. Cleaved sp 2 bonds can either form sp 3 bonds, or through oxidation form both sp 3 bonds and C-O or C=O bonds. The combination of decreasing carbon, but increasing oxidation is indicative that the PMMA is preferentially removed. However, the unchanged sp 3 and oxygen content (despite the disruption to the graphene and the overall removal of carbon) indicates that two competing processes are present. The removal of PMMA is compensated for by the oxidation of the graphene. Until now, we have shown that from Raman results, I(D)/I(G) ratio grows as the water partial pressure increases, indicating more defects are forming in graphene. Together with the XPS data, it suggests that oxidation, triggered by the EUV radiation in the presence of water, results in the graphene being etched, forming cracks and holes, which is similar to the effect observed during thermal oxidation of graphene. In the case of EUV exposure, the oxidation process originates from the dissociation of water by EUV. However, to determine the relative contributions of EUV-induced oxidation and EUV-induced bond cleaving, the EUV-induced plasma must be separated from the graphene surface. I: Atomic concentration of C1s component, the total carbon thickness, and FWHM of the pristine sample, and samples exposed to EUV with and without adsorbed water. The total thickness is calculated based on the angle resolved XPS measurements assuming the carbon density of 2.1 g/cm 3 . (a) unexposed sample (b) exposed sample FIG. 6: SEM images of the pristine sample, and the sample exposed to EUV with adsorbed water showing PMMA residue, cracks and holes. The PMMA appears as the darker patches in both images. Sample C. Exposing graphene with hydrocarbon contamination Samples of single layer graphene on Cu substrate with a hydrocarbon contamination layer were prepared. The hydrocarbon layer is used as barrier layer between the residual water and graphene surface. In this way, it was possible to study the damage to graphene, while minimizing the reaction rate between graphene and water plasma. Four graphene samples were exposed to EUV for 30 min with background gas conditions that should be either reducing or oxidizing. One of the graphene samples was exposed without modifying the background gas (10 −9 mbar, mostly water), a second sample was exposed in a water partial pressure of 1x10 −5 mbar, a third was exposed in a hydrogen partial pressure of 1x10 −5 mbar, and the last was exposed in an oxygen partial pressure of 1x10 −5 mbar. All samples were kept at 289 K by backside cooling during the exposure. Fig. 7 shows the Raman spectrum and XPS results for an unexposed sample. From the Raman spectra, we can see that the single layer graphene is still visible after fluorescence background subtraction, despite the presence of hydrocarbon contamination. The XPS results clearly show that, in addition to the sp2 contribution, there is a significant amount of sp 3 , C-O, and C=O, which is attributed to the presence of hydrocarbon. Assuming a density of 2.1 g/cm 3 , the hydrocarbon contamination layer was found to be about 0.7 nm thick from XPS measurement (see Tab. II). This hydrocarbon layer acts as a barrier layer between the graphene and the background residual gases in the exposure chamber. In this way, reactions between the EUV-induced plasma and the graphene can be excluded. Fig. 8a shows the Raman spectra of the reference (unexposed) and exposed graphene samples with naturally accumulated hydrocarbon contamination. It is noted that all the exposed samples show a clear D peak with approximately the same intensity. The G peak, however, is broadened due to a more disordered carbon network after exposure. Analysis is complicated by the fact that all samples, except the one exposed to oxygen, show an increase in hydrocarbon carbon (see Tab. II), which contributes to, not only the broadening, but also the increased intensity of the G peak. In Tab. II, it is also apparent that the sp 2 atomic concentration is greatly reduced after exposure, from 53.3 at.% to 34.0 at.%, indicating that sp 2 bonds are being broken and sp 3 bonds are being formed. The C-O and C=O peaks are obviously more pronounced in the exposed samples shown in the XPS results in Fig. 8b due to EUV induced oxidation of the covering hydrocarbon carbon layer. These results show that direct contact of the EUVinduced plasma and the graphene is not a necessary requirement for damage to graphene. The I(D)/I(G) 8: (a) Raman spectra for the reference and exposed graphene samples on Cu substrate with naturally accumulated hydrocarbon contamination. (b) XPS results for the graphene sample exposed to oxygen ratio (0.2) found in Fig. 8 is much lower compared with the I(D)/I(G) ratio (1.9) found in Fig. 1 for the sample exposed to EUV with partial water pressure of 1x10 −5 mbar, indicating that the hydrocarbon carbon layer is an effective barrier layer. In addition, in Fig. 8a, the I(D) intensities are almost the same for all exposed samples, indicating that the process is largely independent of the background gases. Considering these facts, the induced defects are predominantly due to photon and/or photoelectron induced bond cleaving, rather than the EUV-induced plasma. The photoelectrons from the Cu substrate may play an important role in defect gener-ation in graphene, since the Cu substrate has a relatively high photoelectron yield 24 . These photoelectrons may directly attribute in breaking the carbon sp 2 bonds. It is important to note that the damage to graphene caused by plasma depends very strongly on the concentration and composition of the background gases, while, on the other hand, the damage to graphene caused by photon and photoelectron is only dependent on the photon flux. It is, therefore, impossible to determine the relative dominance of these two mechanisms, without specifying the background conditions, photon flux and even the type of substrate (photoelectron yield). Judging from the data in Fig. 8, the contribution to I(D)/(G) from photon and photoelectrons is about 0.2. Considering the experimental results shown with Fig. 1 (where the I(D)/(G) is about 1.9), we can conclude that, in those experiments, EUV-induced plasma is the main source for the defect generation. This finding is reinforced by the fact that the photoelectron yield for the SiO 2 substrate is much less than that from the Cu substrate. IV. CONCLUSION We have studied the damage mechanism of CVD grown single layer graphene under EUV irradiation. We found that the residual water in the vacuum chamber, EUV photons, and/or photoelectrons, all contribute to defect generation in graphene during EUV exposure. The experimental data demonstrate that, under EUV radiation in the presence of water, defects were generated through oxidation, resulting in the graphene being etched, forming cracks and holes, which is similar to thermal oxidation of graphene. The oxidation process originates from the dissociation of water by EUV, and the fact that the EUV photons directly break the sp 2 bonds forming sp 3 bonds, which leads to defects in graphene. The photoelectrons emitted from the substrate can either cause oxidation via dissociating water molecules on the graphene surface, or directly break sp 2 bonds, both of which will induce defects in graphene. Our results help understand lifetime considerations for graphene devices in the presence of hard radiation. Furthermore, the EUV-induced oxidation of graphene provides a possible route to resist-free patterning of graphene. FIG. 1 : 1(color online) (a) The Raman spectra of the graphene samples on SiO2/Si substrate exposed to EUV under different water partial pressure. (b) The I(D)/I(G) ratio as function of the water partial pressure. The x axis is in log scale. FIG. 3 : 3(color on line) Temperature-programmed desorption spectra for H2O desorption on the graphene surface (a) TPD spectra for various H2O doses; (b) TPD spectra of the graphene sample under different EUV exposure time with water dosing time of 1 L. The heating rate is 0.5 K/s. The curves are manually offset to the have the same background. FIG. 5: (color online) The XPS spectra of the graphene sample exposed to EUV with adsorbed water onto graphene. The solid curve indicates the C1s peak while the four dot-dash curves are fitted curves. FIG. 7 : 7(color online) Raman spectrum after fluorescence background subtraction (a) and XPS results (b) for the reference sample of monolayer graphene on Cu substrate with naturally accumulated hydrocarbon contamination. TABLE TABLE II : IIAtomic concentration and total carbon thickness of the reference and exposed graphene samples with naturally accumulated hydrocarbon contamination. The error margin of the data is ±1 at.%.Binding Energy [eV]Counts a.u.Sample Reference Exposed to background gas Exposed to H2O Exposed to H2 Exposed to O2 C sp 2 (at.%) 53.3 34.0 36.3 36.2 35.7 C sp 3 (at.%) 10.0 13.0 10.6 11.0 10.1 C-O (at.%) 1.9 5.0 5.4 7.0 5.1 C=O (at.%) 2.0 11.2 8.6 6.7 6.4 O 1s (at.%) 13.0 22.8 23.2 22.8 23.9 Cu2p3 (at.%) 19.8 14.0 15.9 16.3 18.9 Total thickness (nm) 0.98 1.24 1.10 1.06 0.92 1000 1200 1400 1600 1800 2000 0 1 2 3 4 x 10 7 wavenumber cm−1 counts/s reference background O2 H2 H2O (a) 283 284 285 286 287 288 289 290 200 400 600 800 1000 1200 1400 1600 C1s sp3 C−O C=O sp2 (b) FIG. A. K. Geim and K. S. Novoselov, "The rise of graphene," Nature materials6, 183-191 (2007). ACKNOWLEDGMENTSThe authors would like to thank Mr. Goran Milinkovic, Mr. Luc Stevens, Mr. John de Kuster, and Dr. Edgar Osorio for the help with sample preparation and experimental measurements. 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Wireless sensor networks consist of hundreds to thousands of sensor nodes and are widely used in civilian and security applications. One of the serious physical attacks faced by the wireless sensor network is node clone attack. Thus two node clone detection protocols are introduced via distributed hash table and randomly directed exploration to detect node clones. The former is based on a hash table value which is already distributed and provides key based facilities like checking and caching to detect node clones. The later one is using probabilistic directed forwarding technique and border determination. The simulation results for storage consumption, communication cost and detection probability is done using NS2 and obtained randomly directed exploration is the best one having low communication cost and storage consumption and has good detection probability.

10.14445/22315381/ijett-v8p253

1403.2548

f5d5560da15fcbfd1685a521c09c311011340105

Detection of Node Clones in Wireless Sensor Network Using Detection Protocols Neenu George [email protected] Anna University CoimbatoretamilnaduIndia T K Parani [email protected] Anna University CoimbatoretamilnaduIndia Detection of Node Clones in Wireless Sensor Network Using Detection Protocols International Journal of Engineering Trends and Technology #1 II Year M.E Student, #2 Assistant Professor #1, #2 ECE Department, Dhanalakshmi Srinivasan College of Engineering,wireless sensor networks (wsn)distributed hash tablerandomly directed exploration Wireless sensor networks consist of hundreds to thousands of sensor nodes and are widely used in civilian and security applications. One of the serious physical attacks faced by the wireless sensor network is node clone attack. Thus two node clone detection protocols are introduced via distributed hash table and randomly directed exploration to detect node clones. The former is based on a hash table value which is already distributed and provides key based facilities like checking and caching to detect node clones. The later one is using probabilistic directed forwarding technique and border determination. The simulation results for storage consumption, communication cost and detection probability is done using NS2 and obtained randomly directed exploration is the best one having low communication cost and storage consumption and has good detection probability. I.INTRODUCTION A wireless sensor network (wsn) is a high and new technology consists of spatially distributed autonomous sensors to monitor physical or environmental conditions and to pass data through the network to a main location. It is built of hundreds or thousands of nodes and each node act as sensor. Wireless sensor network consists of base stations and number of wireless sensors. These sensors node network has transceiver, micro controller, electronic circuit and energy source. Sensor networks have significant constrains and the individual sensor nodes are typically inexpensive, tiny, distributed, low power and low complexity nodes which used lightweight processors and cheap hardware components of low tamper résistance. And these sensor nodes are often deployed in hostile environments, are highly independent and require only a minimum amount of supervision. The cost of these sensor nodes depend on resources such as energy, memory, speed, bandwidth etc..They are widely used in physical and environmental situations. The wireless sensor network avoids the use of lot of wiring, can accommodate any devices at any time, and is accessed through centralized monitor and highly flexible [8]. The main goal of wsn is to reduce power consumption and to optimize computing resources. The bandwidth range of wsn is radio frequency. Wsn are ad hoc networks (wireless nodes that self organize into an infrastructure less network). In contrast to other adhoc network, wsn need essentially sensing and data processing. It has many more nodes and is densely deployed [8]. Hardware must be cheap and nodes are more prone to failures. Communication scheme is many to one (data collected at base station) rather than peer to peer and nodes are static. The main problems deals with wsn are easy to hack, low speed of communication, high cost and interference. Because of this hackers many attacks affects the wsn. Among those attacks serious and dangerous one is node clone attack. In this attack the adversary may capture some nodes in the network when they are in hostile environment and extract the secret credentials data and information from nodes, reprograms or modifies the data and creates replicas or clones of such nodes in the network. Then these compromised nodes plays active in network and thus the adversary may gain the control over the network [1]. Thus security of network had lost and more over these cloned nodes can create more attacks like DoS inside the network which corrupts the information [2]. If these clones are left undetected, the network is unshielded to attackers and thus extremely vulnerable [3]. Therefore in this paper, an effective two novel node clone detection protocols are proposed to detect the node clones. The previous works incurs more communication cost and required to transmit more messages resulted in the reduction of life time. The first one is distributed hash table (DHT) which is based on hash table value of (key, record) by which a fully decentralized, key-based caching and checking system is constructed to catch nodes. DHT enables sensor nodes to construct the over lay network. The key plays vital role in DHT mechanism which determines the destination node of the message [1]. But this DHT incurs same communication cost as previous, have some storage consumption and strong detection probability. Second one is distributed detection protocol, named randomly directed exploration (RDE) in which probabilistic directed forwarding technique along with random initial direction and border determination. Every node contains signed version of neighbor list and the detection round is initiated by sending claiming message by the nodes to randomly selected neighbors [5]. The communication cost is reduced by using border determination. And this protocol has to store only the list of neighbor nodes so consumes less memory. So the RDE stands with low communication cost, less storage consumption and high detection probability. The RDE and DHT protocols are only used to detect the node clones in the wsn.. The rest of paper is organized as follows. First the previous counter measures are discussed in Section II. Then, present preliminaries in Section III. After wards detailed description about DHT and its performance in Section IV. The RDE is detailed in Section V with its performance. Finally conclude the work in Section VI. II.PREVIOUS WORKS The earliest method to detect node clones was prevention schemes and key plays the main role which provided to nodes by mobile trusted agents. The private key of node comprises of location and identity. But the problems arise here are attackers may takes some time to compromise the nodes (compromising time) in the network. As the compromising time decreases the number of clone nodes increases thus badly affects the security of the network. And also prevention scheme is applicable to only some specific applications. The assumption made on trusted agents is not too strong [7]. In the centralized detection method a base station is connected to each node. Each node sends a list of its neighbor nodes and location to base station. The communication cost is limited by constructing subsets of nodes. Even though communication cost is reduced the life time expectancy of the network is decreased due to the communication burden of the nodes near to the base station [4]. III. PRELIMINARIES A. Network Model The network used is of large scale and have 'n' number of resource constrained sensor nodes. Each node has unique ID and a corresponding private key. The public key kα is the node ID and private key is kα -1 . Message M signed by the node α using private key is [M] kα -1 . The location and current relative time of every node is determined by secure localization protocol and secure time synchronization scheme respectively [1]. And these are not specifying since they are not so important to proposed protocols. As per previous approaches the base station is not powerful in this model, instead of that an initiator plays as a trusted role for initiating the detection round procedures. During node clone detection the sensor nodes are assumed to be stationary. So the node clone can be determined by the collision of location for one node ID [5]. B. Adversary Model The sensor networks are more vulnerable to attacks in hostile environment. The adversary can capture some nodes, can modify or reprogram it and obtains all the secret credentials data. Thus the compromised node creates replicas or clones of such mischievous nodes and adversary may gain control of the whole network by deploying these replicas in place that are decided intelligently. Adversary is always aware of detection protocol and manages to conceal the existence of clone [5]. Adversary interferes with the detection scheme in three ways. First, cloned nodes may not participate in the detection rounds. Second, cloned nodes may drop or modify the messages. Lastly they take some time to compromise the nodes is limited [1]. C. Performance Metrics The performance metrics used to compare both protocols are (i) Communication cost: the average number of messages sent per node is used to represent Communication cost. (ii) Storage consumption: low cost sensors have limited amount memory. Average cache table size per node represents storage consumption. (iii) Detection probability: average number of witness nodes per node represents detection probability. Table is the node clone detection protocol which provides decentralization scheme with the key based caching and checking. Distributed Hash Table is based on a hash table of (key, record) pair which is already distributed. The distributed hash table enables the sensor nodes to form an overlay network. The key plays vital role in distributed hash table and key determines where to send the message from source node i.e. the destination node is determined by the key and source doesn't know anything about the destination node. The detection round initiated by initiator by sending an action message (involves nonce, seed, and time). Then every observer nodes constructs claiming message for each neighbor node, referred as examinee and sends the message with probability pc to reduce the communication over work. The key which determine the destination node of message is the hash value of concatenation of seed and examinee ID [1]. During distributed hash table detection round a claiming message will transmitted to destination node which will cache ID-location pair and check for node clone detection. Distributed Hash Table is a decentralized distributed system which provides a key based look up service. (Key, record) pairs are stored in the table any active node can store and retrieve records associated with specific keys. Thus distributed hash table maintain mapping from keys to records among nodes. Chord is used and choose chord as a distributed hash table implementation to demonstrate protocol. Massive virtual ring is formed by chord in which every node is located at one point, and owning a segment of the periphery. Hash function is used to achieve pseudo randomness on output by mapping an arbitrary input into a b-bit space (in the ring).Chord coordinate is assigned for each node and can join the network. Here a node's Chord point's coordinate is the hash value of the node's MAC address [1].one segment that ends at the node's Chord point is related to every node, and all records whose keys fall into that segment will be transmitted to and stored in that node [5].Every node maintains a finger table of size t= O (log n) to facilitate a binary-tree search. The finger table for a node with responsible for holding the t keys. between 10 and 20. The DHT enable sensor nodes to construct a chord overlay network. Cloned node may not participate in this overlay network construction [1]. And this overlay network construction is independent of node clone detection. Nodes possess the information of their direct predecessor and successor in the Chord ring and also caches information of its consecutive successors in its successors table [6]. The communication cost is thus reduced by this cache mechanism and it enhances systems robustness. Selection of inspectors is done using the facility of the successors The average path length between two random nodes by l which varies from O(log n) to O(√n).On the basis of Chord's properties the number of transfers in the Chord overlay network is c log n , where c is a constant number, usually less than 1. Therefore, the average path hop length of a message is cl log n [1]. There are pcdn claiming messages in total for a round of detection. Thus shown in fig 1(a) the average number of messages sent per node is given by p cdcl logn. Since the pc, d, and c are constant, the asymptotic communication cost of the DHT-based protocol is between O (log 2 n) and O (√nlog n). Storage Consumption: In particular, protocol shows strong resilience against message-discarding by cloned nodes. In fact, the more cloned nodes, the less the size of cache tables for integrity nodes as storage consumption and the more witnesses as security level shown in figure 2 (a). Good pseudo-randomness of the Chord system, on average, every node stores one record in its cache table associated with one examinee's ID as its destination, regardless of the number of claiming messages per examinee. Let p r denote the probability of a predecessor receiving a specific claiming message, then the probability of a predecessor holding a record for an examinee is To overcome these problems a new node clone detection protocol introduced namely randomly directed exploration. Here the each node only needs to know and buffer a neighbor list having all neighbors ID and locations. During detection round each node constructs claiming message with signed version of neighbor list and then deliver message to others which will compares with its own neighbor list to detect node clone. If there exists any node clone, one witness node successfully catches the clone and notifies the entire network by broadcasting. The efficient way to achieve randomly directed exploration needs some mechanisms and routing protocols. First the claiming message needs to provide maximum hop limit and it is sent to random neighbors. Then the further message transmission will maintain a line and this transmission line property enables a message to go through a network as fast as possible [6]. The communication cost of this protocol is low and it is limited by the border determination mechanism. And the assumption made here is that each node knows about its neighbors locations. IV.DISTRIBUTED HASH TABLE Distributed Hash Detection round Initially the node clone detection round is activated by the initiator. At the right mentioned action time, each node creates its own neighbor list (ID of neighbor and location). Then that node act as an observer for all its neighbors and starts to generate claiming messages. The claiming message involves node ID, location and its neighbor list [6]. The claiming message by node is constructed by Mα=ttl, idα, Lα, neighbor list where ttl is time to live. Algorithm 3: rde-processmessage Mα: An intermediate node processes a message 1: verify the signature of Mα 2: compare its own neighbor-list with the neighbor-list in Mα 3: if found clone then 4: broadcast the evidence; 5: ttl<=ttl-1 6: if ttl ≤ 0 then 7: discard Mα 8: else 9: next node<=get next node (Mα) {See Algorithm 4} 10: if next node =NIL then 11: discard Mα 12: else 13: forward Mα to next node [6] The intermediated nodes will change the value of ttl during transmission. In each time, the node transmits message to a random neighbor. When an intermediate node β receives a claiming message Mα, it launches rde-processmessage Mα. During the processing the node clone is detected by comparing the neighbor list of node which acts as inspector β with neighbor list in the message. If clone detected then the witness node β will broadcast an evidence message M evidence= (Mα,Mβ) to notify the whole network such that the cloned nodes are removed from the network [6]. Node decreases the message's ttl by 1 and discards the message if ttl reaches zero during routing; otherwise it will query Algorithm 4 to determine the next node receiving the message. Algorithm 4: get next node (Mα): To determine the next node that receives the message 1: determine ideal angle, target zone, and priority zone 2: if no neighbors within the target zone then 3: return NIL 4: if no neighbors within the priority zone then 5: next node<= the node closest to ideal angle 6: else 7: next node<= a probabilistic node in the priority zone, with respect to its probability proportional to angle distance from priority zone border 8: return next node [6]. Deterministic directed transmission: The ideal direction can be calculated when node receives a claiming message from previous node and the next destination node should be closest to the ideal direction for the best effect of line transmission. Network border determination: The communication cost is reduced by taking network shape into consideration. Due to physical constrains in many sensor network applications, there exist outside borders. The claiming message can be directly discarded when reaching some border in the network. To determine a target zone then no neighbor is found in this zone, target range is used along with ideal direction, the current node will conclude that the message has reached a border, and thus throw it away. Probabilistic directed transmission: priority range along with the ideal direction is used to specify a priority zone, in which the next node will be selected. The deterministic directed candidate within the target zone will be selected as the next node when no nodes are located in that zone,. If there are several nodes in the priority zone, their selection probabilities are proportional to their angle distances to priority zone border. As a result, to reduce detection probability dramatically the adversary may remove some nodes in strategic locations Claiming messages transmissions from a cloned node's neighbors are highly correlated, which affects the protocol communication and security performance [1]. Those drawbacks are overcome, by the probabilistic directed mechanism, and the protocol performance is improved significantly E.Performance Analysis of RDE Communication cost: The RDE's communication cost depends on the routing parameter settings. On average, there are r claiming messages sent by each observer, and each message transmits at most ttl hops , r is a constant small number, say 1 for a dense network, but ttl is generally related to the network size . So ttl=√n because there are nodes in the network, and by the line property of protocol routing, it is very likely for any two nodes to be reachable within √n hops for a normal network topology [6]. In other words, ttl=√n would be sufficient for messages to go across the network. The upper bound of communication cost in the randomly directed exploration protocol is O√n and its shown in fig 4 (a) Detection probability: Relieving message-discarding and protecting witness are achieved by random initial direction and probabilistic directed transmission. By them, there is no critical location to affect message transmission, which limits the capacity of message-discarding, and every neighbor of a cloned node has similar potential to become witness so it is hard for the adversary to get rid of witness in advance [1]. The RDE protocol's detection probability is determined by the number of nodes that are reached when randomly drawing lines where each has a random initial angular and fixed number of nodes along this direction with the border limitation. Let h denote the reachable node number; ᶿ, it is a function of (an initial angular),ttl (the number of maximum hops), and v (the number of the claiming messages). Therefore, for a network with n nodes, the detection probability is given by P RDE=h (ttl,ᶿ,v)/n shown in fig 4(b). Storage consumption: The RDE protocol is exceedingly memory-efficient. It does not rely on broadcasting; thus, no additional memory is required to suppress broadcasting flood. The protocol does not demand intermediate nodes to buffer claiming messages, all memory requirement lies on the neighbor-list, which, in fact, is a necessary component for all distributed detection approaches. Therefore, the protocol consumes almost minimum memory shown in fig 4 (c). VI CONCLUSION Sensor nodes lack tamper-resistant hardware and are subject to the node clone attack. So two distributed detection protocols are presented: One is based on a distributed hash table, which forms a Chord overlay network and provides the key-based routing, caching, and checking facilities for clone detection, and the other uses probabilistic directed technique to achieve efficient communication overhead for satisfactory detection probability. While the DHT-based protocol provides high security level for all kinds of sensor networks by one deterministic witness and additional memory-efficient, probabilistic witnesses, the randomly directed exploration presents outstanding communication performance and minimal storage consumption for dense sensor networks. From the analysis and simulation results, the randomly directed exploration protocol outperforms all other distributed detection protocols in terms of communication cost and storage requirements, while its detection probability is satisfactory, higher than that of line-selected multicast scheme. Figure 3 . 3simulation results of DHT detection on number of cloned nodes V.RANDOMLY DIRECTED EXPLORATION The problems associated with the dht are it incurs more communication cost because of the chord overlay network and thus it is sensitive to energy and storage consumption. Figure 4 4simulation results of RDE on varying size networ .. TABLE I DISTRIBUTED IDETECTION PROTOCOLS COMPARISON, WHERE n IS NETWORK SIZE, d NODE DEGREEProtocols Nodes requirements Communication cost Memory cost Detection Cost Node to network broadcasting Neighbors information O (n) O(d) Strong Randomized multicast All nodes data O (n) O(d√n) Acceptable Line selected All nodes data O (√n) O(d√n) Acceptable RED Knowledge of network geography O (√n) O(d√n) Strong DHT DHT nodes information O (log n √n) O(d) Strong RDE Neighbors information O (√n) O(d) Good Chord intermediate nodes will forwards claiming message to its destination node. Only the source node, Chord intermediate nodes, and the destination node need to process a message, whereas other nodes along the path simply route the message to temporary targets. Algorithm 1 for handling a message and If the algorithm returns NIL, then the message has arrived at its destination. Else the message will forwarded to the next node with the ID that is returned by Algorithm[1]. Algorithm 1: dht_ handle message(Mα4β) handle a message in the DHT-based detection, where y is the current node's Chord coordinate, finger[i]Message for node clone detection is examined by Algorithm 1 and Algorithm 2 compares the message with previous inspected messages that are buffered in the cache table[1]. All records in the cache table should have different examinee ID. If there exist two messages Mα4β and Mα'4β' satisfying idβ = idβ' and Lβ ≠Lβ shows that exists clone and then the witness node broadcasts the evidence to notify the whole network. All integrity nodes verify the evidence message and stop communicating with the cloned nodes. The witness does not need to sign the evidence message.table. Detection round stages (i)The initial stage of detection round is done by activating all nodes by releasing an action message by initiator MACT=nonce, seed, time, {nonce||seed||time} k -1 initiator During each rounds the value of nonce increases monotonously and it intended to prevent the DoS attacks. (ii) By receiving the action message each node verifies the value of nonce with previous values and verifies the signature of the message. If both are valid node will updates the nonce and stores the seed. The node act as observer to generate claiming message for each neighbor at the designated action time and transmits the message through the overlay network with respect to the claiming probability pc. Mα4β=idβ, Lβ, idα, Lα, { idβ ||Lβ ||idα ||Lα||nonce} k -1 α. where, Lα, Lβ, are locations of α and β , respectively. (iii) is the first node on the ring that succeeds key((y+2 b-I mod 2 b ),I £ [1,t] ,successors [j] is the next j th successor j £[1,g][1]. Output: NIL if the message arrives at its destination; otherwise, it is the ID of the next node that receives the message in the Chord overlay network[1]. 1: key<=H (seed||idβ) 2: if key £ [predecessor] then {has reached destination} 3: inspect Mα4β {act as an inspector, see Algorithm 2} 4: return NIL 5: for i=1 to g do 6: if key £(y, successors [i]) then {destination is in the next Chord hop} 7: inspect Mα4β {act as an inspector, see Algorithm 2} 8: return successors [i] 9: for j= 1 to t do {for normal DHT routing process} 10: if key £ [(y+2 b-I mod 2 b ,y)], then 11: return finger [j] 12: return successor [g] Algorithm 2: inspect Mα4β: Inspect a message to check for clone detection in the DHT-based detection protocol 1: verify the signature of Mα4β 2: if idβ found in cache table then 3: if idβ has two distinct locations {found clone, become a witness} 4: broadcast the evidence 5: else 6: buffer Mα4β into cache table D. 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Perrig, and V. Gligor, "Distributed detection of node replication attacks in sensor networks," in Proc. IEEE Symp. Security Privacy, 2005, pp. 49-63 Chord:a scalable peer to peer look up protocol for internet applications. I Stoica, R Morris, D L Nowell, D , R Karger, M F Kaashoek, Hari Balakrishnan, IEEE/ACM Trans.Netw. 111I.Stoica,R.Morris,,D.L.Nowell,D,R.Karger,M.F.Kaashoek,Hari Balakrishnan,"Chord:a scalable peer to peer look up protocol for internet applications,'IEEE/ACM Trans.Netw.,vol.11,no.1,pp,17-32,Feb.2003. Randomly directed exploration: an efficient node clone detection protocol in wireless sensor network. Zhijun Li And Guang Gong, IEEE Trans. 11Zhijun Li And Guang Gong, "Randomly directed exploration: an efficient node clone detection protocol in wireless sensor network" ,in proc 5 th IEEE Trans.Volume 11,pp34-4..2009. A new protocol for securing wsn against node replication attacks. C Bekara And, M L Maknavicius, third IEEE International Conference on wireless and mobile computing,networking and communications. C.Bekara And M.L.Maknavicius,"A new protocol for securing wsn against node replication attacks,"in third IEEE International Conference on wireless and mobile computing,networking and communications,2007,pp 59-59

We construct a theory for low-energy quantum transport in normal|superconductor junctions involving the recently discovered iron-based high-Tc superconductors. We properly take into account both Andreev bound surface states and the complex Fermi surface topology in our approach, and investigate the signatures of the possible order parameter symmetries for the FeAs-lattice. Our results could be helpful in determining the symmetry of the superconducting state in the iron-pnicitide superconductors. PACS numbers: 74.20.Rp, 74.50.+r, 74.70.Dd Introduction. Very recently, a family of iron-based superconductors with high transition temperatures was discovered, with a concomitant avalanche of both experimental and theoretical activity.1,2,3,4,5,6,7,8,9,10,11,12. The highest T c measured so far in this class of materials is 55 K, and many experimental reports indicate signatures of unconventional superconducting pairing. However, it remains to be clarified what the exact symmetry is for both the orbital-and spin-part of the Cooper pair wavefunction -there has for instance been reports of both nodal 11 and fully gapped 10 order parameters (OPs) in the literature up to now.Probing the low-energy quantum transport properties of superconducting materials has proven itself as a highly useful tool to access information about the symmetry of the superconducting OP.13The conductance spectra of normal|superconductor (N|S) junctions often contains important and clear signatures of the orbital structure of the OP. For instance, when the OP contains nodes in the tunneling direction with a sign-change across the nodes on each side of the Fermi surface, the conductance will display a large zero-bias conductance peak (ZBCP) due to the presence of Andreev surface bound states. 14 Two recent studies 10,11 utilized the method of point-contact spectroscopy in order to study the symmetry of the superconducting OP in LaO 0.9 F 0.1−δ FeAs and SmFeAsO 0.85 F 0.15 , respectively. The findings were in stark contrast. Namely, the large ZBCP found in LaO 0.9 F 0.1−δ FeAs gave evidence of a nodal order parameter, while the data of SmFeAsO 0.85 F 0.15 clearly indicated a nodeless OP. In both of these studies, the Blonder-Tinkham-Klapwijk 15 (BTK) framework was used to analyze the data theoretically, using the extension to anisotropic pairing by Tanaka and Kashiwaya 16 . In this model, one considers a cyndrical or spherical Fermi surface with a free-electron dispersion relation, which does not account for the non-trivial multiband Fermi surface topology and dispersion relation in the iron-pnicitides. One might argue that the extended BTK model nevertheless may suffice to describe the transport properties of these materials qualitatively, but this statement clearly warrants a detailed investigation.In this Rapid Communication, we construct a theory of low-energy quantum transport properties of the iron-based high-T c superconductors by considering a N|S junction relevant for point-contact spectroscopy and scanning-tunnelingmicroscopy measurements. In doing so, we model fairly accu-

10.1103/physrevb.79.020501

0811.1775

03a8e133e39560094d72298935be58c868bbfeee

Theory of Andreev reflection in junctions with iron-based High-T c superconductors 11 Nov 2008 Jacob Linder Department of Physics Norwegian University of Science and Technology N-7491TrondheimNorway Asle Sudbø Department of Physics Norwegian University of Science and Technology N-7491TrondheimNorway Theory of Andreev reflection in junctions with iron-based High-T c superconductors 11 Nov 2008(Dated: Received November 11, 2008) We construct a theory for low-energy quantum transport in normal|superconductor junctions involving the recently discovered iron-based high-Tc superconductors. We properly take into account both Andreev bound surface states and the complex Fermi surface topology in our approach, and investigate the signatures of the possible order parameter symmetries for the FeAs-lattice. Our results could be helpful in determining the symmetry of the superconducting state in the iron-pnicitide superconductors. PACS numbers: 74.20.Rp, 74.50.+r, 74.70.Dd Introduction. Very recently, a family of iron-based superconductors with high transition temperatures was discovered, with a concomitant avalanche of both experimental and theoretical activity.1,2,3,4,5,6,7,8,9,10,11,12. The highest T c measured so far in this class of materials is 55 K, and many experimental reports indicate signatures of unconventional superconducting pairing. However, it remains to be clarified what the exact symmetry is for both the orbital-and spin-part of the Cooper pair wavefunction -there has for instance been reports of both nodal 11 and fully gapped 10 order parameters (OPs) in the literature up to now.Probing the low-energy quantum transport properties of superconducting materials has proven itself as a highly useful tool to access information about the symmetry of the superconducting OP.13The conductance spectra of normal|superconductor (N|S) junctions often contains important and clear signatures of the orbital structure of the OP. For instance, when the OP contains nodes in the tunneling direction with a sign-change across the nodes on each side of the Fermi surface, the conductance will display a large zero-bias conductance peak (ZBCP) due to the presence of Andreev surface bound states. 14 Two recent studies 10,11 utilized the method of point-contact spectroscopy in order to study the symmetry of the superconducting OP in LaO 0.9 F 0.1−δ FeAs and SmFeAsO 0.85 F 0.15 , respectively. The findings were in stark contrast. Namely, the large ZBCP found in LaO 0.9 F 0.1−δ FeAs gave evidence of a nodal order parameter, while the data of SmFeAsO 0.85 F 0.15 clearly indicated a nodeless OP. In both of these studies, the Blonder-Tinkham-Klapwijk 15 (BTK) framework was used to analyze the data theoretically, using the extension to anisotropic pairing by Tanaka and Kashiwaya 16 . In this model, one considers a cyndrical or spherical Fermi surface with a free-electron dispersion relation, which does not account for the non-trivial multiband Fermi surface topology and dispersion relation in the iron-pnicitides. One might argue that the extended BTK model nevertheless may suffice to describe the transport properties of these materials qualitatively, but this statement clearly warrants a detailed investigation.In this Rapid Communication, we construct a theory of low-energy quantum transport properties of the iron-based high-T c superconductors by considering a N|S junction relevant for point-contact spectroscopy and scanning-tunnelingmicroscopy measurements. In doing so, we model fairly accu- rately the Fermi surface topology and the associated quasiparticle dispersions, in order to see how this affects the results as compared with the usual BTK-paradigm. We consider several possible OP symmetries which may be realized in the ironpnicitides. Theory and results. We adopt the minimal two-band model derived in Ref. 8 (Fig. 1a) , in which the normal-state Hamiltonian reads H N = kσ φ † kσ ǫ kx − µ ǫ kxy ǫ kxy ǫ ky − µ φ kσ ,(1) where the fermion basis φ kσ = [d kxσ , d kyσ ] T contains the annihilation operators for electrons in the d xz -and d yz -orbitals with spin σ and wavevector k, respectively. We have also defined ǫ kx = −2t 1 c x − 2t 2 c y − 4t 3 c x c y , ǫ kxy = −4t 4 s x s y , ǫ ky = −2t 2 c x − 2t 1 c y − 4t 3 c x c y , with c j = cos(k j a), s j = sin(k j a), j = x, y, and a is the lattice constant. By diagonalizing the above Hamiltonian, one obtains H N = kσφ † kσ diag{Ω + k , Ω − k }φ kσ , Ω ± k = (ǫ kx + ǫ ky )/2 − µ ± (ǫ kx − ǫ ky ) 2 /4 + ǫ 2 kxy (2) where the new basisφ kσ = [γ + kσ , γ − kσ ] T consists of new fermion quasiparticle operators in the bands + and − which are hybrids of the d xz -and d yz -orbitals. The Fermi surface topology is given by Ω ± k = 0, and gives an electron-like band (+) and hole-like band (−) shown in Fig. 1b for the choice t 1 = −1, t 2 = 1.3, t 3 = t 4 = −0.85, µ = 1.54, all measured in units of |t 1 |. Our choice of parameter set is motivated by the fact that it reproduces the same Fermi surface structure as LDA band structure calculations 17 , and was also employed in Ref. 18? . The new fermion operators are related to the old basis φ kσ by ζ k = ǫ kxy /[(ǫ kx − ǫ ky )/2 + (ǫ kx − ǫ ky ) 2 /4 + ǫ 2 kxy ], φ † kσ P k =φ † kσ , P k = (1 + ζ 2 k ) −1/2 × 1 −ζ k ζ k 1 .(3) We now introduce a superconducting pairing between the long-lived quasiparticles γ λ kσ , λ = ±, which then automatically accounts for both inter-and intra-band pairing in the original fermion basis φ k : H SC = kλ ∆ λ k (γ λ k↑ ) † (γ λ −k↓ ) † + h.c. .(4) In this way, we may diagonalize the total Hamiltonian H = H N + H SC by introducing a final fermion basis η λ k = [c λ k↑ , c λ −k↓ ] T describing the quasiparticle excitations in the superconducting state. After discarding unimportant constants, we find that H = kσλ σE λ k (c λ kσ ) † c λ kσ , E λ k = [(Ω λ k ) 2 + |∆ λ k | 2 ] 1/2 . This result is formally identical to a two-band superconductor with gaps ∆ λ k and normal-state dispersions Ω λ k , λ = ±. The belonging wavefunctions which describe the quasiparticle excitations read Ψ λ k = [u λ k , v λ k e −ıφ λ k ] T e ıλk λ ·r , [v λ k e ıφ λ k , u λ k ] T e −ıλk λ ·r , (u λ k ) 2 = 1 − (v λ k ) 2 = 1 2 (1 + E 2 − |∆ λ k | 2 /E),(5) for quasiparticles with positive excitation energies E ≥ 0. Here, k λ denotes the Fermi momentum for band λ while e ıφ λ k = ∆ λ k /|∆ λ k |. We have now effectively described the superconducting state as a two-band model with gaps ∆ ± k and normal-state dispersions Ω ± k . This has allowed us to obtain a simple form for the wavefunctions in Eq. (5) that are to be used in the scattering problem below. The trade-off for this advantage, however, is that the k-dependence of the gap functions ∆ ± k in general will become quite complicated. To see this, we may transform Eq. (4) back to the original fermion basis φ k by means of our expression for P k in Eq. (3) to find that: H SC = k ∆ kx d † kx↑ d † −kx↓ + ∆ ky d † ky↑ d † −ky↓ + ∆ kxy (d † kx↑ d † −ky↓ − d † ky↑ d † −kx↓ ) + h.c. ,(6) where ∆ kx and ∆ ky are the intra-orbital gaps while ∆ kxy is the inter-orbital gap, defined as ∆ kx = (∆ + k + ζ 2 k ∆ − k )/ν + k , ∆ ky = (∆ − k + ζ 2 k ∆ + k )/ν + k , ∆ kxy = ζ k (∆ + k − ∆ − k )/ν + k , with ν ± k = (1 ± ζ 2 k ) . We see that the inter-orbital pairing vanishes in the case where ∆ + k = ∆ − k . However, we emphasise that our model does account for inter-orbital pairing ∆ kxy , and that ∆ kxy = 0 whenever ∆ + k = ∆ − k . Assuming spin-singlet and even-frequency pairing, there are three possible s-wave symmetries {∆ 0 , ∆ 0 (c x + c y ), ∆ 0 c x c y } and two possible dwave symmetries {∆ 0 (c x − c y ), ∆ 0 s x s y } for the superconducting order parameters ∆ kx and ∆ ky in terms of the square lattice harmonics. The gaps in the ± quasiparticle hybridized bands are then obtained as ∆ + k = (∆ kx − ζ 2 k ∆ ky )/ν − k and ∆ − k = (∆ ky − ζ 2 k ∆ kx )/ν − k . Note that the extended swave symmetry ∼ c x c y changes sign on the electron-and hole-Fermi surfaces, similarly to the s ± -scenario suggested in Ref. 5 . We are now in a position to evaluate the conductance of the system. The presence of a Fermi-vector mismatch between the normal and superconducting side of the junction is assumed to be manifested through an effective decrease in the junction transmission. Since the Fermi velocity may be different in the two bands with normal-state dispersions Ω ± k , we allow for different barrier parameters Z ± in the two bands. For a specified pairing symmetry, there are then four fitting parameters present: the barrier strength Z λ and gap magnitude ∆ λ 0 for band λ = ±. By generalizing the results of Refs. 15,16 to a two-band model which also takes into account the non-trivial Fermi surface topology in Fig. 1a, we obtain the following expression for the normalized tunneling conductance: G(eV )/G 0 = λ,ky f (k y )σ λ S (eV )/[2f (k y )σ λ N ], where σ λ N = [1 + (Z λ ) 2 ] −1 and σ λ S (eV ) = σ λ N [1 + σ λ N |Γ λ + (k, eV )| 2 + (σ λ N − 1) × |Γ λ + (k, eV )Γ λ − (k, eV )| 2 ] / |1 + (σ λ N − 1) × Γ λ + (k, eV )Γ λ − (k, eV )ρ λ (k)| 2 , Γ λ ± (k, eV ) = eV − (eV ) 2 − |∆(±λk x , k y )| 2 |∆ λ (±λk x , k y )| , ρ λ (k) = ∆ λ (−λk x , k y )[∆ λ (λk x , k y )] * |∆ λ (−λk x , k y )∆ λ (λk x , k y )| ,(7) where f (k y ) = cos(k y a/2) is a weighting function that models the directional dependence of the incoming quasiparticles. The strategy is now to sum the conductance over allowed values k y ∈ [−π/a, π/a] for the electron-(λ = 1) and hole-like (λ = −1) Fermi surfaces, and solve for k x from Eq. (2) by Ω λ k = 0 for a given k y . In what follows, we choose equal value for the barrier tranparencies Z + = Z − ≡ Z and gap magnitudes ∆ + 0 = ∆ − 0 ≡ ∆ 0 in the two bands for simplicity, and add a small imaginary number δ to the quasiparticle energy to model inelastic scattering: eV → eV + ıδ, δ/∆ 0 = 10 −2 . As in Ref. 18 , we choose ∆ 0 = 0.1. Clearly, it is possible to study a rich variety of interplays between the two quasiparticle bands in terms of different symmetries for the d xz -and d yzorbitals and with different gap magnitudes. Here, however, our main aim is to investigate how the conductance spectra are influenced by the non-trivial Fermi surface topology and dispersion relations, and see how this compares with the cylindrical/spherical Fermi-surface and free-particle dispersion scenario employed in the usual BTK-paradigm. In particular, this is relevant to the interpretation of the point-contact spectroscopy measurements of Refs. 10,11 . There is, however, an important cavaet with regard to which conclusion one may draw with regard to the symmetry of the superconducting OP from the tunneling data of Refs. 10,11 . In these works, polycrystalline samples were used, while the orbital/nodal structure of the OP can only be convincingly probed in single crystal specimens. This is because tunneling into polycrystalline samples may lead to intrinsic averaging effects which distort the contribution from anisotropic OPs. In Fig. 2, we plot the conductance for tunneling along the (100)-direction for several OP symmetries. As seen, the d xywave case stands out from the rest as it features a considerable ZBCP. Comparing with the experimental data of Ref. 11, we would conclude that a nodal d-wave OP is likely to be realized in LaO 0.9 F 0.1−δ FeAs. The results of Ref. 10 seem to be most consistent with either s-wave or extended s-wave pairing, as only one gap is seen in the spectra. For the s-wave and d xywave cases, the standard BTK approach appears to suffice in order to qualitatively say something about the OP symmetry. However, the results are quite different from the usual BTKapproach when considering the extended s-wave and d x 2 −y 2wave symmetries. More specifically, we find satellite features at subgap energies, including sharp peaks. These features most likely pertain to the specific band-structure which we consider here (see Fig. 1), and are thus not possible to capture within the conventional BTK-treatment with the cylindrical Fermi surface approximation. In fact, the DOS in our minimal two-band model is a highly non-monotonic function of energy and contains two van Hove singularities. 8,17 . Let us also consider the case where there is one fully gapped OP and one nodal OP to see what fingerprints this combination leaves in the conductance spectra. In Fig. 3, we plot the conductance for the case where ∆ kx is fully gapped, while ∆ ky has a nodal symmetry. For concreteness, we consider s-wave + d xy -wave pairing and d x 2 −y 2 -wave + d xy -wave pairing in Fig. 3(a) and (b), respectively. As seen, the nodal OP gives rise to a ZBCP while there are several satellite features in addition to the large coherence peak at the gap edge. The plots are qualitatively similar regardless of whether the fully gapped OP is s-wave or d x 2 −y 2 -wave, while the features in the conductance are qualitatively more pronounced in the s-wave case due to the better gapping of the Fermi surface. Finally, we consider the evolution of the conductance spectra upon changing the doping level µ. The Fermi surface topology evolves with a change in µ as shown in Fig. 3(c): the electron-pockets increase in size while the hole-pockets decrease in size upon increasing µ. To see how the subgap features obtained in Ref. 2 evolve upon modifying µ, consider Fig. 3(d) where we consider the c x +c y symmetry with Z = 3. As seen, the satellite features shown in Fig. 2 are still present and qualitatively the same, but they are shifted to different bias voltages. Summary. In summary, we have developed a theory for Andreev reflection in the iron-based high-T c superconductors. Starting with a tight-binding model on a square lattice to model the puckered FeAs planes, we have investigated several order parameter (OP) symmetries and the resulting conductance spectra. Taking fully into account the Fermi surface topology and the quasiparticle dispersion relation, we have investigated scenarios where the symmetry of the superconducting OP in both bands is the same and where it is different, i.e. one is fully gapped and the other is nodal. We find that the standard Blonder-Tinkham-Klapwijk (BTK) formalism should give qualitatively correct results for the case where the OP symmetries on both bands are either isotropic s-wave or d-wave. However, the results differ considerably for the extended s-wave symmetries, as we find satellite features at subgap energies which are absent within the usual BTK treatment. Our results may be useful in the context of analyzing quantum transport data of tunneling in normal|superconductor junctions involving the iron-pnicitides. FIG. 1 : 1(Color online) (a) Illustration of the two-dimensional FeAsplane with the dxz-and dyz-orbitals and hopping between them, as proposed in Ref. 8 . (b) Sketch of the Fermi surface topology for the long-lived quasiparticle excitations in a minimal two-band model (see main text for parameter values). online) Plot of the conductance spectra for tunneling along the (100)-axis in an iron-pnicitide N|S junction for several possible order parameter symmetries. Only in the dxy-wave case ∆ = ∆0sxsy is there a considerable ZBCP. Note that different scale on the voltageaxis for this case due to the narrowness of the ZBCP. High (low) values of the parameter Z denotes low (high) transmissivity interfaces. FIG. 3 : 3(Color online) (a) and (b): Plot of the conductance spectra for tunneling along the (100)-axis in an iron-pnicitide N|S junction for the case of one fully gapped OP and one nodal OP. (c) Evolution of the Fermi surface topology for µ = {1.24, 1.54, 1.84} in the direction of the arrow. (d) Conductance spectra for the cx + cy symmetry with Z = 3 for different doping levels. . Y Kamihara, T Watanabe, M Hirano, H Hosono, J. Am. Chem. Soc. 1303296Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008). . H H Wen, G Mu, L Fang, H Yang, X Y Zhu, arXiv:0803:3021H. H. Wen, G. Mu, L. Fang, H. Yang, and X. Y. Zhu, arXiv:0803:3021. . X H Chen, arXiv:0803:3603X. H. Chen et al., arXiv:0803:3603. . Z A Ren, arXiv:0803.4283arXiv: 0804.2582Z. A. Ren et al., arXiv: 0803.4283; Z. A. Ren et al., arXiv:0804.2053; Z. A. Ren et al., arXiv: 0804.2582 (2008). . I I Mazin, D J Singh, M D Johannes, M H Du, arXiv:0803.2740I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, arXiv:0803.2740. . R H Liu, arXiv:0804.2105R. H. Liu et al., arXiv:0804.2105. . V Cvetkovic, Z Tesanovic, arXiv:0804.4678V. Cvetkovic and Z. Tesanovic, arXiv:0804.4678. . S Raghu, X.-L Qi, C.-X Liu, D J Scalapino, S.-C Zhang, Phys. Rev. B. 77220503S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C. Zhang, Phys. Rev. B 77, 220503(R) (2008). . A J Drew, arXiv:0805.1042A. J. Drew et al., arXiv:0805.1042. . T Y Chen, Z Tesanovic, R H Liu, X H Chen, C L Chien, Nature. 4531224T. Y. Chen, Z. Tesanovic, R. H. Liu, X. H. Chen, C. L. Chien, Nature 453, 1224 (2008). . L Shan, Y Wang, X Zhu, G Mu, L Fang, C Ren, H.-H Wen, Europhys. Lett. 8357004L. Shan, Y. Wang, X. Zhu, G. Mu, L. Fang, C. Ren, H.-H. Wen, Europhys. Lett. 83, 57004 (2008) . V Stanev, J Kang, Z Tesanovic, arXiv:0809.0014V. Stanev, J. Kang, and Z. Tesanovic, arXiv:0809.0014. . G Deutscher, Rev. Mod. Phys. 77109G. Deutscher, Rev. Mod. Phys. 77, 109 (2005). . C.-R Hu, Phys. Rev. Lett. 721526C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994); . J Yang, C.-R Hu, Phys. Rev. B. 5016766J. Yang and C.-R. Hu, Phys. Rev. B 50, 16766 (1994). . G E Blonder, M Tinkham, T M Klapwijk, Phys. Rev. B. 254515G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982). . Y Tanaka, S Kashiwaya, Phys. Rev. Lett. 743451Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995); . S Kashiwaya, Y Tanaka, M Koyanagi, K Kajimura, Phys. Rev. B. 532667S. Kashiwaya, Y. Tanaka, M. Koyanagi, and K. Kajimura, Phys. Rev. B 53, 2667 (1996). . G Xu, arXiv:0803:1282G. Xu et al., arXiv:0803:1282. . M M Parish, J Hu, B A Bernevig, Phys. Rev. B. 78144514M. M. Parish, J. Hu, and B. A. Bernevig, Phys. Rev. B 78, 144514 (2008) . 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This paper is concerned with blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system. We first obtain several blow-up results and the blow-up rate of strong solutions to the system. We then present a global existence result for strong solutions to the system. 2000 Mathematics Subject Classification: 35G25, 35L05

10.1186/1687-2770-2013-158

1206.4134

471a87783afeb53cd57c1b25e1024d0a27076aa5

Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system 19 Jun 2012 Jingjing Liu Department of Mathematics and Information Science Zhengzhou University of Light Industry 450002ZhengzhouChina Blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system 19 Jun 2012Periodic two-component Dullin-Gottwald-Holm systemblow-upblow-up rateglobal existence This paper is concerned with blow-up phenomena and global existence for the periodic two-component Dullin-Gottwald-Holm system. We first obtain several blow-up results and the blow-up rate of strong solutions to the system. We then present a global existence result for strong solutions to the system. 2000 Mathematics Subject Classification: 35G25, 35L05 Introduction In this paper, we consider the following periodic two-component Dullin-Gottwald-Holm (DGH) system:                      m t − Au x + um x + 2u x m + γu xxx + ρρ x = 0, t > 0, x ∈ R, ρ t + (uρ) x = 0, t > 0, x ∈ R, u(0, x) = u 0 (x), x ∈ R, ρ(0, x) = ρ 0 (x), x ∈ R,u(t, x + 1) = u(t, x), t ≥ 0, x ∈ R,ρ(t, x + 1) = ρ(t, x), t ≥ 0, x ∈ R,(1.1) where m = u − u xx , A > 0 and γ are constants. The system (1.1) has been recently derived by Zhu et al. in [1] by follow Ivanov's approach [2]. It was shown in [1] that the DGH system is completely integrable and can be written as a compatibility condition of two linear systems Ψ xx = −ξ 2 ρ 2 + ξ m − A 2 + γ 2 + 1 4 Ψ and Ψ t = 1 2ξ − u + γ Ψ x + 1 2 u x Ψ, where ξ is a spectral parameter. Moreover, this system has the following two Hamiltonians E(u, ρ) = 1 2 (u 2 + u 2 x + (ρ − 1) 2 )dx and F (u, ρ) = 1 2 (u 3 + uu 2 x − Au 2 − γu 2 x + 2u(ρ − 1) + u(ρ − 1) 2 )dx. For ρ = 0 and m = u − α 2 u xx , (1.1) becomes to the DGH equation [3] u t − α 2 u txx − Au x + 3uu x + γu xxx = α 2 (2u x u xx + uu xxx ), where A and α are two positive constants, modeling unidirectional propagation of surface waves on a shallow layer of water which is at rest at infinity, u(t, x) standing for fluid velocity. It is completely integrable with a bi-Hamiltonian and a Lax pair. Moreover, its traveling wave solutions include both the KdV solitons and the CH peakons as limiting cases [3]. The Cauchy problem of the DGH equation has been extensively studied, cf. [4,5,6,7,8,9,10,11,12,13,14,15,16]. For ρ ≡ 0, γ = 0, the system (1.1) becomes to the two-component Camassa-Holm system [2] m t − Au x + um x + 2u x m + ρρ x = 0, ρ t + (uρ) x = 0, (1.2) where ρ(t, x) in connection with the free surface elevation from scalar density (or equilibrium) and the parameter A characterizes a linear underlying shear flow. The system (1.2) describes water waves in the shallow water regime with nonzero constant vorticity, where the nonzero vorticity case indicates the presence of an underlying current. A large amount of literature was devoted to the Cauchy problem (1.2), see [17,18,19,20,21,22,23,24,25]. The Cauchy problem (1.1) has been discussed in [1]. Therein Zhu and Xu established the local well-posedness to the system (1.1), derived the precise blow-up scenario and investigated the wave breaking for the system (1.1). The aim of this paper is to study further the blow-up phenomena for strong solutions to (1.1) and to present a global existence result. Our paper is organized as follows. In Section 2, we briefly give some needed results including the local well posedness of the system (1.1), the precise blow-up scenarios and some useful lemmas to study blow-up phenomena and global existence. In Section 3, we give several new blow-up results and the precise blow-up rate. In Section 4, we present a new global existence result of strong solutions to (1.1). Notation Given a Banach space Z, we denote its norm by · Z . Since all space of functions are over S, for simplicity, we drop S in our notations if there is no ambiguity. Preliminaries In this section, we will briefly give some needed results in order to pursue our goal. With m = u − u xx , we can rewrite the system (1.1) as follows:                      u t − u txx − Au x + γu xxx + 3uu x − 2u x u xx − uu xxx + ρρ x = 0, t > 0, x ∈ R, ρ t + (uρ) x = 0, t > 0, x ∈ R, u(0, x) = u 0 (x), x ∈ R, ρ(0, x) = ρ 0 (x), x ∈ R, u(t, x + 1) = u(t, x), t ≥ 0, x ∈ R, ρ(t, x + 1) = ρ(t, x), t ≥ 0, x ∈ R. (2.1) Note that if G(x) := cosh(x−[x]−1/2) 2 sinh(1/2) , x ∈ R is the kernel of (1 − ∂ 2 x ) −1 , then (1 − ∂ 2 x ) −1 f = G * f for all f ∈ L 2 (S), G * m = u. Here we denote by * the convolution. Using this identity, we can rewrite the system (2.1) as follows:                      u t + (u − γ)u x = −∂ x G * u 2 + 1 2 u 2 x + (γ − A)u + 1 2 ρ 2 , t > 0, x ∈ R, ρ t + (uρ) x = 0, t > 0, x ∈ R, u(0, x) = u 0 (x), x ∈ R, ρ(0, x) = ρ 0 (x), x ∈ R, u(t, x + 1) = u(t, x), t ≥ 0, x ∈ R, ρ(t, x + 1) = ρ(t, x), t ≥ 0, x ∈ R, (2.2) The local well-posedness of the Cauchy problem (2.1) can be obtained by applying the Kato's theorem. As a result, we have the following well-posedness result. Lemma 2.1. ([1]). Given an initial data (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, there exists a maximal T = T ( (u 0 , ρ 0 ) H s ×H s−1 ) > 0 and a unique solution (u, ρ) ∈ C([0, T ); H s × H s−1 ) ∩ C 1 ([0, T ); H s−1 × H s−2 ) of (2.1). Moreover, the solution (u, ρ) depends continuously on the initial data (u 0 , ρ 0 ) and the maximal time of existence T > 0 is independent of s. Consider now the following initial value problem q t = u(t, q), t ∈ [0, T ), q(0, x) = x, x ∈ R,(2.3) where u denotes the first component of the solution (u, ρ) to (2.1). Lemma 2.2. ([1]). Let (u, ρ) be the solution of (2.1) with initial data (u 0 , ρ 0 ) ∈ H s ×H s−1 , s ≥ 2. Then Eq.(2.3) has a unique solution q ∈ C 1 ([0, T ) × R; R). Moreover, the map q(t, ·) is an increasing diffeomorphism of R with q x (t, x) = exp t 0 u x (s, q(s, x))ds > 0, (t, x) ∈ [0, T ) × R. Lemma 2.3. ([1] ). Let (u, ρ) be the solution of (2.1) with initial data (u 0 , ρ 0 ) ∈ H s ×H s−1 , s ≥ 2, and T > 0 be the maximal existence. Then we have ρ(t, q(t, x))q x (t, x) = ρ 0 (x), (t, x) ∈ [0, T ) × S. Moreover, if there exists a x 0 ∈ S such that ρ 0 (x 0 ) = 0, then ρ(t, q(t, x 0 )) = 0 for all t ∈ [0, T ). Next, we will give two useful conservation laws of strong solutions to (2.1). Lemma 2.4. ([1]). Let (u, ρ) be the solution of (2.1) with initial data (u 0 , ρ 0 ) ∈ H s ×H s−1 , s ≥ 2, and T > 0 be the maximal existence. Then for all t ∈ [0, T ), we have S (u 2 + u 2 x + ρ 2 )dx = S (u 2 0 + u 2 0,x + ρ 2 0 )dx := E 0 . Lemma 2.5. Let (u, ρ) be the solution of (2.1) with initial data (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, and T > 0 be the maximal existence. Then for all t ∈ [0, T ), we have S u(t, x)dx = S u 0 (x)dx. Proof. By the first equation in (2.1), we have d dt S u(t, x)dx = S u t dx = S (u txx + Au x − γu xxx − 3uu x + 2u x u xx + uu xxx − ρρ x )dx = 0 This completes the proof of the lemma. Then, we state the following precise blow-up mechanism of (2.1). Lemma 2.6. ([1]). Let (u, ρ) be the solution of (2.1) with initial data (u 0 , ρ 0 ) ∈ H s ×H s−1 , s ≥ 2,f 2 (x) ≤ e + 1 2(e − 1) f 2 H 1 , where the constant e+1 2(e−1) is sharp. (ii) For every f ∈ H 3 (S), we have max x∈[0,1] f 2 (x) ≤ c f 2 H 1 , with the best possible constant c lying within the range (1, 13 12 ]. Moreover, the best constant c is e+1 2(e−1) . Lemma 2.9. ([28]). If f ∈ H 3 (S) is such that S f (x)dx = a 0 2 , then for every ǫ > 0, we have max x∈[0,1] f 2 (x) ≤ ǫ + 2 24 S f 2 x dx + ǫ + 2 4ǫ a 2 0 . Moreover, max x∈[0,1] f 2 (x) ≤ ǫ + 2 24 f 2 H 1 (S) + ǫ + 2 4ǫ a 2 0 . Blow-up phenomena In this section, we discuss the blow-up phenomena of the system (2.1). Firstly, we prove that there exist strong solutions to (2.1) which do not exist globally in time. Theorem 3.1. Let (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, and T be the maximal time of the solution (u, ρ) to (2.1) with the initial data (u 0 , ρ 0 ). If there is some x 0 ∈ S such that ρ 0 (x 0 ) = 0 and u ′ 0 (x 0 ) = inf x∈S u ′ 0 (x) < − e + 1 2(e − 1) E 0 + |γ − A| 8(e + 1) e − 1 E 1 2 0 , then the corresponding solution to (2.1) blows up in finite time. Proof. Applying Lemma 2.1 and a simple density argument, we only need to show that the above theorem holds for some s ≥ 2. Here we assume s = 3 to prove the above theorem. Define now m(t) := inf x∈S [u x (t, x)], t ∈ [0, T ). By Lemma 2.7, we let ξ(t) ∈ S be a point where this infimum is attained. It follows that m(t) = u x (t, ξ(t)) and u xx (t, ξ(t)) = 0. Differentiating the first equation in (2.2) with respect to x and using the identity ∂ 2 x G * f = G * f − f , we have u tx + (u − γ)u xx = − 1 2 u 2 x + 1 2 ρ 2 + u 2 + (γ − A)u − G * (u 2 + 1 2 u 2 x + 1 2 ρ 2 + (γ − A)u). (3.1) Since the map q(t, ·) given by (2.3) is an increasing diffeomorphism of R, there exists a x(t) ∈ S such that q(t, x(t)) = ξ(t). In particular, x(0) = ξ(0). Note that u ′ 0 (x 0 ) = inf x∈S u ′ 0 (x), we can choose x 0 = ξ(0). It follows that x(0) = ξ(0) = x 0 . By Lemma 2.3 and the condition ρ 0 (x 0 ) = 0, we have ρ(t, ξ(t))q x (t, x) = ρ(t, q(t, x(t)))q x (t, x) = ρ 0 (x(0)) = ρ 0 (x 0 ) = 0. Thus ρ(t, ξ(t)) = 0. Valuating (3.1) at (t, ξ(t)) and using Lemma 2.7, we obtain dm(t) dt ≤ − 1 2 m 2 (t) + 1 2 u 2 + (γ − A)u − (γ − A)G * u, (3.2) here we used the relations G * (u 2 + 1 2 u 2 x ) ≥ 1 2 u 2 and G * ρ 2 ≥ 0. Note that G L 1 = 1. By Lemma 2.4 and Lemma 2.8, we get u 2 L ∞ ≤ e + 1 2(e − 1) u 2 H 1 ≤ e + 1 2(e − 1) E 0 , |(γ − A)u| ≤ |γ − A| u L ∞ ≤ |γ − A| e + 1 2(e − 1) E 1 2 0 and |(γ − A)G * u| ≤ |γ − A| G L 1 u L ∞ ≤ |γ − A| e + 1 2(e − 1) E 1 2 0 . It follows that dm(t) dt ≤ − 1 2 m 2 (t) + K, (3.3) where K = e+1 4(e−1) E 0 + 2|γ − A| e+1 2(e−1) E 1 2 0 . Since m(0) < − √ 2K, Lemma 2.10 implies lim t→T m(t) = −∞ with T = 2u ′ 0 (x 0 ) 2K − (u ′ 0 (x 0 )) 2 . Applying Lemma 2.6, the solution blows up in finite time. Theorem 3.2. Let (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, and T be the maximal time of the solution (u, ρ) to (2.1) with the initial data (u 0 , ρ 0 ). Assume that S u 0 (x)dx = a 0 2 . If there is some x 0 ∈ S such that ρ 0 (x 0 ) = 0 and for any ǫ > 0, u ′ 0 (x 0 ) = inf x∈S u ′ 0 (x) < − ǫ + 2 24 E 0 + ǫ + 2 4ǫ a 2 0 + |γ − A| 2(ǫ + 2) 3 E 0 + 4(ǫ + 2) ǫ a 2 0 , then the corresponding solution to (2.1) blows up in finite time. Proof. By Lemma 2.5, we have S u(t, x)dx = a 0 2 . Using Lemma 2.4 and Lemma 2.9, we obtain u 2 L ∞ ≤ ǫ + 2 24 E 0 + ǫ + 2 4ǫ a 2 0 , |(γ − A)u| ≤ |γ − A| u L ∞ ≤ |γ − A| ǫ + 2 24 E 0 + ǫ + 2 4ǫ a 2 0 and |(γ − A)G * u| ≤ |γ − A| G L 1 u L ∞ ≤ |γ − A| ǫ + 2 24 E 0 + ǫ + 2 4ǫ a 2 0 . Following the similar proof in Theorem 3.1, we have dm(t) dt ≤ − 1 2 m 2 (t) + K, (3.4) where K = ǫ+2 48 E 0 + ǫ+2 8ǫ a 2 0 + |γ − A| ǫ+2 6 E 0 + ǫ+2 ǫ a 2 0 . Following the same argument as in Theorem 3.1, we deduce that the solution blows up in finite time. Letting a 0 = 0 and ǫ → 0 in Theorem 3.2, we have the following result. Corollary 3.1. Let (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, and T be the maximal time of the solution (u, ρ) to (2.1) with the initial data (u 0 , ρ 0 ). Assume that S u 0 (x)dx = 0. If there is some x 0 ∈ S such that ρ 0 (x 0 ) = 0 and u ′ 0 (x 0 ) = inf x∈S u ′ 0 (x) < − E 0 12 + 2|γ − A| E 0 3 , then the corresponding solution to (2.1) blows up in finite time. Remark 3.1. Note that the system (2.1) is variational under the transformation (u, x) → (−u, −x) and (ρ, x) → (ρ, −x) even γ = 0. Thus, we can not get a blow up result according to the parity of the initial data (u 0 , ρ 0 ) as we usually do. Next, we will give more insight into the blow-up mechanism for the wave-breaking solution to the system (2.1), that is the blow-up rate for strong solutions to (2.1). Theorem 3.3. Let (u, ρ) be the solution to the system (2.1) with the initial data (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, satisfying the assumption of Theorem 3.1, and T be the maximal time of the solution (u, ρ). Then, we have lim t→T (T − t) inf x∈S u x (t, x) = −2. Proof. As mentioned earlier, here we only need to show that the above theorem holds for s = 3. Define now m(t) := inf x∈S [u x (t, x)], t ∈ [0, T ). By the proof of Theorem 3.1, we have there exists a positive constant K = K(E 0 , γ, A) such that − K ≤ d dt m + 1 2 m 2 ≤ K a.e. on (0, T ). (3.5) Let ε ∈ (0, 1 2 ). Since lim inf t→T m(t) = −∞ by Theorem 3.1, there is some t 0 ∈ (0, T ) with m(t 0 ) < 0 and m 2 (t 0 ) > K ε . Since m is locally Lipschitz, it is then inferred from (3.5) that m 2 (t) > K ε , t ∈ [t 0 , T ). (3.6) A combination of (3.5) and (3.6) enables us to infer 1 2 + ε ≥ − dm dt m 2 ≥ 1 2 − ε a.e. on (0, T ). (3.7) Since m is locally Lipschitz on [0, T ) and (3.6) holds, it is easy to check that 1 m is locally Lipschitz on (t 0 , T ). Differentiating the relation m(t) · 1 m(t) = 1, t ∈ (t 0 , T ), we get d dt ( 1 m ) = − dm dt m 2 a.e. on (t 0 , T ), with 1 m absolutely continuous on (t 0 , T ). For t ∈ (t 0 , T ). Integrating (3.7) on (t, T ) to obtain ( 1 2 + ε)(T − t) ≥ − 1 m(t) ≥ ( 1 2 − ε)(T − t), t ∈ (t 0 , T ), that is, 1 1 2 + ε ≤ −m(t)(T − t) ≤ 1 1 2 − ε , t ∈ (t 0 , T ). By the arbitrariness of ε ∈ (0, 1 2 ) the statement of Theorem 3.3 follows. Global Existence In this section, we will present a global existence result. By Lemma 2.7, we let ξ(t) ∈ S be a point where this infimum is attained. It follows that m(t) = u x (t, ξ(t)) and u xx (t, ξ(t)) = 0. Since the map q(t, ·) given by (2.3) is an increasing diffeomorphism of R, there exists a x(t) ∈ S such that q(t, x(t)) = ξ(t). Set m(t) = u x (t, ξ(t)) = u x (t, q(t, x(t))) and α(t) = ρ(t, ξ(t)) = ρ(t, q(t, x(t))). Valuating (3.1) at (t, ξ(t)) and using Lemma 2.7, we obtain m ′ (t) = − 1 2 m 2 (t) + 1 2 α 2 (t) + f and α ′ (t) = −m(t)α(t),(4.1) where f = u 2 + (γ − A)u − G * (u 2 + 1 2 u 2 x + 1 2 ρ 2 + (γ − A)u). By Lemma 2.4, Lemma 2.8 and By Lemmas 2.2-2.3, we know that α(t) has the same sign with α(0) = ρ 0 (x 0 ) for every x ∈ R. Moreover, there is a constant β > 0 such that |α(0)| = inf x∈S |ρ 0 (x)| ≥ β > 0 because of ρ 0 (x) = 0 for all x ∈ S. Next, we consider the following Lyapunov positive function w(t) = α(0)α(t) + α(0) α(t) (1 + m 2 (t)), t ∈ [0, T ). (4.2) Letting t = 0 in (4.2), we have w(0) ≤ ρ 0 2 L ∞ + 1 + u ′ 0 (x) 2 L ∞ := c 2 . Differentiating (4.2) with respect to t and using (4.1), we obtain w ′ (t) = α(0) α(t) · 2m(t)(f + 1 2 ) ≤ α(0) α(t) (1 + m 2 (t))(|f | + 1 2 ) ≤ w(t)(c 1 + 1 2 ). By Gronwall's inequality, we have w(t) ≤ w(0)e (c 1 + 1 2 )t ≤ c 2 e (c 1 + 1 2 )t for all t ∈ [0, T ). On the other hand, w(t) ≥ 2 α 2 (0)(1 + m 2 (t)) ≥ 2β|m(t)|, ∀ t ∈ [0, T ). Thus, This completes the proof by using Lemma 2.6. |m(t)| ≤ 1 2β w(t) ≤ and T > 0 be the maximal existence. Then the solution blows up in finite time if and only if lim inf t→T − {inf x∈S u x (t, x)} = −∞.Lemma 2.7. ([26]). Let t 0 > 0 and v ∈ C 1 ([0, t 0 ); H 2 (R)). Then for every t ∈ [0, t 0 ) there exists at least one point ξ(t) ∈ R with m(t) := inf x∈R {v x (t, x)} = v x (t, ξ(t)), and the function m is almost everywhere differentiable on (0, t 0 ) with d dt m(t) = v tx (t, ξ(t)) a.e. on (0, t 0 ). Lemma 2 . 210. ([29]). Assume that a differentiable function y(t) satisfiesy ′ (t) ≤ −Cy 2 (t) + K (2.4)with constants C, K > 0. If the initial datum y(0) = y 0 < − K C , then the solution to (2.4) goes to −∞ before t tend to Theorem 4. 1 . 1Let (u 0 , ρ 0 ) ∈ H s × H s−1 , s ≥ 2, and T be the maximal time of the solution (u, ρ) to (2.1) with the initial data (u 0 , ρ 0 ). If ρ 0 (x) = 0 for all x ∈ S, then the corresponding solution (u, ρ) exists globally in time.Proof. Define now m(t) := inf x∈S [u x (t,x)], t ∈ [0, T ). ∞ + 2|γ − A| u L ∞ + G L ∞ u t ∈ [0, T ). 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Dark matter may be discovered through its capture in stars and subsequent annihilation. It is usually assumed that dark matter is captured after a single scattering event in the star, however this assumption breaks down for heavy dark matter, which requires multiple collisions with the star to lose enough kinetic energy to become captured. We analytically compute how multiple scatters alter the capture rate of dark matter and identify the parameter space where the effect is largest. Using these results, we then show how multiscatter capture of dark matter on compact stars can be used to probe heavy m X TeV dark matter with remarkably small dark matter-nucleon scattering cross-sections. As one example, it is demonstrated how measuring the temperature of old neutron stars in the Milky Way's center provides sensitivity to high mass dark matter with dark matter-nucleon scattering cross-sections smaller than the xenon direct detection neutrino floor.

10.1103/physrevd.96.063002

1703.04043

f19fe4f650ebc9b62b538ee9808a359996b3aa44

Multiscatter stellar capture of dark matter 29 Aug 2017 Joseph Bramante Department of Physics University of Notre Dame 225 Nieuwland Hall Notre Dame 46556IndianaUSA Perimeter Institute for Theoretical Physics 31 Caroline St. NN2L 2Y5WaterlooOntarioCanada Antonio Delgado Department of Physics University of Notre Dame 225 Nieuwland Hall Notre Dame 46556IndianaUSA Adam Martin Department of Physics University of Notre Dame 225 Nieuwland Hall Notre Dame 46556IndianaUSA Multiscatter stellar capture of dark matter 29 Aug 20172 Dark matter may be discovered through its capture in stars and subsequent annihilation. It is usually assumed that dark matter is captured after a single scattering event in the star, however this assumption breaks down for heavy dark matter, which requires multiple collisions with the star to lose enough kinetic energy to become captured. We analytically compute how multiple scatters alter the capture rate of dark matter and identify the parameter space where the effect is largest. Using these results, we then show how multiscatter capture of dark matter on compact stars can be used to probe heavy m X TeV dark matter with remarkably small dark matter-nucleon scattering cross-sections. As one example, it is demonstrated how measuring the temperature of old neutron stars in the Milky Way's center provides sensitivity to high mass dark matter with dark matter-nucleon scattering cross-sections smaller than the xenon direct detection neutrino floor. Dark matter may be discovered through its capture in stars and subsequent annihilation. It is usually assumed that dark matter is captured after a single scattering event in the star, however this assumption breaks down for heavy dark matter, which requires multiple collisions with the star to lose enough kinetic energy to become captured. We analytically compute how multiple scatters alter the capture rate of dark matter and identify the parameter space where the effect is largest. Using these results, we then show how multiscatter capture of dark matter on compact stars can be used to probe heavy m X TeV dark matter with remarkably small dark matter-nucleon scattering cross-sections. As one example, it is demonstrated how measuring the temperature of old neutron stars in the Milky Way's center provides sensitivity to high mass dark matter with dark matter-nucleon scattering cross-sections smaller than the xenon direct detection neutrino floor. I. INTRODUCTION The nature of dark matter remains an outstanding mystery of our cosmos. Terrestrial direct detection experiments have become exceptionally sensitive to dark matter in the mass range GeV − 10 TeV. While there are proposals for probing lighter dark matter, finding heavy dark matter, which has a lower particle flux through terrestrial detectors, presents a special challenge. Compact stars, which have a much larger fiducial mass than terrestrial detectors, provide an alternative means to probe dark matter. Specifically, pairs of dark matter particles captured via interactions with the star can annihilate, leaving a distinct thermal trace. Prior studies of dark matter's accumulation in stars have considered the case that dark matter capture occurs after dark matter scatters once off a stellar constituent (e.g. nucleus, nucleon, electron). This is appropriate when the scattering cross section between dark matter and the constituent is small, leading to a mean path length that is large compared to the size of the star, so that at most one scatter is expected [1,2]. In this paper, we consider the case where the single scatter approximation breaks down and the dark matter is predominantly captured by scattering multiple times. We derive equations suitable for computing multiscatter capture of dark matter in stars, and as one application, show that observations of neutron stars in our galaxy would be sensitive to super-PeV mass dark matter that annihilates to Standard Model (SM) degrees of freedom, for dark matter-nucleon scattering cross-sections smaller than the xenon direct detection atmospheric neutrino floor. To become captured while transiting through a star, dark matter must slow to below the stellar escape speed by recoiling against stellar constituents. During a single transit through the star, if the number of such interactions exceeds unity, N ≈ nσR ≥ 1,(1) dark matter will be slowed (and possibly captured) by multiple scatters. Here n is the number density of stellar constituents, R is the radius of the star, and σ is the cross-section for dark matter to scatter off a stellar constituent. In white dwarfs, σ is typically the cross-section for scattering off nuclei (σ N X ), while in neutron stars σ is typically the cross-section for scattering off nucleons (σ nX ). One might also consider dark matter which predominantly scatters with electrons, in which case σ would be the dark matter-electron cross-section. Often the stellar mass is related to the number of scattering sites by M m N n , with m the mass of a scattering site and N n the number of scattering sites per star. Keeping the stellar mass (or, equivalently, N n ) fixed while varying the star's size, Eq. (1) implies that the typical number of dark matter scatters inside a star scales as N ∝ N n σR 4 3 πR 3 ∼ N n σ R 2 .(2) As explored hereafter, this means that multiscatter capture is particularly relevant for dark matter accumulating in compact stars, i.e. white dwarfs and neutron stars. Specifically, fixing σ and comparing our Sun with an equivalent mass white dwarf (R ∼ 10 −2 R sun ) or neutron star (R ∼ 10 −5 R sun ), the smaller size of the compact stars leads to a 10 4 enhancement in the average number of scatters for white dwarfs relative to the Sun, and a 10 10 relative enhancement for neutron stars. While multiscatter can occur for dark matter of any mass, multiscatter capture is most important for heavy dark matter. This is primarily for two reasons. First, in order to be captured, the dark matter must lose a sufficient amount of its energy through collisions with scattering sites in the star. The fraction of the dark matter's energy lost in each collision depends on the scattering angle, but is proportional to the constituent mass m divided by the dark matter mass m X in the limit that m X m. Therefore, heavier dark matter loses less energy per scatter, making gravitational capture after a single scatter less likely and multiscatter capture more important. Second, the range of dark matter-nucleon cross-sections for which heavy (PeV-EeV) dark matter capture in neutron stars proceeds predominantly through multiscatter energy losses, happens to coincide with dark matter-nucleon cross-sections just beyond the reach of next-generation direct detection experiments. Furthermore, it will be demonstrated in Section IV that PeV-EeV mass dark matter can be captured by multiple (∼ 10 − 10 3 times) scatters in neutron stars even for the dark matter-nucleon cross-sections below the xenon direct detection "neutrino floor," σ nX ∼ 10 −45 cm 2 (m X /PeV) [3]. For these reasons, a primary focus of this paper will be dark matter with mass m χ TeV. The dark matter masses just mentioned are well above the canonical WIMP mass scale of about 100 GeV. Dark matter with a weak scale mass has received deserved attention in the past decade because it can reproduce the observed dark matter abundance as a thermal relic. Considerable experimental efforts have bounded the nucleon scattering cross-section for weak-scale mass (m X ∼ 100 GeV) dark matter to σ nX 10 −46 cm 2 , Ref. [4][5][6]. On the other hand, it has been shown that if one deviates from the minimal cosmological scenario, dark matter models with heavier masses m X ∼ TeV − EeV are predicted, e.g. [7][8][9][10][11], either as a result of extra sources of entropy that dilute the thermal overabundance or because dark matter is very weakly coupled to the SM and it never thermalizes. As weak-scale mass dark matter has become increasingly constrained, the prospect of very heavy dark matter, which can still have a nearly "weak" scale cross-section with nucleons (σ ∼ 10 −40 cm 2 ) deserves more attention. However, as a consequence of reduced dark matter flux, direct detection experiments have sensitivities that drop off with 1/m X at high masses, and new methods to probe heavy dark matter are necessary. As we will show, neutron stars in our galaxy are powerful probes of heavy, weakly interacting dark matter. Some prior work has considered multiscatter dark matter capture in the Earth and Sun [12][13][14], where the gravitational potential of the capturing body, nuclear coherence, and relativistic effects could be reasonably neglected. Hereafter we treat single and multiple scatter capture rates and provide an equation valid for capture in the limit that the escape velocity of the capturing body greatly exceeds dark matter's halo velocity. The organization of the rest of this paper is as follows: in Section II, we present our main points and the parametric dependence of multiscatter dark matter capture in compact stars. A detailed derivation of multiscatter capture is given in Section III. Using the derived multiscatter capture formulae, Section IV finds prospects for old neutron stars near the galactic center to constrain heavy dark matter that annihilates to Standard Model particles. In Section V, we conclude. II. PARAMETRICS OF MULTISCATTER CAPTURE In order to calculate the parametric dependence of multiscatter capture, we are going to first examine the dark matter single-scatter capture rate, and then investigate how the rate changes when one accounts for more than one collision. We will find that, for heavy enough dark matter, the mass capture rate of dark matter on compact stars depends linearly on σ and inversely on m X . This ∼ σ/m X scaling of the mass capture rate arises for heavier dark matter, because more scatters (which scale up with σ) are needed for heavier particles to be captured by the star. Dark matter capture in a star depends upon the flux F of dark matter through the star and the probability Ω that collision(s) with the star will deplete the dark matter's energy enough that it becomes gravitationally bound. The flux in turn depends upon the number density of dark matter in the halo n X = ρ X m X , the relative motion of the star with respect to the dark matter halo (v star ), the distribution of dark matter speeds in the dark matter halo, and the escape speed of the dark matter halo (v halo esc ). The probability to capture (Ω) depends on the speed of the dark matter, set by the initial speed plus the amount of speed it has gained falling into the star's gravitational well. Additionally, the probability depends on the density of scattering sites in the star (n T ), the cross section of dark matter to scatter off scattering sites (σ), and the fraction of scattering phase space where sufficient energy is lost. Both the velocity gained by falling into the star and the number density are, in principle, functions of where inside the star the collision occurs. Combining the flux and capture probability yields a differential capture rate, which must be integrated over dark matter initial velocities and trajectories through the star. Schematically, the differential capture rate is d C dV d 3 u = dF (n X , u, v star , v halo esc ) Ω(n T (r), w(r), σ, m n , m X ),(3) where u is the dark matter velocity far from the star (the halo velocity) and w 2 (r) = u 2 + v 2 esc (r) is the speed of the dark matter after it has fallen to a distance r from the star's center (either inside or outside of the star). To focus on the parametrics of dark matter capture, for simplicity we assume no motion of the star relative to the dark matter thermal distribution in the halo (v star → 0) and an infinite escape speed for the dark matter halo (v halo esc → ∞). We also fix the escape speed of dark matter in the star to the escape speed at the star's surface (v esc (r) = v esc (R)), and for the moment omit general relativistic and nuclear physics corrections. With these provisos, a constant-density star in the rest frame of the dark matter halo with stellar escape velocity v 2 esc ∼ 2GM/R has a single-scatter dark matter capture rate derived in Appendix A C 1 = √ 24πG ρ X m X M R 1 v Min 1, σ σ sat 1 − 1 − e −A 2 A 2 .(4) Note that the capture rate scales with dark matter density ρ X and inversely with the dark matter halo velocityv. Here, G is Newton's constant, M is the mass of the star, σ is dark matter's crosssection with a stellar constituent (nucleus, nucleon, electron). The exponential factor 2 and m is the mass of the particle (nucleus, nucleon, electron) dark matter scatters against. Increasing the cross-section past a certain threshold will guarantee that most transiting dark matter scatters with the star at least once, though it may not lose enough energy to be captured. This threshold cross-section is customarily defined as σ sat = πR 2 /N n , where N n is the number of scattering sites, and the "Min" function evaluates to unity once at least one capture is probable. The parenthetical term in Eq. (4) takes into account dark matter that scatters but does not lose sufficient energy to be gravitationally captured. A 2 ≡ 3 2 v 2 esc v 2 β − , where β ± ≡ 4m X m/(m X ± m) To better understand the origin of the parenthetical piece of Eq. (4), let us examine the energetics of gravitational capture. To be captured after a single collision, the energy lost by the dark matter must be greater than its initial kinetic energy in the galactic halo. The energy loss is proportional to the reduced mass of the dark matter -constituent system, µ n and the speed of the dark matter at the collision site. In the limit that the star's escape velocity is much greater than the halo velocity (w = u 2 + v 2 esc v esc ) the capture requirement is ∆E 2 µ 2 n m v 2 esc z ≥ 1 2 m X u 2 ,(5) where z is a kinematic variable ∈ [0, 1] related to the scattering angle. Assuming dark matter is much heavier than the stellar constituents and turning the above requirement above into a condition on u, u < u max = β + z v esc .(6) In the full capture treatment (Appendix A), for dark matter with Boltzmann distributed velocities from 0 to u max and scattering angles z ∈ [0, 1], we consider kinematic phase space where dark matter is moving slowly enough to be captured after a single collision. The limit of this phase space is set by u max , which is evident in the form of the A 2 exponential factor in Eq. (4). Note that when m X m, a limit that will be appropriate throughout this paper, β ± both reduce to 4 m/m X . The origin and form A 2 term are important because A 2 governs the dependence of C 1 on the dark matter mass. When A 2 is large, corresponding to a maximum capture speed much larger than than average dark matter speed, the parenthetical term in Eq. (4) evaluates to 1, and the sole dark matter mass dependence lies in the number density ρ X m X . In this case, the single scatter capture rate scales as C 1 ∝ σ m X , (A 2 1) (7) implying a mass capture rate m X C 1 ∝ σ that is independent of the dark matter mass. However, if A 2 is small, implying a maximum capture speed less than a typical dark matter halo velocityv, we can expand the entire parenthetical expression in Eq. (4), and find that the capture rate scales as C 1 ∝ ρ X m X σ A 2 ∝ σ m 2 X , (A 2 1) (8) implying a mass capture rate scaling m X C 1 ∝ σ/m X that depends inversely on the dark matter mass. To see where the mass capture rate transitions from being constant to being m X -dependent in compact stars, we can insert appropriate values for v esc . For a solar mass white dwarf v esc c ∼ 2 × 10 3 km/s, while a solar mass neutron star has v esc c ∼ 2 × 10 5 km/s; both of these escape speeds are far greater than the average dark matter halo speedvc ∼ 220 km/s, therefore A 2 will only be less than one if the dark matter is much heavier than m. Specifically, taking A 2 = 1 to be the transition value, and solving for m X , we find the transition occurs at m X ∼ TeV in a solar mass white dwarf (assuming scattering off of carbon) and m X ∼ PeV for a solar mass neutron star (assuming scattering off a neutron). To see how the parametric dependence of Eq. (4) changes in the case of multiple scatters, let us revisit the energetics of gravitational capture. For the moment, let us assume that dark matter participates in N ≥ 1 collisions during its transit of the star and that each collision results in an average energy loss ∆E i = β + E i 2 .(9) If the dark matter initially entered the star with energy E 0 , the energy after N 'average' collisions is E N = E 0 1 − β + 2 N ,(10) or a net energy deposit of ∆E N = E 0 − E N . Assuming, as in the single scatter case, that the initial dark matter kinetic energy is E 0 ∼ 1/2 m X v 2 esc and plugging ∆E N into the capture condition Eq. (5), we can solve for the maximum halo velocity u that can be captured u ≤ v esc 1 − 1 − β + 2 N 1/2(11) In the limit that m X m and β + → 4m/m X , the leading order term in the binomial expansion of the right side of Eq. (11) approximates the full expression. In that limit, the maximum allowed velocity simplifies to u ≤ N β + 2 v esc ∼ = 2 N m m X v esc (12) up to corrections of O (N m) 2 m 2 X . As we will show in more detail in the next section, in the limit of v esc v the probability to capture after N scatters can be expressed in a form very similar to (4) but with A 2 -the factor in the exponential -modified to A 2 N ≡ 3 v 2 esc v 2 N m m X .(13) As discussed following Eq. (4), if this exponential factor is large then the m X -dependence in the capture rate from the A 2 term is suppressed. Meanwhile, if the factor is small, the exponential can be approximated by an expansion, resulting in a capture rate ∝ n X A 2 ∝ σ/m 2 X . Comparing Eq. (4) to Eq. (13), we see that multiple scattering has added a factor of N to the A 2 term. The N dependence in the numerator of Eq. (13) means that for N 1, the dark matter mass needs to be larger (for a given v esc ,v and m) before the exponential factor becomes small. Stated another way, if the dark matter scatters N times, the capture rate will behave as C N ∼ σ/m X out to masses N times higher than if dark matter only scatters once. Note that this discussion has involved only the energetics of slowing down a heavy dark matter particle to beneath a star's escape speed and not whether the dark matter interacts with stellar constituents strongly enough to participate in multiple scatters in the first place. Following from Eq. (1), the likelihood to participate in multiple scatters roughly depends on the path length of the dark matter 1/nσ compared to the size of the star. We will flesh out this dependence in the next section. III. MULTISCATTER CAPTURE Having examined the parametric scaling of multiscatter capture in the previous section, in this section we derive the multiscatter dark matter capture rate. Our notation follows that of [12], which considered capture by the Earth's iron core, where the acceleration of incoming dark matter due to Earth's gravity, and -more broadly -general relativistic effects, could be neglected. In the large N limit, the treatment presented here also allows for more efficient computation of the multiscatter capture rate, by obviating the N -fold kinematic phase-space integral in [12]. For multiscatter capture it is convenient to define the optical depth τ = 3σ 2σsat , σ sat = πR 2 Nn , the average number of times a dark matter particle with dark matter -nuclear cross section σ will scatter when traversing the star. 1 The probability for dark matter with optical depth τ to participate in N actual scatters is given by Poisson(τ, N ). However, this expression can be improved to incorporate all incidence angles of dark matter. Defining y as the cosine of the incidence angle of dark matter entering the star, the full probability is p N (τ ) = 2 1 0 dy ye −yτ (yτ ) N N ! .(15) 1 To understand the 3 2 factor in the optical depth, observe that the cross section for which 1 scatter occurs over a distance of 2R, (where R is the radius of the star) is 1 = n σ (2R) = Nn (4/3)πR 3 σ (2R) = 3 Nn 2π R 2 σ (14) → σ = 2 3 πR 2 Nn = 2 3 σsat, The optical depth is normalized so that τ = 1 when dark matter typically scatters once as it passes through the star. While it incorporates all incidence angles, this expression still makes the assumption that the dark matter takes a straight path through the star. In practice, the straight path assumption will produce conservative bounds on dark matter capture, marginally under-predicting the capture rate. Incorporating the likelihood for dark matter to participate in N scatters, the differential dark matter capture rate after exactly N scatters looks similar to the single scatter formula (see Appendix A), with the probability to capture after N scatters g N (w) adjusted to take into account the kinematics of N collisions and replacing σ σsat → p N (τ ) 2 , C N = πR 2 p N (τ ) ∞ 0 f (u) du u w 2 g N (w).(16) The velocity distribution f (u) of dark matter particles in the galactic halo is given in Eq. (A1). In writing the velocity distribution as f (u) we have retained the assumptions from the single capture case that the escape velocity of the dark matter halo is infinite and the velocity of the star relative to the dark matter is zero. We have also maintained that the density of the star is uniform and ignored the radial dependence of the escape velocity. 3 It is convenient to shift the integral to w, where w 2 = u 2 + v 2 esc . The capture rate for N scatters then becomes C N = π R 2 p N (τ ) ∞ ve dw f (u) u 2 w 3 g N (w),(17) and the total capture rate is the sum over all N of the individual C N C tot = ∞ N =1 C N .(18) In actual computations, the sum in Eq. (18) will be cut off at some finite N max where p Nmax (τ ) ≈ 0. Finally, we need to evaluate g N (w), the probability that the speed of the dark matter after N collisions drops below the escape velocity. This probability, which we analyzed dimensionally in Section II, depends solely on dark matter's initial velocity, the amount of energy lost in each scatter, and the escape velocity of the star. For dark matter with initial kinetic energy at the star's surface E 0 = m X w 2 /2, the energy lost in a single scattering event is given by ∆E = zβ + E 0 , where z is related to the scattering angle, z ∈ [0, 1], and we again note that β + ≡ 4m X m/(m X + m) 2 . Iterating for N scatters, the dark matter energy and velocity decrease to E N = N i=1 (1 − z i β + ) E 0 , v N = N i=1 (1 − z i β + ) 1/2 w.(19) If the velocity after N scatters is less than the escape velocity, the dark matter is captured. Phrased as a condition on the initial velocities w that we are integrating over, the capture probability is g N (w) = 1 0 dz 1 1 0 dz 2 · · · 1 0 dz N Θ v esc N i=1 (1 − z i β + ) −1/2 − w ,(20) 2 The multi-scatter capture rate (16) can be obtained by setting n(r) = Nn 4 3 πR 3 in Eq. (A7), integrating r from 0 to R and making the substitutions g1(w) → gN (w) and σ σ sat → pN (τ ). 3 To estimate how much the constant density assumption alters the neutron star capture rate, consider an approximate neutron star density profile (ADP) ρ ADP N S (r) = 2.6 × 10 38 GeV/cm 3 10 km r , which matches a 1.5 M , R = 10 km neutron star. This can be compared to a constant density (CD) profile, such a neutron star would have ρ CD N S 4 × 10 38 GeV/cm 3 . We can calculate the integrated optical depth dτi = n(r)σnX d , where is the path of the dark matter particle. Calculating this integrated optical depth for a dark matter particle that passes within a kilometer of the center of the neutron star, we find that for the constant density and approximate density profile cases, for trajectories passing deep within the neutron star, the optical depth can increase by up to fifty percent. This would somewhat aid capture in the multiscatter regime. Therefore, the bounds derived in this paper are somewhat conservative. where the dz i integrals sum over all possible scattering trajectories (angles) at each step. This condition requires an integral for every scatter, and becomes computationally cumbersome to evaluate for large N . Therefore, as a further approximation, let us replace the z i with their average value. Provided the differential dark matter-nuclear cross section is independent of scattering angle -valid in most scenarios of spin-independent elastic scattering -z i = 1/2 and g N (z) simplifies 4 , g N (w) = Θ v esc N i=1 (1 − z i β + ) −N/2 − w .(21) As in the single scatter case, the capture probability restricts the range of dark matter velocities that allow for capture. To illustrate the relationship between dark matter's halo speed and the number of scatters it takes to slow down to below the star's escape speed, we recast Eq. (21) as contours in u − N space in Fig. 1 below, for typical neutron star and white dwarf parameters (see caption). The fact that dark matter with a given mass and speed requires more scatters to be captured in a white dwarf is due to the fact that the velocity at infinity (u) is a larger fraction of the star's escape speed than for a neutron star. This gives the impression that multiscatter is more important for dark matter capture in white dwarfs. However, the number of scatters needed to slow down to sub-escape velocities is not the only factor in the problem; capture also depends on whether the dark matter-nuclei cross section is large enough for the dark matter to interact scatter multiple times as it transits the star. The strength of the dark matter -constituent interaction is encapsulated in the optical depth τ which, as we have seen, is proportional to 1/R 2 and therefore much larger for neutron stars. Using the simplified form for g N (w), we can evaluate remaining integral in Eq. (17): C N = π R 2 p N (τ ) √ 6 n X √ πv (2v 2 + 3 v 2 esc ) − (2v 2 + 3 v 2 N ) exp − 3(v 2 N − v 2 esc ) 2v 2 ,(22) with v N = v esc (1 − β + /2) −N/2 . In the limit that v esc v and m X m n , this becomes C N = √ 24 π p N (τ ) G n X M R 1 v 1 − 1 − 2A 2 Nv 2 3 v 2 esc e −A 2 N ; A 2 N = 3 v 2 esc N m v 2 m X ,(23) where the last expression follows the format of the single scatter capture equation Eq. (4). Note that the reason C 1 according to this formula does not precisely match Eq. (4) is that we integrated over all possible energy loss fractions (dz 1 ) when deriving the latter, but assume average energy loss in the former. As expected, the capture rate for N scatters has a similar form as the single capture rate, up to a factor of N in the exponential factor A 2 N . Following the logic presented in Sec. II, the factor of N implies the C N ∝ 1/m X scaling persists out to higher m X than in the single scatter case. However, while the behavior of an individual C N is easy to see given m X , m and v esc , the mass scaling of the full capture rate is more subtle as it involves the sum over all C N , each weighted by p N (τ ). Having reviewed the general form of the multiple scatter capture rate, we can now apply it to white dwarfs and neutron stars. Each of these applications involves subtleties not present in Eq. (17). White dwarfs are compact stars (R ∼ 10 4 km, M ∼ 10 57 GeV) that are supported by electron degeneracy pressure. Their suitability as potential laboratories to capture and thereby constrain various dark matter candidates has been studied previously in the single-scatter regime [15][16][17][18][19]. At the upper end of the mass range, white dwarfs are largely composed of carbon and oxygen, so m = m N ∼ O(10 GeV) in the capture equations above. Dark matter possessing spin-independent (e.g. scalar or vector current) interactions with nuclei will scatter coherently off the nucleons within carbon/oxygen if the momentum exchange is low enough, while higher energy exchanges will be sensitive to the substructure of the nucleus and correspondingly suppressed. This loss of coherence is expressed by a form factor. Including the form factor suppression, the multiscatter accumulation rate will be given by Eq. (18) with the cross-section substitution σ → σ WD N X σ nX m 4 N m 4 n F 2 ( E R ),(24) where, in the case of scattering off carbon, the mass of the stellar constituent is m N 12 m n 11.1 GeV, and F 2 ( E R ) is the Helm form factor evaluated at the average recoil energy E R [20]. The average recoil energy is defined as E R E max R 0 dE R E R F 2 (E R ) E max R 0 dE R F 2 (E R ) ,(25) where we make the approximation that v esc is much greater than the halo velocity and therefore E max R 4m N v 2 esc . For recoil energies relevant for heavy dark matter scattering off carbon in a solar mass white dwarf (v esc 0.01, E R MeV), the form factor evaluates to F 2 ( E R ) ∼ 0.5. In addition to affecting the overall scattering cross section, the form factor also impacts the weighting of different momentum exchanges (scattering angles) in each scatter, previously encapsulated in the variable z i . Higher momentum exchanges are suppressed by the form factor as they correspond to reduced dark matter-nucleus scattering coherence. As a result, lower energy scatters -where a smaller fraction the dark matter's kinetic energy is deposited in each scatter -are more common. To account for this, we make the substitution z i = E R /E max R (instead of z i = 1 2 ) in Eq. (21). In deriving z i , we have assumed that the relative velocity of the dark matter and nucleus remains constant (at ∼ v esc ) during the capture process. This assumption is valid so long as the dark matter halo velocity is much smaller than its velocity during capture u w ∼ v esc , implying that the speed of the dark matter remains approximately constant during capture. To understand this, note that as soon as the dark matter velocity decreases by an O(1) factor from w ∼ v esc , its speed will be well below the escape velocity, since u v esc . The mass capture rates for heavy dark matter in a white dwarf, computed using both the single and multiple capture expressions and two different assumptions about the size of the dark matternucleon cross section are shown in Fig. 2. The contours in Figs. 2 display the capture rate for up to N ≤ 1, 10, 100... scatters, using Eq. (18). As the dark matter-nucleon cross-section increases, the difference in mass capture rate for N = 1 versus N ≤ 1000 scatters increases dramatically. This is a consequence of the fact that, as the dark matter-nucleon cross-section becomes large enough, most trajectories through the white dwarf will involve multiple scattering events and so the rate for capture after a single scatter more substantially under-predicts the total capture rate. We can also see that, as the number of scatters increases, the "turnover mass" (the mass at which the capture rate diminishes) also increases. As explored in Section II, this is because lighter dark matter requires fewer scatters to be captured, since the fractional energy loss of the dark matter per scatter is ∼ 2m N /m X . The quoted per-nucleon scattering cross-sections in Fig. 2, σ nX = 10 −34 and 10 −36 cm 2 , which were chosen to be large enough so that multiple scatters are relevant, are typically excluded by direct detection searches for spin-independent DM-nucleon scattering [4][5][6]. One might consider whether white dwarfs could be used to constrain spin-dependent DM-nucleon interactions, which are less constrained by direct detection searches. Unfortunately, white dwarfs are composed of mainly spin-free nuclei (e.g. carbon 12, oxygen 16), and so a precise determination of the fraction of spin > 0 nuclei in a given white dwarf would need to be determined to set bounds on spin-dependent dark matter, something that is beyond the scale of this work. Another scenario for which large dark matter-nucleon cross-sections are not yet excluded and could potentially be probed by white dwarf observations is inelastic dark matter [17,21], provided the dark matter settles to the core of the white dwarf (i.e. thermalizes) within the age of the universe. Turning to neutron stars, a 1.5 solar mass neutron star has escape speed 2GM/R ∼ 2 3 [22] and is supported by neutron degeneracy pressure. The extreme velocities and densities mean we must modify Eq. (17) to account for two general relativistic corrections when considering dark matter capture on a neutron star. First, the amount of dark matter crossing the star's surface will be increased because of an enhancement from the star's gravitational potential. It can be shown [23] that for a dark matter particle with velocity u and impact parameter b, if the particle barely grazes the surface of the star, then C X ∝ b 2 = (2GM R/u 2 )[1 − 2GM/R] −1 , where the square-bracketed term accounts for the general relativistic enhancement to dark matter crossing the star's surface. Accordingly, the dark matter capture rate (with m = m n , of course) is modified to, C N → C N 1 − 2GM R ,(26) to account for general relativity-enhanced capture. 5 The second general relativistic correction we need is to account for the gravitational blueshift of the dark matter's initial kinetic energy, in the rest frame of a distant observer. In the absence of general relativistic corrections, the dark matter must lose its initial halo kinetic energy E i = 1 2 m X u 2 via scattering with the star in order to become gravitationally bound to the star. However, from the rest frame of a distant observer, this initial kinetic energy will be enhanced by a factor χ = [1 − (1 − 2GM/R) 1/2 ] under the influence of the star's gravitational potential. This can be accounted for by making the substitution in Eq. (21) v esc → 2χ. (27) In practice, the gravitational and kinetic energy blueshift effects alter the dark matter capture rate in neutron stars by less than a factor of two. Given the degeneracy of the neutrons that the dark matter must collide with, one may worry that Pauli blocking also comes into play when deriving the capture rate. Specifically, in order to scatter with the constituents of a neutron star, dark matter must excite them to momenta larger than their Fermi momentum, typically p F,NS ∼ 0.1 GeV [23]. However, as the incoming dark matter has been accelerated to semi-relativistic speeds in the gravitational well of the neutron star, this requirement is easily satisfied provided the dark matter is heavy. Plugging in numbers, in the limit m X m n the average momentum exchanged in any scatter is Q ∼ √ 2 m n v esc ∼ 0.7 GeV p F,N S ; see e.g. [24,25] for more discussion. with R = 10 km, in a background dark matter density of ρ X = 0.3 GeV/cm 3 with halo velocity dispersion v 220 km/s is assumed. Note that the dark matter mass where the mass capture rate shifts from m X C X ∝ const to m X C X ∝ 1/m X shifts to higher values as we include more scatters In Fig. 3 we show the mass capture rate of dark matter on a neutron star for a range of dark matter masses and a dark matter-nucleon cross-sections where τ 1. Figure 3 has all of the same qualitative features as Fig. 2: the mass capture rate increases dramatically once multiple scatters are included, and exhibits a 1/m X dependence in the large m X limit. However, comparing Figs. 2 and 3, it is evident that multiscatter capture is relevant for white dwarfs when σ nX ∼ 10 −35 cm 2 , while multiscatter capture on neutron stars becomes important for σ nX ∼ 10 −45 cm 2 . Because the latter cross-section is closer to the cross-section presently probed by direct detection experiments [4,5], we will focus on neutron star probes of dark matter in the next section. While our focus here will be on dark matter which annihilates inside and thereby heats neutron stars, there are many other ways multiscatter stellar capture could be used to probe dark matter, including neutron star implosions [16,23,24,[26][27][28][29][30][31][32][33][34][35], monopole-induced nucleon decay [36,37], white dwarf heating [17,38,39], Type Ia supernova ignition [18,19], neutrino signatures of superheavy dark matter [40,41], and dark matter-powered stars [42][43][44][45]. IV. PROBING HEAVY DARK MATTER WITH OLD NEUTRON STARS Dark matter that is captured in neutron stars may annihilate to Standard Model particles, thereby heating and increasing the apparent luminosity of old neutron stars. Consequently, the temperature of old neutron stars can be used to probe the dark matter-nucleon cross-section, provided that one bounds or measures the temperature of old stars in regions of sufficiently high dark matter density. Because it harbors a high density of dark matter, the galactic center is an obvious target [15,26,46,47]. While old neutron stars at the galactic center are being vigorously sought by the current generation of radio telescopes [48,49], to date none have been found, although they are expected to be within reach of next generation radio telescopes like FAST and SKA [50]. Here we determine the potential bounds on dark matter annihilating to SM particles in old neutron stars in the galactic center. Prior work [26,47] has explored this bound on dark matter using single scatter capture. This document extends these bounds to higher masses using multiple scatter capture, assuming that DM annihilates to Standard Model particles, and that an old, colder neutron star is resolved in the galactic center at some time in the future. The process by which dark matter heats neutron stars involves several steps. First, each captured dark matter particle must thermalize with the host neutron star through successive scatters off neutrons. This thermalization process is complicated by the fact that dark matter momentum will drop after each scatter, and eventually the momentum exchanged between dark matter and the neutrons becomes small enough that Pauli blocking can no longer be ignored. A full calculation of thermalization within neutron stars incorporating Pauli blocking was performed in Ref. [25] and showed that the time to thermalize is much less than the age of the neutron star. As one example, for m X > 100 GeV dark matter with a cross-section σ nX > 10 −48 cm 2 (well below the values where multscatter becomes important), thermalization occurs in less than a thousand years. Once thermalized, the dark matter settles into a spherical volume V th within the star. Approximating the neutron star as having a constant density core ρ N S , V th can be related to the star's temperature T by V th = 4 3 πr 3 th , r th = 9T /4πGρ N S m X (see e.g. [18]) within the star. The next step is to understand how N X (t) -the number of dark matter particles residing in V th -evolves with time. Assuming the thermalization time is rapid compared to other timescales, the number of dark matter particles increases as new particles are captured, and decreases as pairs of dark matter particles meet and annihilate. This can be phrased as a simple differential equation for N X (t) [31], with solution: N X (t) = C X V th σ a v tanh   C X σ a v V th t   ,(28) where C X is the net capture rate, t is time over which collection has occurred, and σ a v is the thermally-averaged self-annihilation cross-section of the dark matter (DM DM → SM fields). Once t > V th C X σav , the dark matter population plateaus, and there is an equilibrium between the rate at which dark matter is annihilated and the rate at which it is captured. Assuming all dark matter passing through a neutron star is captured (which implies the longest equilibration time), this equilibration time is [47]: t eq 10 4 yrs 10 2 GeV m X 1/4 10 3 GeV/cm 3 ρ X 1/2 T N S 3 × 10 4 K 3/4 10 −45 cm 3 /s σ a v 1/2 ,(29) where T N S is the temperature of the neutron star, and this equilibration time assumes that all DM passing through a R = 10 km, 1.5 M NS with central density ρ N S ∼ 4 * 10 14 g/cm 3 is captured. The temperatures for the oldest observed neutron stars (age > 100 million years) are projected to be T 3 × 10 4 K [26]. Plugging this temperature into Eq. (29) and assuming our local dark matter density ρ X = 0.3 GeV/cm 3 , we find the equilibration time is t eq ≤ 10 million years for 100 GeV dark matter with annihilation cross sections of σ a v 10 −48 cm 3 /s. This value is already far less than the age of the oldest neutron stars, and increasing the dark matter mass, density or annihiliation cross section leads to even shorter times; for a benchmark point closer to our region of interest, PeV dark matter in the galactic center (ρ X = 10 3 GeV/cm 3 ) will equilibrate in a 3 × 10 4 K neutron star in as little as 1000 years if σ a v = 10 −45 cm 3 /s. Because this dark matter self-annihilation cross-section is already quite small, hereafter we assume the dark matter annihilation rate rapidly reaches equilibrium with the capture rate. Within the parameter space where thermalization and equilibration times are short compared to the typical neutron star lifetime, the annihilation rate is equivalent to the capture rate, and the rate of energy release is simply the mass capture rate m X C X . We can define an effective neutron star temperature arising from dark matter annihilations by equating the energy release rate to the apparent luminosity, 6 m X C X = L DM = 4πσ 0 R 2 T 4 N S 1 − 2 GM R 2 ,(30) where σ 0 = π 2 /60 is the Stefan-Boltzmann constant, and the parenthetical term accounts for the gravitational redshift of light departing the high curvature environment of a neutron star. Read left to right, Eq. (30) defines a minimum temperature for an old neutron star (provided our assumptions of thermalization and equilibration) for a given dark matter mass, density, and capture cross section. N p N (τ ) 1 − 1 − 2A 2 Nv 2 3 v 2 esc e −A 2 N = const T 4 N S ρ X ,(31) where the constant on the right hand side is a combination of G, σ,v and the mass and size of the neutron star. The sum over N makes this formula a bit opaque, however we know from Sec. II that the left hand side of Eq. (31) is roughly linear in the dark matter-nucleon cross section σ and is either independent of the dark matter mass or ∝ 1/m X depending on whether the dark matter is lighter or heavier than a PeV. Solving Eq. (31) for σ, these two regions translate into bounds that are σ ∝ const (for m X < PeV) or σ ∝ m X (for m X > PeV). To get a feeling for the type of bound that can be set in this way, in Fig. 4 below we show σ could be excluded as a function of m X should we observe an old neutron star with temperature T N S ∼ 3 × 10 4 K in the galactic center (ρ X = 10 3 GeV/cm 3 ). L U X -P a n d a X mX (GeV) . Potential sensitivity to dark matter from annihilation to SM particles, heating a 1.5 M neutron star in the galactic center (ρ X = 10 3 GeV/cm 3 , about 10 parsecs from the Galactic Center) to a core temperature of ∼ 3 × 10 4 K, along with interpolations of the current LUX bounds and the neutrino floor (one atmospheric neutrino event on xenon [3]) for comparison. Here the parameters of the surrounding dark matter density and neutron star temperature have been chosen conservatively; observation of a colder neutron star, or a larger dark matter density would both deepen sensitivity. The curve labeled "1 scatter" uses Eq. (4) to set the bound, while the multiscatter curve uses the multiscatter formulae derived in this document. Note that multiscatter capture allows for heavier dark matter to be discovered or bounded, for cross-sections below the direct detection neutrino floor. There are several interesting features in Figure 4. First, a shift in the cross section bound around m X ∼ PeV is evident; this was the mass at which multiple scatter capture becomes relevant, as derived in Section II. Second, should a neutron star matching the criteria be found, the DM-nucleon cross section bound it implies would dominate over the existing xenon direct detection bound for all dark matter heavier than m X ∼ TeV. Furthermore, while comparing potential neutron star heating bounds to current xenon bounds may seem unfair, for m X > 0.1 PeV, the cross sections ruled out by neutron star heating are beneath the so-called 'neutrino floor' cross section, where direct detection experiments encounter an irreducible background. Given that direct detection experiments are approaching the multi-ton scale and the feasibility of further size increase are far from obvious, observing a cold neutron star may be the best path towards sub-neutrino floor bounds, and further study into how well current and planned telescopes can identify cold neutron stars in environments like the galactic center are warranted [51]. The dependency of the neutron star bound on the temperature of the observed star and the ambient dark matter density where the star is located are clear from the right hand side of Eq. (31), provided one does not deviate too much from the benchmark values of T N S = 3 × 10 4 K, ρ X = 10 3 GeV/cm 3 . For example, observing a T N S ∼ 1.5 × 10 4 K neutron star in the galactic center would strengthen the bound in Fig. 4 by a factor of ∼ 10. For larger temperature or density deviations, the parametrics is not as simple, since the capture rate cannot be increased indefinitely by increasing the DM-nucleus cross section. Specifically, once σ reaches the point where all dark matter (at all halo velocities) is captured, further increasing σ will not change anything. This 'saturation' cross section will depend on the mass of the dark matter. While a detailed study of the feasibility of constraining neutron stars at various temperatures in the galactic center has not yet been undertaken, we note that observations of > 10 4 K neutron stars within a parsec of the galactic center appear to be within the scope of existing X-ray observatories [52], and would lead to the strongest bound on the dark matter-neutron cross section for m X > PeV. V. CONCLUSIONS The existence of dark matter has been established by a number of cosmological and astrophysical observations. It is, therefore, one of the most compelling arguments for physics beyond the Standard Model, since there is no candidate for dark matter within the Standard Model. This has inspired vigorous experimental searches for non-gravitational dark matter interactions, including underground detectors looking for dark matter smacking against nuclei, and satellites searching for annihilation of dark matter into Standard Model particles. These searches are most sensitive to dark matter masses up to a few TeV. One complementary way to look for heavier dark matter is though its accumulation in stars. Most studies addressing dark matter accumulation in stars have supposed that capture occurs after a single scatter. In this paper we explored multiscatter capture and found it is particularly relevant for high mass dark matter, which, even for cross-sections below present constraints, will typically scatter multiple times in a neutron star before being captured. We have derived analytical formulae for this process and we have proven that the dark matter-nucleon cross-section bounds obtained at large dark matter masses will have the same parametric dependence as xenon direct detection experiments. Note that while the σ ∝ m X scaling at high masses for direct detection experiments is a result of decreased local dark matter number density at high masses (n X ∼ ρ X /m X ), the same parametric dependence that arises for heavy dark matter capture in compact stars results from needing more scattering events to capture higher mass dark matter, as explained in Section II. We have used the resulting formalism to point out bounds on heavy dark matter, which could be obtained through thermal observation of old neutron stars in the galactic center. The resulting bounds are stronger at high dark matter masses, than the reach of next generation direct detection experiments. For m X 100 TeV the cross-section bound on dark matter that annihilates to Standard Model particles from a T ∼ 10 4 K neutron star near the galactic center, lies below xenon direct detection cross-sections at which atmospheric neutrinos will begin to provide a substantial background, known as the xenon direct detection neutrino floor. There are additional applications of multiscatter capture, some of which are listed at the end of Section III, which we leave to future work. Then g 1 (w) is the product of Eqs. (A9) and (A10). Inserting this into Eq. (A7), and integrating over the incoming Boltzmann distribution of DM (u), the total capture rate as a function of radius is C 1 = 96 π n X v R 0 dr r 2 n(r)σ(r)v 2 (r) 1 − 1 − e −A 2 (r) A 2 (r) ,(A11) where we have indicated that the number density of scattering sites n(r), the escape velocity, v(r), the Boltzmann variable A 2 ≡ 3v 2 (r)/2v 2 β − , and the scattering cross-section, as a consequence of form factor suppression at higher velocities, all depend on the radius of the mass shell, r. In the limit that we ignore radial dependence, and set v(r) v esc (R) Eq. (4) results. Figure 1 . 1Number of scatters needed to capture dark matter as a function of dark matter's halo speed (i.e. the speed at long distance from the star). The left plot shows the relation assuming a solar mass white dwarf made entirely of carbon (m N ∼ 10 GeV) and with radius R = 0.1 R sun . The right plot shows the relation for a solar mass neutron star with radius R = 10 km, which for the moment neglects relativistic corrections. The lines correspond to 10 TeV-100 PeV mass dark matter, as indicated. Figure 2 . 2Mass capture rate of dark matter on a constant density white dwarf, for a per-nucleon scattering cross-section of σ nX = 10 −38 (left panel) and 10 −36 cm 2 (right panel). Following Eq. (24), these per-nucleon cross sections translate to dark-matter carbon cross sections of σ nX ∼ 10 −34 and ∼ 10 −32 cm 2 . In both panels we have taken the target star to be a 1 solar mass white dwarf composed of carbon 12, with R = 10 4 km, in a background dark matter density of ρ X = 0.3 GeV/cm 3 with halo velocity dispersionv 220 km/s. Figure 3 . 3Mass capture rate of dark matter on neutron star, for a per-nucleon scattering cross-section of σ nX = 10 −44 and 10 −42 cm 2 . A constant density, 1.5 solar mass neutron star composed of neutrons, Figure 4 4Figure 4. Potential sensitivity to dark matter from annihilation to SM particles, heating a 1.5 M neutron star in the galactic center (ρ X = 10 3 GeV/cm 3 , about 10 parsecs from the Galactic Center) to a core temperature of ∼ 3 × 10 4 K, along with interpolations of the current LUX bounds and the neutrino floor (one atmospheric neutrino event on xenon [3]) for comparison. Here the parameters of the surrounding dark matter density and neutron star temperature have been chosen conservatively; observation of a colder neutron star, or a larger dark matter density would both deepen sensitivity. The curve labeled "1 scatter" uses Eq. (4) to set the bound, while the multiscatter curve uses the multiscatter formulae derived in this document. Note that multiscatter capture allows for heavier dark matter to be discovered or bounded, for cross-sections below the direct detection neutrino floor. Read right to left, Eq. (30) forms a bound. Specifically, if an old neutron star is observed to have surface temperature T N S , Eq. (30) dictates what regions of ρ X , m X and σ are allowed and which regions would overheat the observed neutron star. Plugging Eq. (18) into Eq. (30), we can reframe the expression as We have checked numerically that for N 5, the approximate expression in Eq. (21) matches the full expression Eq.(20) to within less than a percent for the applications presented in Section IV. Technically, the general relativistic effects are most straightforwardly introduced into the differential capture rate dCN /dr, which, upon integration, yield Eq. (26) plus corrections. Given that we are already making an approximation in assuming straight trajectories through the star, we will neglect these corrections to Eq.(26). This implicitly assumes that the energy of all DM annihilation products go to heating. It can be verified that the scattering length for neutrinos (and all more strongly coupled Standard Model particles) is much less than the neutron star radius. The exact way the temperature will rise requires knowledge of the equation of state of the star, which is beyond the scope of this paper, but would be an interesting topic for future research. In the case of multiscatter capture, the probability for capturing a dark matter particle that traverses the star in N scatters is given by gN (w)pN (τ ), where these are defined in Section III. ACKNOWLEDGMENTSWe thank Masha BaryakhtarAppendix A: Capture in the optically thin limit It is useful to summarize the derivation of dark matter capture[2]on stars:A. Far enough away from the star, dark matter particles in the galactic halo have speeds that are Boltzmann distributed. Half the particles will be moving towards the star, namely those with headings −π/2 < θ < π/2, where θ is the angle between each particle's velocity and a vector pointing at the star center. The total flux of dark matter is defined as F.B. As it traverses the stars gravitational well, the dark matter moves faster in the star's gravitational potential, but conservation of angular momentum implies that its angular momentum with respect to the star remains fixed. Therefore given θ and the particle's initial speed (i.e. altogether the particle's initial velocity), we can determine whether it has an angular momentum small enough that it will intersect a spherical mass shell at radius r from the center of the star.C. The probability that dark matter scatters and is captured while transiting a mass shell of thickness dr, depends on the density of scattering sites n(r), the initial dark matter velocity u, and the dark matter's cross-section with stellar constituents, σ. Integrating the Boltzmann distributed flux and the probability for capture over 0 < u < ∞ for each stellar mass shell, and integrating mass shells over 0 < r < R, determines the total capture rate. (In the case of multiscatter capture covered in Section III, it is convenient to instead simply consider all dark matter that intersects the star at radius R, and then integrate over paths through the star, calculating the multiscatter probability along each path.)We assume dark matter particles surrounding the star will have velocities that follow a Maxwell-Boltzmann distribution. The number density of dark matter particles with velocities ranging from u to u + du iswhere n X is the number density andv the average velocity of the dark matter particles. Here f (u) du gives the distribution of dark matter velocities far from the gravitational well of the star; nearer to the star each dark matter particle will have a total velocity given bywhere v(r) is the escape velocity from the star at radius r.It is useful to at first consider the flux of dark matter particles across a spherical surface large enough that the star's gravitational potential can be neglected. The angle at which dark matter intersects the large surface will increase or diminish its flux across this spherical surface; to account for this, we incorporate a factor of u ·R a = u cos θ, where θ is the angle between the DM velocity vector u and a unit vectorR a normal to the large surface. Then the flux of dark matter particles towards the star, through an infinitesimal area element, is obtained by integrating the product of u cos θ and Eq. (A1) over the range 0 < d(cos θ) < 1, and including a factor of 1/2 to effectively reject the outgoing DM flux,This leads directly to an expression for the flux of dark matter entering a region of size R a , which is large enough to ignore the star's gravitational potential, dF = 4πR 2 a dF = πR 2 a f (u)u du d(cos 2 θ).To incorporate the star's gravitational potential into the capture rate, we must consider what the dark matter flux will be into a spherical shell of radius r, which is the radius of the star or smaller. We define α as the angle between the dark matter particle's velocity vector w and the unit normal vectorr on this small spherical shell. The dark matter's dimensionless angular momentum iswhere the last equality of Eq. (A4) follows from angular momentum conservation. As noted previously, w 2 = u 2 + v 2 (r), and v(r) is the escape velocity at radius r. 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We study the deformations of the Chow group of zero-cycles using Bloch's formula and differential forms. We thereby obtain a new proof of an algebraization theorem for zero-cycles previously obtained using idelic techniques.

10.1093/qmathj/haaa004

1810.01347

7bc5116c663ef8c78a0b27a84ede4c2d05a17ce0

DEFORMATION THEORY OF THE CHOW GROUP OF ZERO-CYCLES 2 Oct 2018 Morten Lüders DEFORMATION THEORY OF THE CHOW GROUP OF ZERO-CYCLES 2 Oct 2018 We study the deformations of the Chow group of zero-cycles using Bloch's formula and differential forms. We thereby obtain a new proof of an algebraization theorem for zero-cycles previously obtained using idelic techniques. Introduction Let A be a henselian discrete valuation ring with uniformising parameter π and residue field k. Let X be a smooth projective scheme over Spec(A) of relative dimension d. Let X n := X × A A/(π n ), i.e. X 1 is the special fiber and the X n are the respective thickenings of X 1 . Let K M * ,X (resp. K M * ,Xn ) be the improved Milnor K-sheaf defined in [14]. One classicaly studies the formal deformations of Chow groups via Bloch's formula setting CH p (X n ) := H p (X 1 , K M p,Xn ) Then one uses the commutative diagram H p (X 1 , K M p,X n+1 ) / / H p (X 1 , K M p,Xn ) / / H p+1 (X 1 , Ω p−1 X 1 ) Throughout this article we assume that either (1) k is of characteristic 0 and A = k[[π]] or that (2) A is the Witt ring W (k) of a perfect field k of ch(k) > 2. In each of these two cases there exists a well-defined exponential map exp : Ω d−1 X 1 → K M d,Xn defined by xdlog(y 1 ) ∧ ... ∧ dlog(y d−1 ) → {1 + xπ n−1 , y 1 , ..., y d−1 } (see [3,Sec. 2] for (1) and [4,Sec. 12] for (2)). In these two cases we therefore get an exact sequence Ω d−1 X 1 → K M d,Xn → K M d,X n−1 → 0 which we use to study the restriction map res Xn : CH d (X) ∼ = / / H d (X, K M d,X ) res Xn / / H d (X 1 , K M d,Xn ) assuming the Gersten conjecture for the Milnor K-sheaf K M * ,X for the isomorphism on the left. The Gersten conjecture holds in case (1) by [14] and in case (2) with finite coefficients if we assume ch(k) to be large enough by [18,Prop. 1.1]. Our main theorem is the following: Theorem 1.1. With the above notation, and assuming the Gersten conjecture for the Milnor K-sheaf K M * ,X , the map res Xn : CH 1 (X) → H d (X 1 , K M d,Xn ) is surjective. In particular the map res : CH 1 (X) → "lim n "H d (X 1 , K M d,Xn ) is an epimorphism in pro-Ab. Theorem 1.1 is also proved in [18] using an idelic method. The approach of this article, using differential forms, is closer to the deformation theory described above. We think that it is of interest as well. The text is organised as follows: in Section 5 we prove our main theorem for d = 2 using, among others, results from Section 2. We then deduce the general case using a Lefschetz theorem which is proved in Section 4. Section 3 is independent and treats the case of d = 1. In the last section we list some open questions. Acknowledgement. This article is part of the author's PhD-thesis. I would like to thank my supervisor Moritz Kerz for many helpful comments and discussions. Local cohomology and some calculations In this section we recall some definitions and calculations in local cohomology which we will need later on. A standard reference for the following is [10, Ch. IV]. Let X be a locally noetherian scheme. To any sheaf of abelian groups F on X we can associate a coniveau complex of sheaves C(F ) := x∈X (0) i x, * H 0 x (X, F ) → x∈X (1) i x, * H 1 x (X, F )) → ... where i x : x → X is the natural inclusion. This complex is also called the Cousin complex of F . E p,q 1 = x∈X (p) H p+q x (X, F ) ⇒ H n (X, F ) one can easily deduce that the property of being CM for F is equivalent to C(F ) being an acyclic resolution of F (see [10,Ch. IV,Prop. 2.6]). In that case one can use C(F ) to calculate the cohomology of F , i.e. H * (X, F ) ∼ = H * (X, C(F )). Locally free sheaves are CM (see [10, p.239]) so in particular the sheaf of differential forms Ω 1 X and its exterior powers Ω a X are CM if X is a smooth variety over a field. Lemma 2.2. Let k be a field and X 1 be a scheme of dimension 1 over Spec(k). Let x ∈ X 1 be a regular closed point and f a local parameter at x. Then [10, p.217] to the triple (x, X 1 , X 1 − x), we get a short exact sequence O X 1 ,x [ 1 f ]/O X 1 ,x ∼ = H 1 x (X 1 , O X 1 ). Proof. We calculate H 1 x (X 1 , O X 1 ) locally as follows: Let X 1,x := Spec(O X 1 ,x ). Applying Motif B ofH 0 (X 1,x , O X 1 | X 1,x ) → H 0 (X 1,x −x, O X 1 | X 1,x −x ) → H 1 x (X 1,x , O X 1 | X 1,x ) → H 1 (X 1,x , O X 1 | X 1,x ). Since H 1 (X 1,x , O X 1 | X 1,x ) = 0, this gives an isomorphism O X 1 ,x [ 1 f ]/O X 1 ,x ∼ = H 1 x (X 1,x , O X 1 | X 1,x ). We now turn to the higher dimensional case. Similar calculations can be found in [2, Sec. 5]. Lemma 2.3. Let k be a field and X 1 be a scheme of dimension d > 1 over Spec(k). Let x ∈ X 1 be a regular closed point and f 1 , ..., f d ∈ m x a local parameter system at x. Then H d x (X, Ω d−1 X 1 ) is generated by elements of the form df 1 ∧ ... ∧d f i ∧ ... ∧ df d f n 1 1 ...f n d d modulo df 1 ∧...∧d f i ∧...∧df d f n 1 1 ...f j ...f n d d over O X 1 ,x . Proof. Let U be an affine neighbourhood of x ∈ X 1 . Let V = {V i := U − V (f i )} be a covering of U − x. Then theČech complex 0 → Ω d−1 X 1 (V i ) → i =j Ω d−1 X 1 (V i ∩ V j ) → ... → Ω d−1 X 1 (V 1 ∩ ... ∩ V d ) gives an isomorphism coker( Ω d−1 (V 1 ∩ ... ∩V i ∩ ... ∩ V d )) → Ω d−1 (V 1 ∩ ... ∩ V d )) ∼ = Γ(U, R d−1 j * (Ω d−1 | U −x )), where j is the inclusion X 1 − x ֒→ X 1 . By Motif B of [10, p.220] and since d ≥ 2 there is an isomorphism R d−1 j * (Ω d−1 | X 1 −x ) ∼ = H d x (X 1 , Ω d−1 X 1 ). In other words, Γ(U, H d x (X, Ω d−1 X 1 )) is generated by elements of the form df 1 ∧...∧d f i ∧...∧df d f n 1 1 ...f n d d modulo df 1 ∧...∧d f i ∧...∧df d f n 1 1 ...f j ...f n d d over O(U). Passing to the limit, we get the desired result. In case (1) of the introduction, i.e. for k a field of characteristic 0, S n = Speck[t]/(t n ), S = Speck [[t]] and X smooth, separated and of finite type over S, there exists a short exact sequence 0 → Ω r−1 X 1 → K M r,Xn → K M r,X n−1 → 0 (see [3,Prop. 2.3]). In particular, K M r,Xn is CM for all n ≥ 1 (see [3,Prop. 3.5]). We now show analogues statements for case (2). Proposition 2.4. Let k be a perfect field with ch(k) = p > 2 and let X be a smooth scheme over A := W (k). Then there is an exact sequence (2.1) 0 → Ω r−1 X 1 /B n−1 Ω r−1 X 1 → K M r,X n+1 → K M r,Xn → 0. Proof. Let R n be an essentially smooth local ring over A/π n . We define a filtration U i K M r (R n ) ⊂ K M r (R n ) by U i K M r (R n ) :=< {1 + π i x, x 2 , .. ., x r | x ∈ R n , x 2 , ..., x r ∈ R * n } > . The U i fit into the following exact sequences: 0 → U n K M r (R n+1 ) → K M r (R n+1 ) → K M r (R n ) → 0. By [4, Proof of Prop. 12.3, Step 3] there is an isomorphism Ω r−1 R 1 /B i−1 Ω r−1 R 1 ∼ = gr i K M r (R n ) ∼ = U i K M r (R n )/U i+1 K M r (R n ) and since U n+1 (K M r (R n+1 )) = 0, this implies that U n (K M r (R n+1 )) ∼ = Ω r−1 R 1 /B n−1 Ω r−1 R 1 and therefore the exact sequence 0 → Ω r−1 R 1 /B n−1 Ω r−1 R 1 → K M r (R n+1 ) → K M r (R n ) → 0.H i x (X 1 , Ω r−1 R 1 /B n−1 Ω r−1 R 1 ) → H i x (X 1 , K M r,X 2 ) → H i x (X 1 , K M r,X 1 ) . By [12, Cor. 3.9, p. 572], the sheaf Ω r−1 X 1 /B n−1 Ω r−1 X 1 is locally free and therefore CM. The sheaf K M r,X 1 is CM by [14] and [15]. The result follows inductively. The relative dimension 1 case Before we state the main proposition of this section, we quickly review the theory of pro-objects. Standard references are [1] and [9]. Let C be a category. The category of pro-objects pro-C in C is defined as follows: A pro-object is a contravariant functor X : I • → C, from a filtered index category I to C, i.e. an inverse system of objects X i in C. We denote X also by "lim"X i or (X i ) i . The morphisms between two objects X = "lim"X i and Y = "lim"Y i ∈ in pro-C are given by Hom(X, Y ) = lim ← − j (lim − → i Hom(X i , Y j )). There is a natural fully faithful embedding of C into pro-C which associates to an object C ∈ C the constant diagram C. This functor has a right adjoint pro-C → C, "lim"X i → lim ← −i X i . If C has finite direct (inverse) limits, then the functor Hom(I • , C) → pro-C commutes with finite direct (inverse) limits. In particular if C has finite direct and inverse limits, then the above functor is exact (see [1, p.163 ]). A criterion for when a map of pro-systems is an isomorphism is given by the following proposition (see [13,Lem. 2 .3]): Proposition 3.1. A level map A → B in pro-C, i.e. a map between pro-systems with the same index category and maps A s → B s for all s ∈ I, is an isomorphism if and only if for all s there exists a t ≥ s and a commutative diagram A t / / B t } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ A s / / B s . We now give a proof of Conjecture 6.1 for d = 1. Theorem 3.2. Let k be a finite field of characteristic p > 2 and A = W (k) the Witt ring of k. Let X be a smooth projective scheme of relative dimension 1 over A. Then the map res : CH 1 (X) ⊗ Z/p i Z → "lim"H 1 (X 1 , K M 1,Xn /p i ) is an isomorphism in the category of pro-systems of abelian groups. Proof. We first note that CH 1 (X) ∼ = Pic(X) and that Pic( X) ∼ = lim ← − Pic(X n ) by [8, Thm. 5.1.4]. Furthermore, H 1 (X 1 , K M 1,Xn ) = H 1 (X 1 , O × Xn ) ∼ = Pic(X n ). It therefore suffices to show that lim ← − Pic(X n ) ⊗ Z/p i Z → "lim"Pic(X n ) ⊗ Z/p i Z is an isomorphism. Using the p-adic logarithm isomorphism 1 + pO Xn ∼ = − → pO Xn , the short exact sequence 1 → (1 + p j O Xn ) → O × Xn → O × X j → 1 induces a short exact sequence 0 → H 1 (X 1 , p j O Xn ) → H 1 (X 1 , O * Xn ) → H 1 (X 1 , O * X j ) → H 2 (X 1 , p j O Xn ) = 0 (the last equality following for dimension reasons). Applying the Functor lim ← −n , we get an exact sequence lim ← − n H 1 (X 1 , p j O Xn ) → lim ← − n Pic(X n ) → Pic(X j ) → lim ← − n 1 H 1 (X 1 , p j O Xn ). Now lim ← − 1 n H 1 (X 1 , p j O Xn ) = 0 since the inverse system (H 1 (X 1 , p j O Xn ) ) n satisfies Mittag-Leffler being an inverse system of finite dimensional vector spaces. Tensoring with Z/p i Z gives the exact sequence lim ← − n H 1 (X 1 , p j O Xn ) ⊗ Z/p i Z → lim ← − n Pic(X n ) ⊗ Z/p i Z → Pic(X j ) ⊗ Z/p i Z → 0. We now apply the exact functor " lim ← −j " to this sequence. By the theorem on formal functions, there is an isomorphism " lim ← − j " lim ← − n H 1 (X 1 , p j O Xn ) ⊗ Z/p i Z ∼ = " lim ← − j "H 1 (X, p j O X ) ⊗ Z/p i Z. Since the image of the inclusion p i+j O X ֒→ p j O X vanishes modulo p i , the same holds for the image of the morphism H 1 (X, p i+j O X ) → H 1 (X, p j O X ). By Proposition 3.1 this implies that " lim ← − j " lim ← − n H 1 (X 1 , p j O Xn ) ⊗ Z/p i Z is pro-isomorphic to zero and therefore that the theorem holds. A Lefschetz theorem In this section we prove a Kodaira vanishing theorem which implies a Lefschetz theorem allowing us later in Section 5 to reduce our main theorem to relative dimension 2. To put the following proposition into context, we recall the Kodaira vanishing theorem. A good reference is [5]. Then H a (X, Ω b X ⊗ A) = 0 for a + b > dimX. In this section let X 1 , unless otherwise stated, be a smooth projective scheme over a field k. Let H ⊂ X 1 be a hyperplane section and L(d) = |dH|, d > 0, be the linear system of hypersurface sections of degree d. We say that a hypersurface section Y 1 ⊂ X 1 is of high or sufficiently high degree if Y 1 ∈ L(d) with d sufficiently large such that certain higher cohomology groups vanish by Serre vanishing. Proposition 4.2. Let Y 1 be a smooth hypersurface of X 1 and d = dimX 1 . If Y 1 is of sufficiently high degree, then H a (X 1 , Ω b Y 1 ⊗ O X 1 O X 1 (Y 1 )) = H a (Y 1 , Ω b Y 1 ⊗ O Y 1 O X 1 (Y 1 )| O Y 1 ) = 0 for a + b > d − 1. Proof. Note that the first equality in the statement follows from the projection formula i * Ω b Y 1 ⊗ O X 1 O X 1 (Y 1 ) = i * (Ω b Y 1 ⊗ O Y 1 i * O X 1 (Y 1 )) for i the inclusion Y 1 ֒→ X 1 . We first show that for ω X 1 = Ω d X 1 and ω Y 1 = Ω d−1 Y 1 , we have that H a>0 (X 1 , ω Y 1 ⊗ O X 1 O X 1 (Y 1 )) = 0 if Y 1 is of high degree. By [11,Ch. II,Prop. 8.20] we know that ω Y 1 ∼ = ω X 1 ⊗ O Y 1 ⊗ O X 1 (Y 1 ). This implies that ω X 1 | Y 1 = ω Y 1 (−Y 1 ) and therefore that the sequences 0 → ω X 1 (Y 1 ) → ω X 1 (2Y 1 ) → ω Y 1 (Y 1 ) → 0 and H a (X 1 , ω X 1 (2Y 1 )) → H a (X 1 , ω Y 1 (Y 1 )) → H a+1 (X 1 , ω X 1 (Y 1 )) are exact. Since by Serre vanishing H a (X 1 , ω X 1 (2Y 1 )) = H a (X 1 , ω X 1 (Y 1 )) = 0 for a > 0 and Y 1 of sufficiently high degree, this implies that if Y 1 is of sufficiently high degree we also have that H a>0 ( X 1 , ω Y 1 (Y 1 )) = H a>0 (Y 1 , ω Y 1 ⊗ O X 1 O X 1 (Y 1 )| O Y 1 ) = 0. We now consider the exact sequence 0 → Ω p−1 Y 1 (−Y 1 ) → Ω p X 1 | Y 1 → Ω p Y 1 → 0 coming from the conormal exact sequence 0 → O Y 1 (−Y 1 ) → Ω 1 X 1 | Y 1 → Ω 1 Y 1 → 0. Tensoring with O Y 1 (2Y 1 ) gives an exact sequence 0 → Ω p−1 Y 1 (Y 1 ) → Ω p X 1 (2Y 1 )| Y 1 → Ω p Y 1 (2Y 1 ) → 0. This implies that the sequence H a (X 1 , Ω b Y 1 (2Y 1 )) → H a+1 (X 1 , Ω b−1 Y 1 (Y 1 )) → H a+1 (X 1 , Ω b X 1 (2Y 1 )) is exact. The proposition follows inductively. We can now deduce the following Lefschetz theorem: φ : H d−1 (Y 1 , Ω d−2 Y 1 ) → H d (X 1 , Ω d−1 X 1 ) which is an isomorphism for d ≥ 4 and surjective for d = 3 if Y 1 is of high degree. Proof. Let i denote the inclusion Y 1 ֒→ X 1 . We define φ to be the composition H d−1 (Y 1 , Ω d−2 Y 1 ) − → H d−1 (Y 1 , R 1 i ! Ω d−1 X 1 ) ∼ = H d Y 1 (X 1 , Ω d−1 X 1 ) → H d (X 1 , Ω d−1 X 1 ) where the first map is induced by the Gysin map g : Ω d−2 Y 1 → R 1 i ! Ω d−1 X 1 , ω → ω ∧ df d f d (see [7, Ch. II, (3.2.13)]) with f d is the regular parameter defining Y 1 . Since H d (X 1 − Y 1 , Ω d−1 X 1 −Y 1 ) = 0 for d ≥ 1, we have that H d Y 1 (X 1 , Ω d−1 X 1 ) ∼ = H d (X 1 , Ω d−1 X 1 ) for d ≥ 2. We are therefore reduced to showing that g induces an isomorphism on H d−1 for d − 1 ≥ 3 and a surjection for d − 1 = 2. We define a filtration g(Ω d−2 Y 1 ) = F 1 ⊂ F 2 ⊂ ... ⊂ ∪ i≥0 F = R 1 i ! Ω d−1 X 1 , letting F i be the subsheaf of R 1 i ! Ω d−1 X 1 locally defined by < ω ∧ df d f n d d |n d ≥ i > . Here ω ∈ Ω d−2 Y 1 . Let gr i R 1 i ! Ω d−1 X 1 := F i+1 /F i . Then gr i R 1 i ! Ω d−1 X 1 ∼ = Ω d−2 Y 1 ⊗ O Y 1 O X 1 (Y 1 )| O Y 1 and the short exact sequence 0 → F i → F i+1 → gr i R 1 i ! Ω d−1 X 1 → 0 induces the following exact sequence on cohomology groups: [11, Ch. III, Prop. 2.9]), the same holds for the maps H a (Y 1 , H d−2 (Y 1 , Ω d−2 Y 1 ⊗ O Y 1 O X 1 (Y 1 )| O Y 1 ) → H d−1 (Y 1 , F i ) → H d−1 (Y 1 , F i+1 ) → H d−1 (Y 1 , Ω d−2 Y 1 ⊗ O Y 1 O X 1 (Y 1 )| O Y 1 ) (4.1) By Proposition 4.2 we have that if Y 1 is of high degree, then H d−1 (Y 1 , Ω d−2 Y 1 ⊗ O X 1 O X 1 (Y 1 )| O Y 1 ) vanishes for d > 2 and H d−2 (Y 1 , Ω d−2 Y 1 ⊗ O X 1 O X 1 (Y 1 )| O Y 1 ) for d > 3. This implies that the maps H a (Y 1 , F i ) → H a (Y 1 , F i+1 ) are isomorphisms for d ≥ 4 and surjective for d = 3 if Y 1 is of sufficiently high degree. Since H a (Y 1 , lim − → F i ) ∼ = lim − → H a (Y 1 , F i ) (seeF 1 = Ω d−2 Y 1 ) → H a (Y 1 , R 1 i ! Ω d−1 X 1 ). In particular, for d = dimX 1 = 3 we get that H d−1 (Y 1 , Ω d−2 Y 1 ) → H d (X 1 , Ω d−1 X 1 ) is surjective and for d = dimX 1 ≥ 4 that H d−1 (Y 1 , Ω d−2 Y 1 ) → H d (X 1 , Ω d−1 X 1 ) is an isomorphism. Corollary 4.4. Let X be as in the introduction and d ≥ 3. Let Y 1 a smooth hypersurface section of X 1 . Let α ∈ H d (X 1 , K M d,Xn ). If Y 1 is of sufficiently high degree and contains the image of α in CH 0 (X 1 ) under the restriction map H d (X 1 , K M d,Xn ) → H d (X 1 , K M d,X 1 ) ∼ = CH 0 (X 1 ), then Y 1 lifts to a smooth projective subscheme Y of X over A and α is in the image of H d−1 (Y 1 , K M d−1,Yn ) → H d (X 1 , K M d,Xn ) . Proof. That Y 1 lifts to a smooth projective subscheme Y of X over A if it is of high degree follows from Serre vanishing. We do the n = 2 case. The general case follows inductively. Consider the commutative diagram H d (X 1 , Ω d−1 X 1 ) / / H d (X 1 , K M d,X 2 ) / / H d (X 1 , K M d,X 1 ) / / 0 H d−1 (Y 1 , Ω d−2 Y 1 ) / / O O H d−1 (Y 1 , K M d−1,Y 2 ) / / O O H d−1 (Y 1 , K M d−1,Y 1 ) O O / / 0 induced by the (right-)exact sequence of sheaves Ω d−1 → K M 2 → K M 2 → 0 on X 1 and Y 1 . The statement follows from Proposition 4.3 and a simple diagram chase. For the sake of completeness we also prove the following Lefschetz theorem. We will not use it though. i * : H q (X 1 , Ω p X 1 ) → H q (Y 1 , Ω p Y 1 ) is an isomorphism for p + q < d − 1 and injective for p + q = d − 1 if Y 1 is of high degree. Proof. We factorise the map i * : Ω p X 1 → i * Ω p Y 1 by Ω p X 1 → i * (Ω p X 1 | Y 1 ) followed by i * (Ω p X 1 | Y 1 ) → i * Ω p Y 1 and show that each of these maps induce isomorphisms, resp. injections, on cohomology in the stated range. We first consider the exact sequence 0 → Ω p X 1 (−Y 1 ) → Ω p X 1 → Ω p X 1 | Y 1 → 0. This induces the exact sequence H q (X 1 , Ω p X 1 (−Y 1 )) → H q (X 1 , Ω p X 1 ) → H q (Y 1 , Ω p X 1 | Y 1 ) → H q+1 (X 1 , Ω p X 1 (−Y 1 )). By Serre duality H q (X, Ω p X 1 (−Y 1 )) ∼ = H d−q (X, Ω d−p X 1 (Y 1 )). This implies that H q (X 1 , Ω p X 1 ) → H q (Y 1 , Ω p X 1 | Y 1 ) is an isomorphism for p + q < d − 1 and injective for p + q = d − 1 if Y 1 is of sufficiently high degree by Serre vanishing. We now consider the exact sequence 0 → Ω p−1 Y 1 (−Y 1 ) → Ω p X 1 | Y 1 → Ω p Y 1 → 0 on Y 1 . This induces the exact sequence H q (Y 1 , Ω p−1 Y 1 (−Y 1 )) → H q (Y 1 , Ω p X 1 | Y 1 ) → H q (Y 1 , Ω p Y 1 ) → H q+1 (Y 1 , Ω p−1 Y 1 (−Y 1 )) which by Serre duality and Proposition 4. 2 implies that H q (Y 1 , Ω p X 1 | Y 1 ) → H q (Y, Ω p Y 1 ) is an isomorphism for p + q < d + 1 and injective for p + q = d + 2 if Y 1 . Remark 4.6. Proposition 4.5 is an analogue of the following theorem (see for example [21,Thm. 1.29]): Let X be an n-dimensional compact complex variety and let Y ֒→ X be a smooth hypersurface such that the line bundle O X (Y ) = (I Y ) * is ample. Then the restriction j * : H k (X, Q) → H k (Y, Q) is an isomorphism for k < n − 1 and injective for k = n − 1. This theorem may be deduced from Kodaira vanishing using the Hodge decomposition for Y C and X C . We note that when working over an arbitrary field k, one seems to need the extra assumption that Y is of high degree to obtain Lefschetz theorems. Main theorem We return to the situation of the introduction. Let A be a henselian discrete valuation ring with uniformising parameter π and residue field k. Let X be a smooth projective scheme over Spec(A) of relative dimension d. Let X n := X × A A/(π n ), i.e. X 1 is the special fiber and the X n are the respective thickenings of X 1 . We assume furthermore that either (1) k is of characteristic 0 and A = k[[π]] or that (2) A is the Witt ring W (k) of a perfect field k of ch(k) > 2. Let us first recall how one can lift a regular closed point x ∈ X 1 to a 1-cycle on X: Let {f 1 , ..., f d } ⊂ O X 1 ,x be a generating set of local parameters and let {f 1 , ...,f d } be lifts of these generators to O X,x . The idealf 1 O X,x + ... +f d O X,x defines a subscheme of SpecO X,x and its closure in X defines a subscheme Z of X. The unique irreducible component of Z containing x is a prime-cycle C ∈ Z 1 (X) which is flat and finite over A. Such liftings are of course not unique. We also introduce the following notation: Let X be a scheme and Z an effective Cartier divisor on X. Let C be a curve in X, i.e. an effective 1-cycle on X. Let (Z, C) x := length O X,x (O X,x /I Z + I C ) be the intersection multiplicity of Z and C at x. We say that Z and C intersect transversally at x if (Z, C) x = 1 and if Z and C are regular at x. If Z and C intersect transversally everywhere, we denote this by Z ⋓ C. In this section we show the following proposition: Proposition 5.1. Let X be of relative dimension 2 over A. Then, assuming the Gersten conjecture for the Milnor K-sheaf K M * ,X , the map res Xn : CH 1 (X) → H 2 (X 1 , K M 2,Xn ) is surjective. In particular the map res : CH 1 (X) → "lim n "H 2 (X 1 , K M 2,Xn ) is an epimorphism in pro-Ab. We need some preparation for the proof. From now on we assume that d = 2 and in particular that dimX 1 = 2. Consider the (right-)exact sequence Ω 1 X 1 → K M 2,X 2 → K M 2,X 1 → 0. We will lift elements which lie in the kernel of res : H 2 (X 1 , K M 2,X 2 ) → H 2 (X 1 , K M 2,X 1 ) in a compatible way to CH 1 (X). The kernel of res is in the image of H 2 (X 1 , Ω 1 X 1 ). Now since Ω 1 X 1 is CM, H 2 (X 1 , Ω 1 X 1 ) is isomorphic to coker(⊕ x∈X (1) 1 H 1 x (X 1 , Ω 1 X 1 ) → ⊕ x∈X (2) 1 H 2 x (X 1 , Ω 1 X 1 )) . In order to proceed, we need to study this cokernel and the occurring local cohomology groups a bit further. By Lemma 2.3 we have that H 2 x (X 1 , Ω 1 X 1 ) is generated by differential forms of the form df 1 f n 1 1 f n 2 2 O X 1 ,x ⊕ df 2 f n ′ 1 1 f n ′ 2 2 O X 1 ,x mod df i f n j j O X 1 ,x for {f 1 , f 2 } a system of local parameters in O X 1 ,x and i, j ∈ {1, 2}. We define subgroups F r :=< df 1 f n 1 1 f n 2 2 + df 2 f n ′ 1 1 f n ′ 2 2 |n 1 + n 2 − 1 ≤ r, n ′ 1 + n ′ 2 − 1 ≤ r > of H 2 x (X 1 , Ω 1 X 1 ) with respect to a system of local parameters. Sometimes we therefore write H 2 x (X 1 , Ω 1 X 1 ) (f 1 ,f 2 ) instead of H 2 x (X 1 , Ω 1 X 1 ) to indicate with respect to which system of local parameters we are working. Then 0 ⊂ F 1 ⊂ F 2 ⊂ ... ⊂ H 2 x (X 1 , Ω 1 X 1 ) (f 1 ,f 2 ) defines an ascending filtration on H 2 x (X 1 , Ω 1 X 1 ). We will call elements of F 1 forms with simple poles. The following lemma shows that this definition is in fact independent of the chosen parameter system, meaning that there is a natural isomorhism between H 2 x (X 1 , Ω 1 X 1 ) (f 1 ,f 2 ) and H 2 x (X 1 , Ω 1 X 1 ) (f ′ 1 ,f ′ 2 ) for two local parameter systems {f 1 , f 2 } and {f ′ 1 , f ′ 2 } inducing isomorphisms on the respective filtrations. This isomorphism is given by considering a differential form with respect to the respective defining parameter systems. Lemma 5.2. (1) Let x ∈ X 1 be a closed point. Then subgroups F r ⊂ H 2 x (X 1 , Ω 1 X 1 ) are independent of the local parameter system we consider them in. (2) In particular, the subgroup F 1 =< df 1 f 1 f 2 O X 1 ,x ⊕ df 2 f 1 f 2 O X 1 ,x >⊂ H 2 x (X 1 , Ω 1 X 1 ) is independent of the chosen local parameter system of O X 1 ,x . We therefore denote it by Λ x . . Let H ⊂ X 1 be a hyperplane section and for an integer d > 0 let L(d) = |dH| be the linear system of hypersurface sections of degree d. Now for d ≫ 0 there exists an F 1 ∈ L(d) such that (1) x ∈ F 1 , (2) F 1 ⋓ D 1 at any y ∈ F 1 ∩ D 1 . For d ′ sufficiently large relative to d, there exists an F 2 ∈ L(d ′ ) such that wheref 1 ,f 2 ∈ O X,x are lifts of f 1 and f 2 . We now show that this lift is mapped to exp(αdf 1 /(f 1 f 2 )) = {f 1 , 1 + π n−1 α/f 2 } ∈ H 2 x (X 1 , K M 2,Xn ) by the restriction map. The map res : CH 1 (X) → H 2 (X, K M 2,X ), induced by the assumption of the Gersten conjecture, sends the cycle V (f 1 ,f 2 + π n−1 α) − V (f 1 ,f 2 ) to ({f 1 ,f 2 + π n−1 α}, −{f 1 ,f 2 }) iň H 1 (X V (f 1 ,f 2 +π n−1 α) − V (f 1 ,f 2 + π n−1 α), K M 2,X ) ⊕Ȟ 1 (X V (f 1 ,f 2 ) − V (f 1 ,f 2 ), K M 2,X ). Finally, the restriction map H 2 (X, K M 2,X ) → H 2 (X, K M 2,Xn ) sends the tuple ofČech-cycles ({f 1 ,f 2 + π n−1 α}, −{f 1 ,f 2 }) to {f 1 , 1 + π n−1 α/f 2 } ≡ {f 1 , 1 + π n−1 α/f 2 } ∈Ȟ 1 (X 1 − x, K M 2,Xn ). In sum this shows in particular that ker[H 2 (X 1 , K M 2,Xn ) → H 2 (X 1 , K M 2,X n−1 )] is in the image of CH 1 (X). The surjectivity of CH 1 (X) → H 2 (X n , K M 2,Xn ) now follows by induction: For n = 1 this is just the surjectivity of CH 1 (X) → CH 0 (X 1 ). For n > 1, let α ∈ H 2 (X 1 , K M 2,Xn ). Let α n−1 be the image of α in H 2 (X 1 , K M 2,X n−1 ). By assumption there is a cycle Z ∈ CH 1 (X) mapping to α n−1 . Denote the image of Z in H 2 (X n , K M 2,Xn ) by α Z . Now α − α Z is in the kernel of H 2 (X 1 , K M 2,Xn ) → H 2 (X 1 , K M 2,X n−1 ) and by the above construction lifts to an element Z ′ ∈ CH 1 (X). Now Z ′ − Z maps to α. Corollary 5.5. Let X be as in Proposition 5.1 but of arbitrary relative dimension d over A. Then, assuming the Gersten conjecture for the Milnor K-sheaf K M * ,X , the map res Xn : CH 1 (X) → H d (X 1 , K M d,Xn ) is surjective. In particular the map res : CH 1 (X) → "lim n "H d (X 1 , K M d,Xn ) is an epimorphism in pro-Ab. Proof. This follows immediately from Corollary 4.4, Proposition 5.1 and standard Bertini arguments. Open problems The proof of Proposition 5.1 can be summed up in the following diagram: Λ x & & ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ lif t y y CH 1 (X) H 2 x (X 1 , Ω 1 X 1 ) H 2 (X, K M 2,X ) / / H 2 (X 1 , K M 2,X 2 ) H 2 (X 1 , K M 2,X 1 ) The question remains if there is a well-defined map Λ x → CH 1 (X)/F n for some filtration .. ⊂ F 2 ⊂ F 1 ⊂ CH 1 (X) making the above diagram commutative. This would make it possible to construct an inverse to the restriction map res : CH 1 (X)/F n → H 2 (X 1 , K M 2,Xn ). Furthermore, if " lim n "F n ⊗Z/p r Z, then this would imply the following conjecture by Kerz, Esnault and Wittenberg: Let us consider the case where X is of relative dimension 1 over A. Let F n be the subgroup of CH 1 (X) generated by all cycles Z vanishing on X n , i.e. Z| Xn = 0. By Lemma 2.2 we know that H 1 x (X 1 , O X 1 ) ∼ = O X 1 ,x [ 1 f ]/O X 1 , x for a local parameter f ∈ O X 1 ,x . In this case we can define a map Let A is as in case (2). Then by an argument similar to that in Section 3, where we showed Conjecture 6.1 for d = 1, we see that "lim n "F n ⊗ Z/p r Z = 0. Conjecture 6.1 would therefore hold if ker[res : CH 1 (X)/F n → H d (X 1 , K M d,Xn )] is generated by the filtrations defined above on smooth relative curves. For a similar conjecture see [18,Conj. 5.7]. Remark 6.2. Let X K be a smooth projective variety over a p-adic field K. In [16,Sec. 10], a relation between Conjecture 6.1 and a question by Colliot-Thélène is postulated: is there a (non-canonical) isomorphism CH 0 (X K ) ∼ = Z ⊕ Z m p ⊕ (finite group) ⊕ (divisible group)? Let us sketch this relationship. Let A be the ring of integers in K. Let X be a smooth and projective scheme of relative dimension d over A. For a smooth projective (over A) subscheme of codimension one Y ⊂ X we expect that the map "lim n "H d−1 (Y 1 , K M d−1,Yn /p r ) → "lim n "H d (X 1 , K M d,Xn /p r ) is an isomorphism for d ≥ 3 and surjective for d = 2 if we choose Y to be of high degree. For n = 1, this follows from class field theory and standard Lefschetz theorems for the étale fundamental group. In order to prove this for arbitrary n, the Lefschetz theorem of Section 4, Proposition 4.3, needs to be improved by one degree. This and the injectivity of res for arbitrary dimension would imply that the map CH 1 (Y )/p r → CH 1 (X)/p r is bijective for d ≥ 3 and surjective for d = 2. Since CH 1 (Y ) → CH 0 (Y K ) and CH 1 (X) → CH 0 (X K ) are surjective, the same statement would hold for CH 0 (Y K )/p r → CH 0 (X K )/p r . γ x : O X 1 ,x [ 1 f ]/O X 1 ,x → CH 1 (X)/F n by α = α 0 f m → V (f m + α 0 π n−1 ) − V (f m ) wheref ∈ O X, For a curve C K over K we know that A 0 (C K ) ∼ = A m ⊕ (finite group) for some m ∈ N (see [19]). This implies, under the above assumptions, the same result for the p-completion of CH 1 (X K ) and therefore a positive answer to the above Question. The corresponding weak Lefschetz theorem for l prime to p saying that the map CH 1 (Y )/l r → CH 1 (X)/l r is bijective for d ≥ 3 and surjective for d = 2 is proved in [20,Cor. 9.6]. Theorem 4. 1 . 1Let X be a complex projective manifold and A an ample invertible sheaf. Proposition 4. 3 . 3Let Y 1 be a hypersurface section of X 1 and d = dimX 1 . Then there is a map Proposition 4 . 5 . 45Let Y 1 be a smooth hypersurface section of X 1 and d = dimX 1 . Let i denote the inclusion Y 1 ֒→ X 1 . Then the map ( 3 ) 3The graded pieces F r+1 /F r are independent of the chosen local parameter system of O X 1 ,x . Conjecture 6.1. [16, Sec. 10] The map res : CH d (X) ⊗ Z/p r Z → "lim n "H d (X 1 , K M d,Xn /p r ) is an isomorphism in pro-Ab if ch(Quot(A)) = 0 and if k is perfect of characteristic p > 0 . x is a lifting of f . That γ x is well-defined can be seen as follows: Let f 1 ,f 2 ∈ O X,x be liftings of f . Then (1 Definition 2.1. A sheaf F on X is called Cohen-Macaulay, or simply CM, if for every x ∈ X it holds that H i x (X, F ) = 0 for i = codim(x). Via the coniveau spectral sequence V (f 1 ,f 2 + π n−1 α) − V (f 1 ,f 2 ) ∈ CH 1 (X) Proof. It suffices to show the proposition for two parameter systems (f 1 , f 2 ) and (f 1 , f ′ 2 ) and f 2 = f ′ 2 + βf 1 . We saw in Section 2 that H 2 x (X 1 , Ω 1 X 1 ) can be calculated locally aŝ H 1 (SpecO X 1 ,x \ {x}, Ω 1 X 1 ) with respect to coverings of of SpecO X 1 ,x \ {x}. Now considering, which lies again in F n 1 +n 2 −1 considering it as an element ofĤ 1 (SpecO. This proves(1).(2)and(3)follow immediately. In order to prove Proposition 5.1, we need to prove key Lemma 5.4. Its proof is inspired by the techniques of[17]from which we cite the following lemma:By Lemma 5.2 we can talk about the pole order of elements of H 2x (X 1 , Ω 1 X 1 ) independent of the parameter system chosen. In particular, the following lemma makes sense:is equivalent to a sum of forms with simple poles in H 2 (X 1 , Ω 1 X 1 ). Proof. Without loss of generality we work with γ = αdf 1x (X 1 , Ω 1 X 1 ). Let D 1 be a regular curve containing x and f ′ 1 a local parameter of D 1 at x. We consider αdf 1. By Lemma 5.2, γ is still in F r+1 for r + 1 = n 1 + n 2 . We may assume that it is of the formWe choose F 3 , ..., F n 2 analogously. Furthermore, we choose F n 2 to be of sufficiently high degree so that H 1 (X 1 , Ω 1 X 1 ((n 1 −1)D 1 +F 1 +...+F n 2 ))) = H 1 (D, Ω 1 X 1 (n 1 D 1 +F 1 +...+F n 2 )| D ⊗O D (−x)) = 0 holds by Serre vanishing. By Lemma 5.3, the last condition implies that the restriction map) is surjective. Let y be the generic point of D 1 . By construction, the diagram. Notice that the map on the right is well-defined. This implies that there is a γ ∈ H 1 y (X 1 , Ω 1. Furthermore for any x ′ ∈ |D 1 | − x, the form d x ′ (γ) has at most simple poles in f 2 at x ′ , i.e. d x ′ (γ) ∈ F n 2 +1 . Now we apply the same construction to the form d x ′ (γ) which completes the proof.Proof of Proposition 5.1. Let x ∈ X be a closed point and X 1,x be the spectrum of the stalk of O X 1 in x. TheČech to derived functor spectral sequenceSince this edge map is functorial in F , we get a commutative diagramfor a closed point x ∈ X 1 . We saw in Lemma 2.3 thatȞ 1 (X 1,x − x, Ω 1 X 1 ) is generated by elements of the form α 1 for a local parameter system (f 1 , f 2 ) ∈ O X 1 ,x and by key Lemma 5.4 we may assume that it has simple poles. The point of the proof is that for forms with simple poles we can write down explicit lifts of these forms to CH 1 (X).Without loss of generality we consider the form αdf 1 /(f 1 f 2 ). We lift αdf 1 /(f 1 f 2 ) to Etale homotopy. M Artin, B Mazur, Lecture Notes in Mathematics. 100Springer-VerlagM. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. Semi-regularity and deRham cohomology. S Bloch, Invent. Math. 17S. Bloch, Semi-regularity and deRham cohomology, Invent. Math., 17 (1972), pp. 51-66. Deformation of algebraic cycle classes in characteristic zero. S Bloch, H Esnault, M Kerz, Algebr. Geom. 1S. Bloch, H. Esnault, and M. Kerz, Deformation of algebraic cycle classes in characteristic zero, Algebr. Geom., 1 (2014), pp. 290-310. p-adic deformation of algebraic cycle classes. Invent. Math. 195, p-adic deformation of algebraic cycle classes, Invent. Math., 195 (2014), pp. 673-722. H Esnault, E Viehweg, Lectures on vanishing theorems. BaselBirkhäuser Verlag20H. Esnault and E. Viehweg, Lectures on vanishing theorems, vol. 20 of DMV Seminar, Birkhäuser Verlag, Basel, 1992. Formal deformation of Chow groups. M Green, P Griffiths, The legacy of Niels Henrik Abel. BerlinSpringerM. Green and P. Griffiths, Formal deformation of Chow groups, in The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 467-509. Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique. M Gros, Mém. Soc. Math. France (N.S87M. Gros, Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. France (N.S.), (1985), p. 87. éléments de géométrie algébrique. III. étude cohomologique des faisceaux cohérents. I, Inst. A Grothendieck, Hautes Études Sci. Publ. Math. 167A. Grothendieck, éléments de géométrie algébrique. III. étude cohomologique des faisceaux co- hérents. I, Inst. Hautes Études Sci. Publ. Math., (1961), p. 167. A Grothendieck, J.-L Verdier ; M. Artin, A Grothendieck, J L Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos. Berlin-New York, 1972. Séminaire de Géométrie Algébrique du Bois-MarieSpringer-Verlag269SGA 4. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-DonatA. Grothendieck and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. R Hartshorne, Lecture Notes in Mathematics. 20Springer-VerlagR. Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Algebraic geometry. Graduate Texts in Mathematics. New York-HeidelbergSpringer-Verlag, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathe- matics, No. 52. Complexe de de Rham-Witt et cohomologie cristalline. L Illusie, Ann. Sci. École Norm. Sup. 124L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. École Norm. Sup. (4), 12 (1979), pp. 501-661. A model structure on the category of pro-simplicial sets. D C Isaksen, Trans. Amer. Math. Soc. 353D. C. Isaksen, A model structure on the category of pro-simplicial sets, Trans. Amer. Math. Soc., 353 (2001), pp. 2805-2841. The Gersten conjecture for Milnor K-theory. M Kerz, Invent. Math. 175M. Kerz, The Gersten conjecture for Milnor K-theory, Invent. Math., 175 (2009), pp. 1-33. Milnor K-theory of local rings with finite residue fields. J. Algebraic Geom. 19, Milnor K-theory of local rings with finite residue fields, J. Algebraic Geom., 19 (2010), pp. 173- 191. A restriction isomorphism for cycles of relative dimension zero. M Kerz, H Esnault, O Wittenberg, Camb. J. Math. 4M. Kerz, H. Esnault, and O. Wittenberg, A restriction isomorphism for cycles of relative dimension zero, Camb. J. Math., 4 (2016), pp. 163-196. Chow group of 0-cycles with modulus and higher-dimensional class field theory. M Kerz, S Saito, Duke Math. J. 165M. Kerz and S. Saito, Chow group of 0-cycles with modulus and higher-dimensional class field theory, Duke Math. J., 165 (2016), pp. 2811-2897. Algebraization for zero-cycles and the p-adic cycle class map. M Lüders, Math. Res. Letters. to appear inM. Lüders, Algebraization for zero-cycles and the p-adic cycle class map, to appear in Math. Res. Letters., (2017). Abelian varieties over p-adic ground fields. A Mattuck, Ann. of Math. 622A. Mattuck, Abelian varieties over p-adic ground fields, Ann. of Math. (2), 62 (1955), pp. 92-119. A finiteness theorem for zero-cycles over p-adic fields. S Saito, K Sato, Ann. of Math. 2With an appendix by Uwe JannsenS. Saito and K. Sato, A finiteness theorem for zero-cycles over p-adic fields, Ann. of Math. (2), 172 (2010), pp. 1593-1639. With an appendix by Uwe Jannsen. C Voisin, Hodge theory and complex algebraic geometry. II. Leila SchnepsCambridgeCambridge University Press77of Cambridge Studies in Advanced Mathematics. english ed.C. Voisin, Hodge theory and complex algebraic geometry. II, vol. 77 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, english ed., 2007. Translated from the French by Leila Schneps. . Mathematik Fakultät Für, [email protected], Germany E-mail addressUniversität RegensburgFakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany E-mail address: [email protected]

Tensor networks are powerful factorization techniques which reduce resource requirements for numerically simulating principal quantum many-body systems and algorithms. The computational complexity of a tensor network simulation depends on the tensor ranks and the order in which they are contracted. Unfortunately, computing optimal contraction sequences (orderings) in general is known to be a computationally difficult (NP-complete) task. In 2005, Markov and Shi showed that optimal contraction sequences correspond to optimal (minimum width) tree decompositions of a tensor network's line graph, relating the contraction sequence problem to a rich literature in structural graph theory. While treewidthbased methods have largely been ignored in favor of dataset-specific algorithms in the prior tensor networks literature, we demonstrate their practical relevance for problems arising from two distinct methods used in quantum simulation: multi-scale entanglement renormalization ansatz (MERA) datasets and quantum circuits generated by the quantum approximate optimization algorithm (QAOA). We exhibit multiple regimes where treewidth-based algorithms outperform domain-specific algorithms, while demonstrating that the optimal choice of algorithm has a complex dependence on the network density, expected contraction complexity, and user run time requirements. We further provide an open source software framework designed with an emphasis on accessibility and extendability, enabling replicable experimental evaluations and future exploration of competing methods by practitioners.

10.1371/journal.pone.0207827

1807.04599

d32f55882b596ec1ebfdc2af181b25972beb5df9

Benchmarking treewidth as a practical component of tensor network simulations Eugene F Dumitrescu Quantum Computing Institute Oak Ridge National Laboratory Oak RidgeTNUnited States of America Allison L Fisher Department of Computer Science North Carolina State University RaleighNCUnited States of America Timothy D Goodrichid Department of Computer Science North Carolina State University RaleighNCUnited States of America Travis S Humble Quantum Computing Institute Oak Ridge National Laboratory Oak RidgeTNUnited States of America Blair D Sullivanid Department of Computer Science North Carolina State University RaleighNCUnited States of America Andrew L Wright Department of Computer Science North Carolina State University RaleighNCUnited States of America Benchmarking treewidth as a practical component of tensor network simulations RESEARCH ARTICLE Tensor networks are powerful factorization techniques which reduce resource requirements for numerically simulating principal quantum many-body systems and algorithms. The computational complexity of a tensor network simulation depends on the tensor ranks and the order in which they are contracted. Unfortunately, computing optimal contraction sequences (orderings) in general is known to be a computationally difficult (NP-complete) task. In 2005, Markov and Shi showed that optimal contraction sequences correspond to optimal (minimum width) tree decompositions of a tensor network's line graph, relating the contraction sequence problem to a rich literature in structural graph theory. While treewidthbased methods have largely been ignored in favor of dataset-specific algorithms in the prior tensor networks literature, we demonstrate their practical relevance for problems arising from two distinct methods used in quantum simulation: multi-scale entanglement renormalization ansatz (MERA) datasets and quantum circuits generated by the quantum approximate optimization algorithm (QAOA). We exhibit multiple regimes where treewidth-based algorithms outperform domain-specific algorithms, while demonstrating that the optimal choice of algorithm has a complex dependence on the network density, expected contraction complexity, and user run time requirements. We further provide an open source software framework designed with an emphasis on accessibility and extendability, enabling replicable experimental evaluations and future exploration of competing methods by practitioners. Introduction Tensor network factorizations provide a framework for controlled approximation which exponentially reduces the memory required to simulate a variety of quantum many-body systems [1,2] and circuits [3,4]. These factorizations do so by representing targeted sub-sectors of the full (exponentially scaling) Hilbert space. The tensors comprising the factorization are placed on the vertices of a graph, one appropriate to the geometry under consideration, and are contracted along the edges as needed to compute physical observables [1]. PLOS Since their early usage as the density matrix renormalization group description for gapped spin chains [1,5], tensor networks have been adapted and reformulated to also describe 2D area-law states [6,7], critical systems [2], lattice gauge theories [8,9], AdS/CFT duality [10], and open quantum systems [11,12]. In addition to describing a wide range of physical phenomena, satisfiability problems [5,13] and quantum computing simulations [3,4] can be formulated as tensor contraction problems. In the case of the latter, simulations of quantum error correcting codes are leading to important insights into fault tolerant quantum computation [14,15]. Additionally, given the tremendous interest in validating increasing complex experimental quantum computations [16], a flurry of simulations have recently appeared in which the underlying tensor network graph emerges from the structure of the algorithm being employed [17][18][19][20]. The overall descriptive power and algorithmic computational complexity, as formalized by the contraction complexity of a tensor network [4], is determined by the tensor network construction's underlying graph structure. For some tensor network algorithms (e.g. the matrix product state formulation) the contraction complexity is fixed and well understood. However, the task of determining the contraction complexity in general, along with computing an optimal contraction sequence witnessing this complexity, is NP-complete [21]. Despite this daunting theoretical complexity, efficient methods exist in practice for obtaining both optimal and 'good enough' contraction sequences. Domain-specific approaches typically search the space of all possible sequences and apply heuristic pruning techniques to reduce the search space [19,21,22]. Effective algorithms in this area incorporate pruning rules proprietary to the target application's data (e.g., MERA networks [22]), which limits their broader applicability. Another standard technique involves transforming the tensor network into a line graph, then computing a perfect elimination ordering and its treewidth, which can be translated into a contraction sequence and complexity for the original network, respectively [4]. In practice, however, engineering issues prevent these methods from being applied effectively. Domain-specific approaches typically suffer from proprietary construction, each assuming a different representation of the tensor networks, and using different code languages, dependencies, and interfaces (often with little-to-no documentation). Additionally, these implementations are typically only tested on the data for which they were designed, providing no expectation for how they might perform and/or scale in different contexts. Treewidthbased approaches further suffer from the graph theory overhead needed to convert their (typical) output of tree decompositions into perfect elimination orderings for the line graph and then contraction sequences for the tensor network. Our primary contribution to the literature is to provide an open source code framework (available at github.com/TheoryInPractice/ConSequences) for integrating all existing contraction sequence algorithms into a common interface designed for extendability and documented for accessibility; further, we have tabulated the performance of several leading contraction sequence algorithms. Our results provide quantum circuit simulation developers an extended benchmark for expected performance on circuits with varying structures and complexities. We use container-based (Docker [23]) wrappers for each contraction sequence algorithm, completely removing code dependency issues, and provide Python-based utilities for converting various input/output formats into standardized internal formats for consistency. We demonstrate the utility of this software by reproducing two previous studies based on domainspecific algorithms, and extending them to include treewidth-based solvers in a broader set of experimental results. We find that modern treewidth solvers from the recent PACE 2017 coding challenge [24] are both faster and have more consistent run times than the domain-specific algorithms. This speed increase allows us to study larger datasets in both experiments, and provide more competitive comparisons of a tensor network simulator [19] against Microsoft's LIQUi|> Hilbert space simulator [25]. In particular, we show that contraction sequence algorithms are no longer the major bottleneck in tensor network simulations, and there is immediate value in work improving the scalability of downstream contraction code. The paper is organized as follows. We begin with relevant definitions and an overview of related work in the Background, then describe the functionality of our code framework and considerations for use and extension in ConSequences: An Accessible, Extendable Framework. In the subsequent MERA Applications section, we reproduce a study by Pfeifer et al. [22], evaluating their algorithm netcon alongside two treewidth algorithms from the PACE 2017 challenge (freetdi [26] and meiji-e [27]) on a dataset including multi-scale entanglement renormalization ansatz (MERA) networks [2]. We extend this initial comparison on a larger corpus of MERA networks, pushing the limits of these exact contraction sequence solvers on a new benchmark. In the Applications with QAOA Circuit Simulation section, we reproduce a study by Fried et al. [19], evaluating another treewidth-based solver (quickbb [28]) against freetdi and meiji-e on quantum circuits formulated with Farhi et al.'s quantum approximate optimization algorithm (QAOA) for MAXCUT on r-regular graphs [29]. In addition to contraction sequence comparisons, we simulate the tensor network with qTorch [19], noting the correlation between simulation time and contraction complexity, and providing an updated comparison with Microsoft's LIQUi|> simulator [25]. We conclude with a summary and directions for future work. Background For a graph G, we use V(G) and E(G) to denote the sets of vertices and edges, respectively, and use G[X] to denote the subgraph induced by a set of vertices X. We say two vertices u, v are adjacent if (u, v) 2 E(G), and call the set of all vertices adjacent to v its neighborhood N(v). The degree of v is |N(v)|, and a graph is r-regular if every vertex has degree r. Formally, a tensor network is represented by a graph whose vertices correspond to tensors and edges denote tensor contractions over tensor indices. A contraction of two tensors corresponds to an edge contraction in the graph, where two vertices u, v with respective neighborhoods N(u), N(v) are replaced with a single vertex uv with neighborhood (N(u) [ N(v))\{u, v}. In the remainder of this section we define contraction complexity and its relationship to notions from structural graph theory including treewidth, then outline the methods used to generate MERA and QAOA tensor networks, which are used as data in our experiments. Contraction complexity and treewidth Simulation of a tensor network requires its contraction down to a single tensor, and the network's structure imposes certain lower bounds on the information that must be kept in the network, fundamentally captured in the notion of contraction complexity: Definition (Contraction Complexity (cc)). A contraction sequence is an ordering of a tensor network's edges, and the complexity of a contraction sequence S is the largest degree of a merged vertex created by contracting the network according to S. The contraction complexity (cc) of a tensor network is the minimum complexity over all possible contraction sequences. The run time complexity for simulating a tensor network is O(χ k ), where χ is a refinement parameter for bond dimensions [22] and k � cc is the cost of the contraction sequenced executed. Since better contraction sequences can yield exponentially-faster simulations, it is of practical interest to quickly find sequences with minimum cost. Unfortunately, computing the contraction complexity optimally is an NP-hard [21] optimization problem with strong ties to structural graph theory under the guise of treewidth and elimination orderings. Definition (Treewidth (tw)). A tree decomposition of a graph G is a tree T with a function f mapping nodes in T to bags (sets) of vertices from G, such that the following conditions hold: 1. All vertices are represented: S t2V(T) f(t) = V(G). 2. All edges are represented: 8(u, v) 2 E(G), 9t 2 V(T) s.t. u, v 2 f(t). 3. Graph vertices induce a (connected) subtree of T: if w 2 f(r) \ f(s) for r, s 2 V(T), w 2 V(G), then w 2 f(t) for all t on the path from r to s in T. The width of a tree decomposition is max t2V(T) |f(t)| − 1, and the treewidth of a graph G, denoted tw(G), is the minimum width over all valid tree decompositions of G. Perhaps surprisingly, treewidth can be viewed as a vertex-centric formulation of the edgecentric contraction complexity, via a transformation of the underlying graph G to its line graph L(G). The line graph is constructed with V(L(G)) = E(G) and E(L(G)) = {(e 1 , e 2 ) | e 1 6 ¼ e 2 2 E(G) s.t. e 1 , e 2 share a common endpoint}. The treewidth of L(G) then captures the same complexity: Theorem (Markov and Shi [4]). The contraction complexity of a graph equals the treewidth of its line graph. The relationship between treewidth and contraction sequences is perhaps easier seen through the characterization of treewidth using elimination orderings, which are permutations of the vertices. Given an elimination order π = v 1 , v 2 , . . ., v n of a graph G, the fill-in graph G π is constructed by iterating over v i from i = 1 to n and adding edges to make the neighbors of v i in G[v i , . . ., v n ] a clique. A graph has treewidth at most k if and only if there exist an elimination ordering π so that each vertex has at most k higher numbered neighbors in G π (see e.g., [30] for a proof). This condition naturally corresponds to the maximum degree of a tensor in the contraction sequence. While this intuition provides an straightforward mapping from a contraction sequence to a tree decomposition, the other direction of Markov and Shi's proof shows how to convert an arbitrary tree decomposition into a contraction sequence with equal complexity. This non-trivial conversion allows treewidth solvers, whose native outputs are only tree decompositions, to be used directly as contraction sequence algorithms. To enable future work, we provide an implementation of this conversion as a modular subroutine in our post-processing utilities. Rapid advances have been made in treewidth solvers in recent years, in large part to the Parameterized Algorithms and Computational Experiments (PACE) Challenge [24,31]. Previous algorithms with practical implementations (such as quickbb [28]) are based on searching the space of elimination orderings and given the connection between contraction sequences and elimination orderings [4], share a strong resemblance to typical domain-specific algorithms [21,22]. However, recent work in separator-based treewidth algorithms has begun to dominate modern benchmarks. The classic Arnborg, Corniel, and Proskurowski dynamic programming algorithm [32], reformulated as a positive-instance dynamic programming (PID) algorithm, has produced the winners of both the PACE 2016 (a Java implementation by Tamaki) and PACE 2017 (a C++ implementation by Larisch and Salfelder, freetdi) challenges [24,26,31]. The 2017 challenge also saw a better scaling implementation (meiji-e [27]) based off of a PID-reformulation of the improved dynamic programming algorithm by Bouchitté and Todinca [33]. MERA tensor networks One class of tensor networks that we examine is the multi-scale entanglement renormalization ansatz (MERA). Given that contraction sequence algorithms only utilize the structure of the underlying graph in these networks, we restrict our presentation here to the important structural notions (visualized with a 1D binary MERA in Fig 1). We direct the interested reader to [34] for a rigorous description beyond the graph structure. Fundamentally, MERA is a scheme for mapping a lattice of operator sites onto a coarser lattice. This mapping is expressed in terms of coarsening layers. The lattices on which MERA acts have an inherent dimension, which we denote d; for simplicity we only consider examples in 1-and 2-dimensions in this paper. The most detailed lattice (L 0 ) contains all sites for operators, and lattice L 1 is produced after one level of coarsening. A coarsening level consists of a layer of unitaries followed by a layer of isometries. To disentangle the sites, in the MERAs we consider, unitary tensors take in 2 d wires (edges) and output 2 d wires for a lattice of dimension d. For the coarsening layer, k: 1 isometry tensors take in k wires and output one wire. In total, going from lattice L i with s sites to L iþ1 requires s 2 unitary tensors, s 2 isometry tensors, and produces a new lattice with s k sites. This structure is then reflected for negative lattice levels, and a wire connects the top and bottom level interface tensors. Once this MERA graph is defined, operators are placed on lattice sites in L 0 and the causal cone is computed by including the operators and any tensor that lies on an ascending (descending) path to the upper (lower) interface tensor (Fig 1). Once the causal cone is computed, every tensor not included in the cone is removed (by unitarity and properties of the isometries), and wires are added from a tensor to its dual mirror such that all tensors have the requisite number of wires. QAOA quantum circuits Another source for data comes from the quantum approximation optimization algorithm (QAOA) [29], a hybrid classical-quantum algorithm for utilizing near-term (* 100 qubit) quantum computers. While applicable to generic satisfiability problems, we restrict ourselves to the MAXCUT optimization problem on r-regular graphs. Notably, when QAOA is applied to the MAXCUT problem, the structure of the input graph is reflected in the quantum circuit. Interested readers should refer to Farhi et al.'s formulation of QAOA [29] for a theoretical treatment, or the qTorch source code [19] for a practical example. ConSequences: An accessible, extendable framework One issue preventing widespread experimentation with (and adoption of) contraction sequence algorithms was the practical problem of installing the software and managing software dependencies. Of the algorithms presented in this paper, one is interfaced with MATLAB and uses C extensions for computationally-difficult sections (netcon), one is written in Java (meiji-e), two are written in C++ (freetdi, qTorch), and one is only distributed as a binary executable for Linux (quickbb). Often these implementations were written as a proof of concept and contain little-to-no documentation, especially regarding the code library dependencies needed to compile the code. Additionally, once the code is compiled, each solver has proprietary input and output format. Algorithms from the treewidth literature may require the input graph to have particular vertex labels, and typically output a tree decomposition or an elimination ordering. Algorithms from the contraction sequence literature may require the input as a quantum circuit in Quantum Assembly format or as a tensor network, and the contraction sequence output may be an ordering of edges in the network or a sequence of contractions that automatically removes resulting self-loops. Addressing both problems at once, we provide an open source framework ConSequences (Fig 2) for running contraction sequence algorithms, designed to be both accessible and extendable. The code is available at github.com/TheoryInPractice/ConSequences, complete with documentation for using existing code and tutorials for extending the functionality. At the core of this framework is a pipeline for pre-processing input, dispatching solvers, and post-processing output. The pre-processing utilities take tensor networks in formats such as Quantum Assembly and various graph formats, then generate graph files in a standardized format. The solver dispatcher manages calls to contraction sequence algorithms, and includes functionality such as parallelism, managing seeds, and handling timeouts. The post-processing utilities then take output from these solvers and generate a full set of output (including tree decompositions, perfect elimination orderings, and contraction sequences), allowing various aspects of the output to be reported and analyzed. To make the framework accessible, the central pipeline is implemented as lightweight Python files, and individual contraction sequence algorithms are wrapped inside of Docker [23] images. Docker images are similar to virtual machines in that they abstract away all dependency issues, but have the distinct advantage of very little CPU and memory footprint on a native Linux system. Users need only install Python and Docker to run the code on Windows, MacOS or Linux; these steps are especially straightforward on Linux and we provide instructions for new users. To make the framework extendable, we provide Docker templates for data generators (e.g., MERA networks) and contraction sequence algorithms. These templates are accompanied by tutorials which guide new users through wrapping up their code in Docker and interacting with our existing structure properly. All experiments were run on three identical workstations, each with a single Xeon E5-2623 v3 processor (8 threads with a 3.0GHz base clock and 10MB cache) and 64GB system memory. Contraction sequence algorithms were run with a single thread and LIQUi|> and qTorch simulations with all threads. Experiments were run one at a time on the workstations, preventing noise from non-uniform cache usage between competing jobs. These workstations ran Fedora 27 with Docker 18.03.1-ce and Python 3.6.5. Algorithm-dependent software requirements (e.g., gcc, MATLAB) are fixed per algorithm in its Docker image wrapper; details are deferred to the code repository. MERA applications In this section we compare exact treewidth solvers to Pfeifer et al.'s netcon algorithm [22], on data from MERA networks. A small corpus of datasets from [22] is initially considered, in which case optimal contraction sequences are found within four seconds by both treewidth Externally-generated domain data is parsed into standardized graph formats with the pre-processing utility. The solver dispatcher then allows the user to compute contraction sequences (or their equivalent, e.g. tree decompositions) using external algorithms in Docker containers, then executes a post-processing utility to output standardized formulations of a contraction sequence. solvers. To analyze algorithmic scaling with an extended benchmark, we generate 1-and 2-dimensional MERA networks with all possible placements of 1-and 2-operators, where we additionally observe that the meiji-e treewidth solver scales better than freetdi when networks become dense and increase in contraction complexity. Initial comparison on netcon benchmark The netcon implementation was developed as the contraction sequence algorithm for a simulation toolset written for MATLAB [22,35,36]. In the vein of previous approaches [21] such as depth-first search and dynamic programming, netcon's core subroutine is a breadth-first search (BFS) over the solution space of all possible contraction sequences. To trim down this exponentially-sized space, the authors introduce two pruning methods for reducing the search space at each step of the BFS: first, if a contraction would cost more than a user-defined threshold, this contraction will not be considered; second, the authors provide a list of criteria for when outer product contractions should not be made. This algorithm is exact (i.e., it finds optimal contraction complexity), but its run time depends heuristically on the effectiveness of the pruning techniques to a particular network's search space. Provided as a MATLAB package, the core subroutines in netcon are implemented in external C code for efficiency. The authors of [22] evaluate their pruning heuristics on seven networks, including Tree Tensor Networks (TTN), Time-Evolution Block Decimation (TEBD), and MERA networks ranging from five to 27 tensors. In this previous work, the authors found that the the pruning techniques resulted in faster results on all networks, with the largest network requiring 36 seconds. We reproduce this experiment on all seven networks (provided in [22]). For netcon we use the MATLAB interface with an external C package, using all optimizations as specified in the accompanying code [22]. We compare against two exact PACE algorithms, freetdi and meiji-e. Refer to the ConSequences section for workstation specifications. As seen in Table 1, both netcon and freetdi require on the order of 0.001 seconds on the first four networks, whereas meiji-e was two orders of magnitude slower. On the three larger networks, however, freetdi was fastest, with both PACE algorithms finishing within four seconds on the largest network. This last data point is of particular interest because it hints at scalability differences between these algorithms. Whereas meiji-e's run time was 5× slower on the 4:1 2D MERA compared to the 9:1 2D MERA, freetdi increased to over 250× slower and netcon over 450× slower. Extended benchmark on large MERA networks To test scalability further, we compared the algorithms' performance on a larger corpus of MERA networks. As seen in Fig 1, the iterative layers of isometries and unitaries in MERA Table 1. Run times for each contraction sequence algorithm when executed on tensor network datasets from Pfeifer et al. [22]. For each tensor network, the number of tensors (|V|), edges (|E|), and optimal contraction complexity (cc) are reported. Instance Run networks allow one to easily generate underlying graphs given a unitary and isometry specification. We provide a MERA generator in our code for 1D lattices with binary isometries and 2D lattices with 4:1 isometries. This generator takes as input the number of coarsening levels and whether the top level of isometries should connect to a common tensor in the style of [34]. Once a MERA network is generated from these parameters, the locations of operators to be evaluated are chosen, a causal cone is computed, and the network is reduced down to the final tensor network by simplifying the network outside the causal cone. Notably, given a fixed set of parameters needed to generator the MERA network, several choices of operator inputs can result in isomorphic (i.e., identical up to relabeling) networks. As seen in Fig 1, placing one operator at various sites can result in both isomorphic and nonisomorphic networks, so care must be taken to generate an appropriate set of representative graphs. In our extended comparison, we generate 1D binary MERA and 2D 4:1 MERA with 1 and 2 operator placements. For 1 operator, we generate a MERA network for every possible operator placement, then compute the unique networks up to isomorphism. For 2 operators, we fix an operator at the first position (0 for 1D and (0,0) for 2D), then range the second operator over all possible positions; again we extract the unique graphs up to isomorphism. Fig 3 summarizes the number of networks up to isomorphism and details some of their nuances. For example, for a 1D 2:1 MERA with six coarsening levels and two operator placements, the network will have between 40 and 86 vertices based on operator placement (see the dark-blue highlighted row in the summary table). In this case, over all placements, jMj ¼ 63 networks were generated, but only jMj ¼ 48 were unique. Looking into these unique networks even further (left table), we find that only one network had 40 vertices, and over half of the unique graphs had over 80 vertices. While these networks may vary in the extreme cases, on average these networks are fairly predictable. Figs 4 and 5 visualize the results of our extended experiment running contraction sequences algorithms on larger MERA networks using ConSequences. In Fig 4, networks are binned by the number of qubits they can support for a quantum simulation (which is determined by the number of possible operator sites) and run time is reported in seconds. For the networks that were included, a timeout of 20 minutes was used; if no algorithm could solve an instance within 20 minutes then the network was not included as a datapoint. From this perspective, netcon appears slower than both treewidth-based solvers, and freetdi generally dominates meiji-e in 1D MERAs. However, on 2D MERAs with one operator, all algorithms seem roughly equal for 16-qubits and 64-qubits, with meiji-e pulling slightly ahead on 256-qubits. The improved scaling of meiji-e is reflected in 2D MERAs with two operators, although here we begin to hit the limit of exact algorithms with a 20 minute timeout. In Fig 5, networks are binned based on their contraction complexity, then run times are reported for each algorithm. From this perspective we find a stipulation for freetdi's dominance on 1D MERAs: the contraction complexity never exceeded 9. Indeed, freetdi performed well on 2D MERAs until the optimal contraction complexity reaches 18, at which point meiji-e starts to outperform both algorithms. These experiments depict that finding the optimal contraction sequence within 20 minutes becomes problematic when the contraction complexity reaches the mid-twenties, which may be useful for predicting when heuristics should be used. We note that while the experiment from [22] contained a graph with contraction complexity of 26, this network only had 27 tensors and 55 edges, whereas our 2D 4:1 MERA networks with contraction complexity 26 could have as many as 393 edges. This fact implies that searchspace-based approaches need pruning techniques that prune exponentially-many sequences in the number of network edges, otherwise the run time will scale without the optimal contraction complexity necessarily increasing. Applications with QAOA circuit simulation In this section we compare exact treewidth solvers from PACE with the quickbb solver used in Fried et al.'s qTorch tensor network simulator [19]. This comparison is run on quantum circuits constructed with the QAOA method for solving MAXCUT on r-regular graphs. We find that exact contraction sequences on these graphs can be computed in less time than needed to execute the tensor contractions, allowing us to discuss the total simulation time without using an untimed preprocessing step. Rerunning the comparison with Microsoft's LIQUi|> simulator from [19], we find that tensor networks are competitive against state-of-the-art simulators, allowing speedups when little information is needed to represent the network (i.e., on sparse graphs), and a potential solution for the 22-qubit limit currently built into LIQUi|>. Computing contraction complexity Before comparing total simulation times, we start by evaluating the times required to find optimal contraction sequences. Because non-optimal contraction sequences lead to exponentiallyslower downstream simulation times, we are particularly interested in exploring the limits of exact solvers. The data for this experiment comes from qTorch's QAOA quantum circuit constructor, which computes MAXCUT on a specified (arbitrary) graph. Reproducing circuits similar to the original qTorch experiments, we use r-regular graphs for r 2 {3, 4, 5}, generated with Net-workX and seeded for easy replicability. Fig 6 visualizes the results from using 25 random graphs for each (r, |V|) pair and a 15 minute timeout. Similar to the sparse MERA graphs, we find that freetdi dominates run times and is up to five orders of magnitude faster than quickbb on the smallest graphs. meiji-e again scales better than freetdi, allowing it to compute optimal contraction sequences for several networks within the timeout that other algorithms could not finish. Notably, quickbb is an anytime algorithm that finds increasingly better perfect elimination orderings, so it always provides a (potentially non-optimal) solution when given a timeout. Similar behavior may be adapted from heuristic versions of PACE submissions (e.g., meiji-e), but this functionality is left as future work. Table 2 overviews the optimal contraction complexity for r-regular graphs, |V| vertices, 25 samples per (r, |V|) pair. Optimal values are guaranteed and found by at least one of freetdi and meiji-e. We find these values have a small standard deviation, implying that the quantum circuit resulting from QAOA computing MAXCUT have fairly predictable contraction complexity, which may be useful for recognizing a priori the time and hardware resources needed to evaluate a tensor network. Simulation run times Computing downstream simulation times using qTorch, we first evaluate how simulation time correlates with contraction complexity. This comparison enables us to quantify how downstream runtime is impacted by the contraction complexity in practice (theoretical analysis predicts a exponential increase due to the size of the merged tensors, but it is possible that some downstream code mitigates this impact). Further, it provides for a finer-grained comparison of the quality of contraction sequences produced by freetdi and meiji-e, as measured by resulting simulation run time. Although both algorithms produce sequences with the same contraction complexity (and thus have the same leading-order term in the simulation time complexity), this term may occur during the contraction sequence between 1 and |V| times. By examining raw simulation times, we may infer that one algorithm tends towards 'higher quality' contraction sequences than another. As shown in Fig 7, simulation time indeed has an exponential dependence on the contraction complexity. Additionally, the contraction sequences produced by freetdi and meiji-e appear to be of comparable quality over the same corpus of networks. We observe that the range of run times for a given contraction complexity is nearly always an order of magnitude, meaning that the variance in run time scales proportional to the total run time. This observation may be useful for reliably predicting simulation time based on a network's known contraction complexity, which may be useful for optimizing future simulators. In our final experiment, we compare qTorch paired with optimal contraction sequences to Microsoft's LIQUi|> simulator. Refer to the ConSequences section for workstation specifications. Previous comparisons to LIQUi|> included non-optimal contraction sequences found by quickbb, which may have caused the downstream simulation to be exponentially more expensive. Additionally, the experiments in [19] were structured so that quickbb was run for 3000 seconds per network ahead of time, which resulted in impractical total run times for lower levels of regularity. Results for this experiment are visualized in Fig 8. Aligning with previous conclusions in [19], both 3-regular and 4-regular graphs are more quickly simulated on tensor networks than LIQUi|>. We additionally find that tensor networks can scale beyond the 22-qubit limit imposed on LIQUi|> (supposedly for exponential system memory usage). Even with 5-regular graphs, qTorch remains competitive as an alternative simulator. Comparing Fig 6 with Fig 8, it is also clear that computing the contraction complexity is no longer the primary bottleneck. Instead, efforts should now be directed to improving the contraction simulation times. Conclusion In summary, ConSequences provides an open source, extendable platform for comparing contraction sequence algorithms for tensor networks. By packaging conversion utilities with containerized solvers, we remove both the theoretical and engineering difficulties preventing practitioners from running any contraction sequence solver on any tensor network. Table 2. Exact contraction complexities found using freetdi and meiji-e on QAOA circuits for computing MAXCUT on r-regular graphs. 25 random regular graphs are generated at each r, |V| level using NetworkX, and algorithms were timed out at 15 minutes. Timed out values were dropped from the data, resulting in less than 25 Samples for some parameter values. Network Optimal Contraction Complexity Additionally, we demonstrate the framework's applicability by reproducing and significantly extending several prior empirical evaluations. With MERA networks, we introduce a more extensive and difficult benchmark dataset which allows identification of solutions that will scale (e.g., freetdi on 1D MERA networks), subtler performance differences between algorithms (e.g., how meiji-e scales better for larger contraction complexities), and areas where new approaches are needed (e.g, 2D, 2-operator MERA, where 193 of 207 networks timed out). With qTorch on QAOA data, we were able to validate the exponential dependence of run time on optimal contraction complexity, and produce a total simulation time more representative of the full pipeline required for simulation. In doing so, we illuminate the urgent need for improved contraction times when the contraction sequence is known. For contraction sequence algorithms, several avenues offer promising future work. With large MERA networks we found that exact solvers for optimal contraction sequences had prohibitively high run times, a difficulty which may require using non-exact heuristics. Several treewidth-based algorithms have heuristic formulations in a different track of the PACE 2017 challenge [24], and our comparison against domain-specific algorithms suggests that these PACE submissions would be the natural starting point for an investigation into heuristics. Additionally, the use of heuristics involves a trade-off between finding a sequence with smaller complexity and the additional search time, which is not a problem for exact solvers. Exploring this trade-off in practical applications may be of interest. [19] with contraction sequences produced by exact treewidth algorithms vs. Microsoft's LIQUID solver [25]. Total simulation time includes both computation of the contraction sequence using ConSequences and tensor network simulation time using qTorch. A timeout of 900 seconds is used for computing the contraction sequence (horizontal dashed line), and a simulation is not run unless an optimal contraction sequence is found by at least one contraction sequence algorithm. LIQUID is limited to simulations up to 22 qubits. Fig 1 . 1(Top) A 1D binary MERA with a 16-site lattice and 3 levels of coarsening; three operator placements are highlighted (red, blue, green). (Bottom) Causal cones and final tensor networks for each of the three highlighted operators. Note that the tensor networks for the left-most (red) and right-most (green) operators are isomorphic to one another, but structurally distinct from the middle (blue) operator's network.https://doi.org/10.1371/journal.pone.0207827.g001 Benchmarking treewidth as a practical component of tensor network simulations PLOS ONE | https://doi.org/10.1371/journal.pone.0207827 December 18, 2018 Fig 2 . 2Visualization of the ConSequences pipeline. https://doi.org/10.1371/journal.pone.0207827.g002 Fig 3 . 3Summary of extended MERA benchmark data. (Center) For each lattice type (1D binary or 2D 4-ary) and number of operators, as the number of levels ℓ varies, we report the number of vertices |V| in the resulting networks, the total number of networks jMj, and the number of unique networks up to isomorphism jMj. (Left, Right) These detailed tables each expand a row from the summary table, specifying the number of networks produced (total jSj and up to isomorphism ĵ S j), for each pair of values for the number of vertices |V| and number of edges |E|. Note that sum of the jSj's in a detailed table sums to the corresponding jMj in the summary table, and likewise for ĵ S j and jMj.https://doi.org/10.1371/journal.pone.0207827.g003 Fig 4 . 4Run times for the contraction sequence algorithms on select MERA networks, binned by number of qubits possible (number of L 0 sites). All algorithms are timed out at 20 minutes (horizontal dashed line), and a network that remained unsolved by every algorithm is not included. 2D MERA with one operator had 48 of 131 networks that did not finish, and the two operator networks had 193 of 207 that did not finish. https://doi.org/10.1371/journal.pone.0207827.g004 Fig 5 . 5Run times for the three algorithms on select MERA networks, binned by optimal contraction complexity. All algorithms are timed out at 20 minutes (horizontal dashed line), and a network that remained unsolved by every algorithm is not included. 2D MERA with one operator had 48 of 131 networks that did not finish, and the two operator networks had 193 of 207 that did not finish.https://doi.org/10.1371/journal.pone.0207827.g005 Fig 6 . 6Run times for freetdi, meiji-e, and quickbb on QAOA circuits for computing MAXCUT on r-regular graphs. 25 random regular graphs are generated at each r, |V| level using NetworkX, and algorithms were timed out at 15 minutes (horizontal line). Fig 7 . 7Simulation time is tightly correlated with the contraction complexity of a network. While exact algorithms freetdi and meiji-e may generate different tree decompositions and thus contraction sequences with the same treewidth, the differences have little impact on simulation times. https://doi.org/10.1371/journal.pone.0207827.g007 Fig 8 . 8Simulation times of the qTorch tensor network simulator ONE | https://doi.org/10.1371/journal.pone.0207827 December 18, 2018 1 / 19 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Dumitrescu EF, Fisher AL, Goodrich TD, Humble TS, Sullivan BD, Wright AL (2018) Benchmarking treewidth as a practical component of tensor network simulations. PLoS ONE 13(12): e0207827. https://doi.org/10.1371/journal. pone.0207827 https://doi.org/10.1371/journal.pone.0207827.t002r |V| Samples Mean S.D. Min 50% Max 3 10 25 5.0 0.6 4 5 6 14 25 5.2 0.6 4 5 6 18 25 6.0 0.8 5 6 7 22 25 6.3 0.9 5 6 8 26 25 6.7 0.7 6 7 8 30 25 7.8 0.6 7 8 9 4 10 25 6.5 0.5 6 7 7 14 25 7.7 0.8 6 8 9 18 25 8.7 0.7 8 9 10 22 25 10.2 0.8 8 10 11 26 24 11.2 1.2 9 12 13 30 5 12.0 0.7 11 12 13 5 10 25 7.7 0.6 6 8 9 14 25 9.6 0.7 8 10 11 18 25 11.3 0.7 10 11 13 22 23 12.7 0.8 11 13 14 26 1 12.0 NaN 12 12 12 PLOS ONE | https://doi.org/10.1371/journal.pone.0207827December 18, 2018 https://doi.org/10.1371/journal.pone.0207827.g006 Benchmarking treewidth as a practical component of tensor network simulations PLOS ONE | https://doi.org/10.1371/journal.pone.0207827 December 18, 2018 13 / 19 https://doi.org/10.1371/journal.pone.0207827.g008 Benchmarking treewidth as a practical component of tensor network simulations PLOS ONE | https://doi.org/10.1371/journal.pone.0207827 December 18, 2018 16 / 19 AcknowledgmentsThis manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy. gov/downloads/doe-public-access-plan).Author ContributionsAnother consideration is that tensor network simulations are increasingly run on high-performance computing (HPC) systems. Modern HPC platforms scale-up performance with parallelism and heterogenous computing accelerators (such as GPUs and FPGAs). 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We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods.

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Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities Apr 2020 Jelena Diakonikolas [email protected] Department of Computer Sciences UW-Madison Halpern Iteration for Near-Optimal and Parameter-Free Monotone Inclusion and Strong Solutions to Variational Inequalities Apr 2020 We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving monotone inclusion problems. These results immediately translate into near-optimal guarantees for approximating strong solutions to variational inequality problems, approximating convex-concave min-max optimization problems, and minimizing the norm of the gradient in min-max optimization problems. Our analysis is based on a novel and simple potential-based proof of convergence of Halpern iteration, a classical iteration for finding fixed points of nonexpansive maps. Additionally, we provide a series of algorithmic reductions that highlight connections between different problem classes and lead to lower bounds that certify near-optimality of the studied methods. Introduction Given a closed convex set U ⊆ R d and a single-valued monotone operator F : R d → R d , i.e., an operator that maps each vector to another vector and satisfies: (∀u, v ∈ R d ) : F (u) − F (v), u − v ≥ 0,(1.1) the monotone inclusion problem consists in finding a point u * that satisfies: 0 ∈ F (u) + ∂I U (u), where I U (u) = 0, if u ∈ U , ∞, otherwise(MI) is the indicator function of the set U ⊆ R d and ∂I U (·) denotes the subdifferential operator (the set of all subgradients at the argument point) of I U . Monotone inclusion is a fundamental problem in continuous optimization that is closely related to variational inequalities (VIs) with monotone operators, which model a plethora of problems in mathematical programming, game theory, engineering, and finance [Facchinei and Pang, 2003, Section 1.4]. Within machine learning, VIs with monotone operators and associated monotone inclusion problems arise, for example, as an abstraction of convex-concave min-max optimization problems, which naturally model adversarial training [Madry et al., 2018, Goodfellow et al., 2014. When it comes to convex-concave min-max optimization, approximating the associated VI leads to guarantees in terms of the optimality gap. Such guarantees are generally possible only when the feasible set U is bounded; a simple example that demonstrates this fact is Φ(x, y) = x, y with the feasible set x, y ∈ R d . The only (min-max or saddle-point) solution in this case is obtained when both x and y are the all-zeros vectors. However, if either x = 0 or y = 0, then the optimality gap max y ′ ∈R d Φ(x, y ′ )− min x ′ ∈R d Φ(x ′ , y) is infinite. On the other hand, approximate monotone inclusion is well-defined even for unbounded feasible sets. In the context of min-max optimization, it corresponds to guarantees in terms of stationarity. Specifically, in the unconstrained setting, solving monotone inclusion corresponds to minimizing the norm of the gradient of Φ. Note that even in the special setting of convex optimization, convergence in norm of the gradient is much less understood than convergence in optimality gap [Nesterov, 2012, Kim andFessler, 2018]. Further, unlike classical results for VIs that provide convergence guarantees for approximating weak solutions [Nemirovski, 2004, Nesterov, 2007, approximations to monotone inclusion lead to approximations to strong solutions (see Section 1.2 for definitions of weak and strong solutions and their relationship to monotone inclusion). We leverage the connections between nonexpansive maps, structured monotone operators, and proximal maps to obtain near-optimal algorithms for solving monotone inclusion over different classes of problems with Lipschitz-continuous operators. In particular, we make use of the classical Halpern iteration, which is defined by [Halpern, 1967]: u k+1 = λ k+1 u 0 + (1 − λ k+1 )T (u k ),(Hal) where T : R d → R d is a nonexpansive map, i.e., ∀u, v ∈ R d : T (u) − T (v) ≤ u − v . In addition to its simplicity, Halpern iteration is particularly relevant to machine learning applications, as it is an implicitly regularized method with the following property: if the set of fixed points of T is nonempty, then Halpern iteration (Hal) started at a point u 0 and applied with any choice of step sizes {λ k } k≥1 that satisfy all of the following conditions: (i) lim k→∞ λ k = 0, (ii) ∞ k=1 λ k = ∞, (iii) ∞ k=1 |λ k+1 − λ k | < ∞ (1.2) converges to the fixed point of T with the minimum ℓ 2 distance to u 0 . This result was proved by Wittmann [1992], who extended a similar though less general result previously obtained by Browder [1967]. The result of Wittmann [1992] has since been extended to various other settings [Bauschke, 1996, Xu, 2002, Kohlenbach, 2011, Körnlein, 2015, Lieder, 2017, and references therein]. Contributions and Related Work A special case of what is now known as the Halpern iteration (Hal) was introduced and its asymptotic convergence properties were analyzed by Halpern [1967] in the setting of u 0 = 0 and T : B 2 → B 2 , where B 2 is the unit Euclidean ball. Using the proof-theoretic techniques of Kohlenbach [2008], Leustean [2007] extracted from the asymptotic convergence result of Wittmann [1992] the rate at which Halpern iteration converges to a fixed point. The results obtained by Leustean [2007] are rather loose and provide guarantees of the form T (u k ) − u k = O( M log(k) ) in the best case (obtained for λ k = Θ( 1 k )), where M ≥ u 0 + T (u 0 ) + u k , ∀k. A tighter result that shows that T (u k ) − u k decreases at rate that is at least as good as 1/ √ k was obtained by Kohlenbach [2011]. The results of Leustean [2007] and Kohlenbach [2011] apply to general normed spaces. The work of Kohlenbach [2011] also provided an explicit rate of metastability that characterizes the convergence of the sequence of iterates {u k } in Hilbert spaces. More recently, Lieder [2017] proved that under the standard assumption that T has a fixed point u * and for the step size λ k = 1 k+1 , Halpern iteration converges to a fixed point as T (u k ) − u k = 2 u 0 −u * k+1 . A similar result but for an alternative algorithm was recently obtained by Kim [2019]. These two results (as well as all the results from this paper) only apply to Hilbert spaces. Unlike Halpern iteration, the algorithm introduced by Kim [2019] is not known to possess the implicit regularization property discussed earlier in this paper. The results of Lieder [2017] and Kim [2019] can be used to obtain the same 1/k convergence rate for monotone inclusion with a cocoercive operator but only if the cocoercivity parameter is known, which is rarely the case in practice. Similarly, those results can also be extended to more general monotone Lipschitz operators but only if the proximal map (or resolvent) of F can be computed exactly, an assumption that can rarely be met (see Section 1.2 for definitions of cocoercive operators and proximal maps). We also note that the results of Lieder [2017] and Kim [2019] were obtained using the performance estimation (PEP) framework of Drori and Teboulle [2014]. The convergence proofs resulting from the use of PEP are computer-assisted: they are generated as solutions to large semidefinite programs, which typically makes them hard to interpret and generalize. Our approach is arguably simpler, as it relies on the use of a potential function, which allows us to remove the assumptions about the knowledge of the problem parameters and availability of exact proximal maps. Our main contributions are summarized as follows: Results for cocoercive operators. We introduce a new, potential-based, proof of convergence of Halpern iteration that applies to more general step sizes λ k than handled by the analysis of Lieder [2017] (Section 2). The proof is simple and only requires elementary algebra. Further, the proof is derived for cocoercive operators and leads to a parameter-free algorithm for monotone inclusion. We also extend this parameterfree method to the constrained setting using the concept of gradient mapping generalized to monotone operators (Section 2.1). To the best of our knowledge, this is the first work to obtain the 1/k convergence rate with a parameter-free method. Results for monotone Lipschitz operators. Up to a logarithmic factor, we obtain the same 1/k convergence rate for the parameter-free setting of the more general monotone Lipschitz operators (Section 2.2). The best known convergence rate established by previous work for the same setting was of the order 1/ √ k [Dang andLan, 2015, Ryu et al., 2019]. We obtain the improved convergence rate through the use of the Halpern iteration with inexact proximal maps that can be implemented efficiently. The idea of coupling inexact proximal maps with another method is similar in spirit to the Catalyst framework [Lin et al., 2017] and other instantiations of the inexact proximal-point method, such as, e.g., in the work of Davis and Drusvyatskiy [2019], Asi and Duchi [2019], Lin et al. [2018]. However, we note that, unlike in the previous work, the coupling used here is with a method (Halpern iteration) whose convergence properties were not well-understood and for which no simple potential-based convergence proof existed prior to our work. Results for strongly monotone Lipschitz operators. We show that a simple restarting-based approach applied to our method for operators that are only monotone and Lipschitz (described above) leads to a parameter-free method for strongly monotone and Lipschitz operators (Section 2.3). Under mild assumptions about the problem parameters and up to a poly-logarithmic factor, the resulting algorithm is iterationcomplexity-optimal. To the best of our knowledge, this is the first near-optimal parameter-free method for the setting of strongly monotone Lipschitz operators and any of the associated problems -monotone inclusion, VIs, or convex-concave min-max optimization. Lower bounds. To certify near-optimality of the analyzed methods, we provide lower bounds that rely on algorithmic reductions between different problem classes and highlight connections between them (Section 3). The lower bounds are derived by leveraging the recent lower bound of Ouyang and Xu [2019] for approximating the optimality gap in convex-concave min-max optimization. Notation and Preliminaries Let (E, · ) be a real d-dimensional Hilbert space, with norm · = ·, · , where ·, · denotes the inner product. In particular, one may consider the Euclidean space (R d , · 2 ). Definitions that were already introduced at the beginning of the paper easily generalize from (R d , · 2 ) to (E, · ), and are not repeated here for space considerations. Variational Inequalities and Monotone Operators. Let U ⊆ E be closed and convex, and let F : E → E be an L-Lipschitz-continuous operator defined on U . Namely, we assume that: (∀u, v ∈ U ) : F (u) − F (v) ≤ L u − v . (1.3) The definition of monotonicity was already provided in Eq. (1.1), and easily specializes to monotonicity on the set U by restricting u, v to be from U . Further, F is said to be: 1. strongly monotone (or coercive) on U with parameter m, if: (∀u, v ∈ U ) : F (u) − F (v), u − v ≥ m 2 u − v 2 ; (1.4) 2. cocoercive on U with parameter γ, if: (∀u, v ∈ U ) : F (u) − F (v), u − v ≥ γ F (u) − F (v) 2 . (1.5) It is immediate from the definition of cocoercivity that every γ-cocoercive operator is monotone and 1/γ-Lipschitz. The latter follows by applying the Cauchy-Schwarz inequality to the left-hand side of Eq. (1.5) and then dividing both sides by γ F (u) − F (v) . Examples of monotone operators include the gradient of a convex function and appropriately modified gradient of a convex-concave function. Namely, if a function Φ(x, y) is convex in x and concave in y, then F ([ x y ]) = [ ∇xΦ(x,y) −∇yΦ(x,y) ] is monotone. The Stampacchia Variational Inequality (SVI) problem consists in finding u * ∈ U such that: (∀u ∈ U ) : F (u * ), u − u * ≥ 0.(SVI) In this case, u * is also referred to as a strong solution to the variational inequality (VI) corresponding to F and U . The Minty Variational Inequality (MVI) problem consists in finding u * such that: (∀u ∈ U ) : F (u), u * − u ≤ 0,(MVI) in which case u * is referred to as a weak solution to the variational inequality corresponding to F and U . In general, if F is continuous, then the solutions to (MVI) are a subset of the solutions to (SVI). If we assume that F is monotone, then (1.1) implies that every solution to (SVI) is also a solution to (MVI), and thus the two solution sets are equivalent. The solution set to monotone inclusion is the same as the solution set to (SVI). Approximate versions of variational inequality problems (SVI) and (MVI) are defined as follows: Given ǫ > 0, find an ǫ-approximate solution u * ǫ ∈ U , which is a solution that satisfies: (∀u ∈ U ) : F (u * ǫ ), u * ǫ − u ≤ ǫ, or (∀u ∈ U ) : F (u), u * ǫ − u ≤ ǫ, respectively. Clearly, when F is monotone, an ǫ-approximate solution to (SVI) is also an ǫ-approximate solution to (MVI); the reverse does not hold in general. Similarly, ǫ-approximate monotone inclusion can be defined as fidning u * ǫ that satisfies: 0 ∈ F (u * ǫ ) + ∂I U (u * ǫ ) + B(ǫ), (1.6) where B(ǫ) is the ball w.r.t. · , centered at 0 and of radius ǫ. We will sometimes write Eq. (1.6) in the equivalent form −F (u * ǫ ) ∈ ∂I U (u * ǫ ) + B(ǫ). The following fact is immediate from Eq. (1.6). Fact 1.1. Given F and U , let u * ǫ satisfy Eq. (1.6). Then: (∀u ∈ {U ∩ B u * ǫ }) : F (u * ǫ ), u * ǫ − u ≤ ǫ, where B u * ǫ denotes the unit ball w.r.t. · , centered at B u * ǫ . Further, if the diameter of U , D = sup u,v∈U u − v , is bounded, then: (∀u ∈ U ) : F (u * ǫ ), u * ǫ − u ≤ ǫD. Thus, when the diameter D is bounded, any ǫ D -approximate solution to monotone inclusion is an ǫapproximate solution to (SVI) (and thus also to (MVI)); the converse does not hold in general. Recall that when D is unbounded, neither (SVI) nor (MVI) can be approximated. We assume throughout the paper that a solution to monotone inclusion (MI) exists. This assumption implies that solutions to both (SVI) and (MVI) exist as well. Existence of solutions follows from standard results and is guaranteed whenever e.g., U is compact, or, if there exists a compact set U ′ such that Id − 1 L F maps U ′ to itself [Facchinei and Pang, 2003]. Nonexpansive Maps. Let T : E → E. We say that T is nonexpansive on U ⊆ E, if ∀u, v ∈ U : T (u) − T (v) ≤ u − v . Nonexpansive maps are closely related to cocoercive operators, and here we summarize some of the basic properties that are used in our analysis. More information can be found in, e.g., the book by Bauschke and Combettes [2011]. Fact 1.2. T is nonexpansive if and only if Id − T is 1 2 -cocoercive, where Id is the identity map. T is said to be firmly nonexpansive or averaged, if ∀u, v ∈ U : T (u) − T (v) 2 + (Id − T )u − (Id − T )v 2 ≤ u − v 2 . Useful properties of firmly nonexpansive maps are summarized in the following fact. Fact 1.3. For any firmly nonexpansive operator T, Id − T is also firmly non-expansive, and, moreover, both T and Id − T are 1-cocoercive. Halpern Iteration for Monotone Inclusion and Variational Inequalities Halpern iteration is typically stated for nonexpansive maps T as in (Hal). Because our interest is in cocoercive operators F with the unknown parameter 1/L, we instead work with the following version of the Halpern iteration: u k+1 = λ k+1 u 0 + (1 − λ k+1 ) u k − 2 L k+1 F (u k ) ,(H) where L k ∈ (0, ∞), ∀k. If L was known, we could simply set L k+1 = L, in which case (H) would be equivalent to the standard Halpern iteration, due to Fact 1.2. We assume throughout that λ 1 = 1 2 . We start with the assumption that the setting is unconstrained: U ≡ E. We will see in Section 2.1 how the result can be extended to the constrained case. Section 2.2 will consider the case of operators that are monotone and Lipschitz, while Section 2.3 will deal with the strongly monotone and Lipschitz case. Some of the proofs are omitted and are instead provided in Appendix A. To analyze the convergence of (H) for the appropriate choices of sequences {λ i } i≥1 and {L i } i≥1 , we make use of the following potential function: C k = 1 L k F (u k ) 2 − λ k 1 − λ k F (u k ), u 0 − u k . (2.1) Let us first show that if A k C k is non-increasing with k for an appropriately chosen sequence of positive numbers {A k } k≥1 , then we can deduce a property that, under suitable conditions on {λ i } i≥1 and {L i } i≥1 , implies a convergence rate for (H). Lemma 2.1. Let C k be defined as in Eq. (2.1) and let u * be the solution to (MI) that minimizes u 0 − u * . Assume further that F (u 1 ) − F (u 0 ), u 1 − u 0 ≥ 1 L 1 F (u 1 ) − F (u 0 ) 2 . If A k+1 C k+1 ≤ A k C k , ∀k ≥ 1, where {A i } i≥1 is a sequence of positive numbers that satisfies A 1 = 1, then: (∀k ≥ 1) : F (u k ) ≤ L k λ k 1 − λ k u 0 − u * . Using Lemma 2.1, our goal is now to show that we can choose L k = O(L) and λ k = O( 1 k ), which in turn would imply the desired 1/k convergence rate: F (u k ) = O( L u 0 −u * k ). The following lemma provides sufficient conditions for {A i } i≥1 , {λ i } i≥1 , and {L i } i≥1 to ensure that A k+1 C k+1 ≤ A k C k , ∀k ≥ 1, so that Lemma 2.1 applies. Lemma 2.2. Let C k be defined as in Eq. (2.1). Let {A i } i≥1 be defined recursively as A 1 = 1 and A k+1 = A k λ k (1−λ k )λ k+1 for k ≥ 1. Assume that {λ i } i≥1 is chosen so that λ 1 = 1 2 and for k ≥ 1 : λ k+1 1−2λ k+1 ≥ λ k L k (1−λ k )L k+1 . Finally, assume that L k ∈ (0, ∞) and F (u k ) − F (u k−1 ), u k − u k−1 ≥ 1 L k F (u k ) − F (u k−1 ) 2 , ∀k. Then, (∀k ≥ 1) : A k+1 C k+1 ≤ A k C k . Observe first the following. If we knew L and set L k = L, λ k = 1 k+1 , and A k = k(k + 1)/2, then all of the conditions from Lemma 2.2 would be satisfied, and Lemma 2.1 would then imply F (u k ) ≤ L u 0 −u * k , which recovers the result of Lieder [2017]. The choice λ k = 1 k+1 is also the tightest possible that satisfies the conditions Lemma 2.2 -the inequality relating λ k+1 and λ k is satisfied with equality. This result is in line with the numerical observations made by Lieder [2017], who observed that the convergence of Halpern iteration is fastest for λ k = 1 k+1 . To construct a parameter-free method, we use that F is L-cocoercive; namely, that there exists a constant L < ∞ such that F satisfies Eq. (1.5) with γ = 1/L. The idea is to start to with a "guess" of L (e.g., L 0 = 1) and double the guess L k as long as F (u k ) − F (u k−1 ), u k − u k−1 < 1 L k F (u k ) − F (u k−1 ) 2 . The total number of times that the guess can be doubled is bounded above by max{0, log 2 (2L/L 0 )}. Parameter λ k is simply chosen to satisfy the condition from Lemma 2.2. The algorithm pseudocode is stated in Algorithm 1 for a given accuracy specified at the input. We now prove the first of our main results. Note that the total number of arithmetic operations in Algorithm 1 is of the order of the number of oracle queries to F multiplied by the complexity of evaluating F at a point. The same will be true for all the algorithms stated in this paper, except that the complexity of evaluating F may be replaced by the complexity of projections onto U . Theorem 2.3. Given u 0 ∈ U and an operator F that is 1 L -cocoercive on E, Algorithm 1 returns a point u k such that F (u k ) ≤ ǫ after at most max{2L,L 0 } u 0 −u * ǫ + max{0, log 2 (2L/L 0 )} oracle queries to F . Algorithm 1: Parameter-Free Halpern -Cocoercive Case Input: L 0 > 0, ǫ > 0, u 0 . If not provided at the input, set L 0 = 1.; λ 1 = 1 2 , k = 0; while F (u k ) > ǫ do k = k + 1; L k = L k−1 ; p k = L k−1 L k λ k−1 1−λ k−1 , λ k = p k 1+2p k ; u k = λ k u 0 + (1 − λ k )(u k−1 − 2F (u k−1 )/L k ); while F (u k ) − F (u k−1 ), u k − u k−1 < 1 L k F (u k ) − F (u k−1 ) 2 do L k = 2 · L k ; p k = L k−1 L k λ k−1 1−λ k−1 , λ k = p k 1+2p k ; u k = λ k u 0 + (1 − λ k )(u k−1 − 2F (u k−1 )/L k ); end end return u k Proof. As F is 1 L -cocoercive, L k ≤ max{2L , L 0 } and the total number of times that the algorithm enters the inner while loop is at most max{0, log 2 (2L/L 0 )}. The parameters satisfy the assumptions of Lemmas 2.1 and 2.2, and, thus, F (u k ) ≤ L k λ k 1−λ k u 0 − u * . Hence, we only need to show that λ k decreases sufficiently fast with k. As L k can only be increased in any iteration, we have that λ k+1 ≤ λ k 1−λ k 1 + 2 λ k 1−λ k = λ k 1 + λ k ≤ λ k−1 1 + 2λ k−1 ≤ · · · ≤ λ 1 1 + kλ 1 = 1 k + 2 . Hence, the total number of outer iterations is at most max{2L,L 0 } u 0 −u * ǫ . Combining with the maximum total number of inner iterations from the beginning of the proof, the result follows. Constrained Setups with Cocoercive Operators Assume now that U ⊆ E. We will make use of a counterpart to gradient mapping [Nesterov, 2018, Chapter 2] that we refer to as the operator mapping, defined as: G η (u) = η u − Π U u − 1 η F (u) , (2.2) where Π U u − 1 η F (u) is the projection operator, namely: Π U u − 1 η F (u) = argmin v∈U 1 2 v − u + F (u)/η 2 = argmin v∈U F (u), v + η 2 v − u 2 . Operator mapping generalizes a cocoercive operator to the constrained case: when U ≡ E, G η ≡ F. It is a well-known fact that the projection operator is firmly-nonexpansive [Bauschke and Combettes, 2011, Proposition 4.16]. Thus, Fact 1.3 can be used to show that, if F is 1 L -cocoercive and η ≥ L, then G η is 1 2η -cocoercive. This is shown in the following (simple) proposition. Proposition 2.4. Let F be an 1 L -cocoercive operator and let G η be defined as in Eq. (1.1), where η ≥ L. Then G η is 1 2η -cocoercive. As G η is 1 2η -cocoercive, applying results from the beginning of the section to G η , it is now immediate that Algorithm 2 (provided for completeness) produces u k with G L k (u k ) ≤ ǫ after at most max{4L,L 0 } u 0 −u * ǫ + max{0, log 2 (4L/L 0 )} oracle queries to F (as each computation of G η requires one oracle query to F ). To Algorithm 2: Parameter-Free Halpern -Cocoercive and Constrained Case Input: L 0 > 0, ǫ > 0, u 0 ∈ U . If not provided at the input, set L 0 = 1.; λ 1 = 1 2 , k = 0; u 0 = Π U (u 0 − F (u 0 )/L 0 ) ,L 0 = F (ū 0 )−F (u 0 ) ū 0 −u 0 ; while G L k (u k ) > ǫ/(1 +L k /L k ) do 1 k = k + 1; 2 L k = L k−1 ; 3 p k = L k−1 L k λ k−1 1−λ k−1 , λ k = p k 1+2p k ; 4 u k = λ k u 0 + (1 − λ k )ū k−1 ; while G L k (u k ) − G L k (u k−1 ), u k − u k−1 < 1 2L k G L k (u k ) − G L k (u k−1 ) 2 do 5 L k = 2 · L k ; 6 p k = L k−1 L k λ k−1 1−λ k−1 , λ k = p k 1+2p k ; 7 u k = λ k u 0 + (1 − λ k )(u k−1 − G L k (u k−1 )/L k ); end 8ū k = Π U (u k − F (u k )/L k ),L k = F (ū k )−F (u k ) ū k −u k , L k = max{L k ,L k } ; end returnū k , u k complete this subsection, it remains to show that G η is a good surrogate for approximating (MI) (and (SVI)). This is indeed the case and it follows as a suitable generalization of Lemma 3 from Ghadimi and Lan [2016], which is provided here for completeness. Lemma 2.5. Let G η be defined as in Eq. (2.2). Denoteū = Π U (u − F (u)/η), so that G η (u) = η(u −ū). If, for some u ∈ U , G η (u) ≤ ǫ, then F (ū) ∈ −∂I U (ū) + B((1 + L loc /η)ǫ), where L loc = F (ū)−F (u) ū−u ≤ L. Proof. As, by definition,ū = argmin v∈U F (u), v + η 2 v−u 2 , by first-order optimality ofū, we have: 0 ∈ F (u) + η(ū − u) + ∂I U (ū). Equivalently: −F (ū) ∈ F (u) − F (ū) − G η (u) + ∂I U (ū). The rest of the proof follows simply by using G η ≤ ǫ and F (u) − F (ū) = L loc u −ū = L loc η G η (u) ≤ L loc η ǫ. Lemma 2.5 implies that when the operator mapping is small in norm · , thenū = Π U (u − F (u)/η) is an approximate solution to (MI) corresponding to F on U . We can now formally bound the number of oracle queries to F needed to approximate (MI) and (SVI). Theorem 2.6. Given u 0 ∈ U and a 1 L -cocoercive operator F , Algorithm 2 returnsū k ∈ U such that 1. G L k (ū k ) ≤ ǫ 2 , max v∈{U ∩Bū k } F (ū k ),ū k − v ≤ ǫ after at most 4 max{4L, L 0 } u 0 − u * ǫ + 2 max{0, log 2 (4L/L 0 )} oracle queries to F ; 2. max v∈U F (ū),ū − v ≤ ǫ after at most 4 max{4L, L 0 } u 0 − u * D ǫ + 2 max{0, log 2 (4L/L 0 )} oracle queries to F. Further, every point u k that Algorithm 2 constructs is from the feasible set: u k ∈ U , ∀k ≥ 0, and a simple modification to the algorithm takes at most max{4L, L 0 } u 0 −u * ǫ + max{0, log 2 (4L/L 0 )} oracle queries to F to construct a point such that G L k (u k ) ≤ ǫ. Proof. By the definition of G η , if u 0 ∈ U , then u k ∈ U , for all k. This follows simply as: u k+1 = λ k+1 u 0 + (1 − λ k+1 ) u k − 1 L k+1 G L k+1 (u k ) = λ k+1 u 0 + (1 − λ k+1 )Π U (u k − F (u k )/L k+1 ). Observe that, due to Line 8 of Algorithm 2, L k ≥L k . The rest of the proof follows using Lemma 2.5, Fact 1.1, and the same reasoning as in the proof of Theorem 2.3. Observe that if the goal is to only output a point u k such that G L k (u k ) ≤ ǫ, then computingū k and F (ū k ) is not needed, and the algorithm can instead use G L k (u k ) > ǫ as the exit condition in the outer while loop. Setups with non-Cocoercive Lipschitz Operators We now consider the case in which F is not cocoercive, but only monotone and L-Lipschitz. To obtain the desired convergence result, we make use of the resolvent operator, defined as J F +∂I U = (Id + F + ∂I U ) −1 . A useful property of the resolvent is that it is firmly-nonexpansive [Ryu and Boyd, 2016, and references therein], which, due to Fact 1.3, implies that P = Id − J F +∂I U is 1 2 -cocoercive. Finding a point u ∈ U such that P (u) ≤ ǫ is sufficient for approximating monotone inclusion (and (SVI)). This is shown in the following simple proposition, provided here for completeness. Proposition 2.7. Let P = Id − J F +∂I U . If P (u) ≤ ǫ, thenū = u − P (u) = J F +∂I U (u) satisfies F (ū) ∈ −∂I U (ū) + B(ǫ). Proof. By the definition of P and J F +∂I U , u − P (u) = (Id + F + ∂I U ) −1 (u). Equivalently: u − P (u) + F (u − P (u)) + ∂I U (u − P (u)) ∋ u. As P (u) ≤ ǫ, the result follows. If we could compute the resolvent exactly, it would suffice to directly apply the result of Lieder [2017]. However, excluding very special cases, computing the exact resolvent efficiently is generally not possible. However, since F is Lipschitz, the resolvent J F +∂I U can be approximated efficiently. This is because it corresponds to solving a VI defined on a closed convex set U with the operator F + Id that is 1-strongly monotone and (L+1)-Lipschitz. Thus, it can be computed by solving a strongly monotone and Lipschitz VI, for which one can use the results of e.g., Nesterov and Scrimali [1977] in Algorithm 4 (Appendix A), for which we prove that it attains the optimal convergence rate without the knowledge of L. The convergence result is summarized in the following lemma, whose proof is provided in Appendix A. Lemma 2.8. Letū * k = J F +I U (u k ), where u k ∈ U and F is L-Lipschitz. Then, there exists a parameterfree algorithm that queries F at most O((L + 1) log( L u k −ū * k ǫ )) times and outputs a pointū k such that ū k −ū * k ≤ ǫ. To obtain the desired result, we need to prove the convergence of a Halpern iteration with inexact evaluations of the cocoercive operator P . Note that here we do know the cocoercivity parameter of P -it is equal to 1/2. The resulting inexact version of Halpern's iteration for P is: u k+1 = λ k+1 u 0 + (1 − λ k+1 )(u k −P (u k )) = λ k+1 u 0 + (1 − λ k+1 )J F +∂I U (u k ), (2.3) whereP (u k ) − P (u k ) = J F +∂I U (u k ) −J F +∂I U (u k ) = e k is the error. To analyze the convergence of (2.3), we again use the potential function C k from Eq. (2.1), with P as the operator. For simplicity of exposition, we take the best choice of λ i = 1 i+1 that can be obtained from Lemma 2.1 for L i = L = 2, ∀i. The key result for this setting is provided in the following lemma, whose proof is deferred to the appendix. Lemma 2.9. Let C k be defined as in Eq. (2.1) with P as the 1 2 -cocoercive operator, and let L k = 2, λ k = 1 k+1 , and A k = k(k+1) 2 , ∀k ≥ 1. If the iterates u k evolve according to (2.3) for an arbitrary initial point u 0 ∈ U , then: (∀k ≥ 1) : A k+1 C k+1 ≤ A k C k + A k+1 e k , (1 − λ k+1 )P (u k ) − P (u k+1 ) . Further, if, ∀k ≥ 1, e k−1 ≤ ǫ 4k(k+1) , then P (u K ) ≤ ǫ after at most K = 4 u 0 −u * ǫ iterations. We are now ready to state the algorithm and prove the main theorem for this subsection. Algorithm 3: Parameter-Free Halpern -Monotone and Lipschitz Case Input: ǫ > 0, u 0 ∈ U ; k = 0, ǫ 0 = ǫ 8 ; u 0 =J F +∂I U (u 0 ), where J F +∂I U (u 0 ) − J F +∂I U (u 0 ) ≤ ǫ 0 ; P (u 0 ) = u 0 −ū 0 ; while P (u k ) > 3ǫ 4 do k = k + 1, λ k = 1 k+1 , ǫ k = ǫ 8(k+1)(k+2) ; u k = λ k u 0 + (1 − λ k )ū k−1 ; u k =J F +∂I U (u k ), where J F +∂I U (u k ) − J F +∂I U (u k ) ≤ ǫ k ; P (u k ) = u k −ū k ; end returnū k , u kO (L+1) u 0 −u * ǫ log (L+1) u 0 −u * ǫ . Proof. Recall thatP (u k ) − P (u k ) = e k and e k = ǫ k = ǫ 8(k+1)(k+2) < ǫ 4 . Hence, as Algorithm 3 outputs a point u k with P (u k ) ≤ 3ǫ 4 , by the triangle inequality, P (u k ) ≤ ǫ. To bound the number of iterations until P (u k ) ≤ 3ǫ 4 , note that, again by the triangle inequality, if P (u k ) ≤ ǫ/2, then P (u k ) ≤ 3ǫ 4 . Applying Lemma 2.9, P (u k ) ≤ ǫ/2 after at most k = 8 u 0 −u * ǫ iterations, completing the proof of the first part of the theorem. For the remaining part, using Lemma 2.8,J F +∂I U (u k ) can be computed (with target error ǫ k ) in O((L + 1) log( (L+1) u k −J F +∂I U (u k ) ǫ k )) = O((L+1) log( (L+1) P (u k ) ǫ )) iterations, as O(log( 1 ǫ k )) = O(log( 1 ǫ )) and P (u k ) = u k − J F +∂I U (u k ) , by definition. It remains to use that P (u k ) = O( u 0 − u * ), which can be deduced from, e.g., Eq. (A.7) in the proof of Lemma 2.9. Similarly as before, P (u k ) ≤ ǫ implies an ǫ-approximate solution to (MI), by Proposition 2.7. When the diameter D is bounded, P (u k ) ≤ ǫ D implies an ǫ-approximate solution to (SVI). Remark 2.11. In degenerate cases where L << 1, instead of using the resolvent of F + ∂I U , one could use the resolvent of F/η + ∂I U for η = O(L), assuming the order of magnitude of L is known (this is typically a mild assumption). Then, each approximate computation of the resolvent would take O((L/η + 1) log( (L/η+1) u 0 −u * ǫ ) oracle queries to F, and we would need to require that P (u k ) ≤ 3ǫ 4η . Thus, the total number of queries to F would be O((L + η) log( (L+η) u 0 −u * ǫ )). Setups with Strongly Monotone and Lipschitz Operators We now show that by restarting Algorithm 3, we can obtain a parameter-free method with near-optimal oracle complexity. To simplify the exposition, we assume w.l.o.g. that L = Ω(1). Theorem 2.12. Given F that is L-Lipschitz and m-strongly monotone, consider running the following algorithm A, starting with u 0 ∈ U : (A) : At iteration k, invoke Algorithm 3 with error parameter ǫ k = 7 16 P (u k−1 ) . Then, A outputs u k ∈ U with P (u k ) ≤ ǫ after at most 1 + log 2 ( u 0 −u * ǫ ) iterations, for any ǫ ∈ (0, 1 2 ]. The total number of queries to F until P (u k ) ≤ ǫ is O (L + L m ) log( u 0 −u * ǫ ) log(L + L m ) . Proof. The first part is immediate, as each call to Algorithm 3 ensures, due to Theorem 2.10, that P (u k ) ≤ 7P (u k−1 ) 16 ≤ 7 P (u k−1 ) 16 + ǫ k 8 ≤ P (u k−1 ) 2 , and P (u 0 ) ≤ 2 u 0 − u * as P is 2-Lipschitz (because it is 1 2 -cocoercive) and P (u * ) = 0. It remains to bound the number of calls to F for each call to Algorithm 3. Using Theorem 2.10 and P (u k ) = Θ( P (u k ) ), each call to Algorithm 3 takes O( L u k−1 −u * P (u k−1 ) log( L u k−1 −u * P (u k−1 ) )) calls to F. De- noteū * k−1 = J F +∂I U (u k−1 ) = u k−1 − P (u k−1 ) . Using Proposition 2.7: F (ū * k−1 ),ū * k−1 − u * ≤ P (u k−1 ) ū * k−1 − u * . On the other hand, as F is m-strongly monotone and u * is an (MVI) solution, m ū * k−1 − u * 2 ≤ F (ū * k−1 ),ū * k−1 − u * . Hence: ū * k−1 −u * ≤ 1 m P (u k−1 ) . It remains to use the triangle inequality and P (u k−1 ) = u k−1 −ū * k−1 to obtain: u k−1 − u * ≤ 1 + 1 m P (u k−1 ) . Lower Bound Reductions In this section, we only state the lower bounds, while more details about the oracle model and the proof are deferred to Appendix A. Lemma 3.1. For any deterministic algorithm working in the operator oracle model and any L, D > 0, there exists an L-Lipschitz-continuous operator F and a closed convex feasible set U with diameter D such that: (a) For all ǫ > 0 such that k = LD 2 ǫ = O(d), max u∈U F (u k ), u k − u = Ω(ǫ); (b) For all ǫ > 0 such that k = LD ǫ = O(d), max u∈{U ∩Bu k } F (u k ), u k − u = Ω(ǫ); (c) If F is 1 L -cocoercive, then for all ǫ > 0 such that k = LD ǫ log(D/ǫ) = O(d), it holds that max u∈{U ∩Bu k } F (u k ), u k − u = Ω(ǫ); (d) If F is m-strongly monotone, then for all ǫ > 0 such that k = L m = O(d), it holds that max u∈{U ∩Bu k } F (u k ), u k − u = Ω(ǫ). Parts (a) and (b) of Lemma 3.1 certify that Algorithm 3 is optimal up to a logarithmic factor, due to Theorem 2.10. This is true because we can run Algorithm 3 with accuracy ǫ D to obtain max u∈U F (u k ), u k − u = O(ǫ) in k = O( LD 2 ǫ log( LD ǫ )) iterations, or with accuracy ǫ to obtain max u∈{U ∩Bu k } F (u k ), u k − u = O(ǫ) in k = O( LD ǫ log( LD ǫ )) iterations (see Proposition 2.7). Part (c) of Lemma 3.1 certifies that Algorithm 2 is optimal up to a log(D/ǫ) factor, due to Theorem 2.6. Part (d) certifies that the restarting algorithm from Theorem 2.12 is optimal up to a factor log(D/ǫ) log(L/m) whenever L = Ω(L/m). Note that L = Ω(L/m) can be ensured by a proper scaling of the problem instance, as any such scaling would leave the condition number L/m unaffected and would only impact the target error ǫ, which only appears under a logarithm. Conclusion We showed that variants of Halpern iteration can be used to obtain near-optimal methods for solving different classes of monotone inclusion problems with Lipschitz operators. The results highlight connections between monotone inclusion, variational inequalities, fixed points of nonexpansive maps, and proximal-point-type algorithms. Some interesting questions that merit further investigation remain. In particular, one open question that arises is to close the gap between the upper and lower bounds provided here. We conjecture that the optimal complexity of monotone inclusion is: (i) Θ( LD ǫ ) when the operator is either L-Lipschitz or 1 L -cocoercive, and (ii) Θ( L m log( LD ǫ )) when the operator is L-Lipschitz and m-strongly monotone. − u * . Assume further that F (u 1 ) − F (u 0 ), u 1 − u 0 ≥ 1 L 1 F (u 1 ) − F (u 0 ) 2 . If A k+1 C k+1 ≤ A k C k , ∀k ≥ 1, where {A i } i≥1 is a sequence of positive numbers that satisfies A 1 = 1, then: (∀k ≥ 1) : F (u k ) ≤ L k λ k 1 − λ k u 0 − u * . Proof. The statement holds trivially if F (u k ) = 0, so assume that F (u k ) > 0. Under the assumption of the lemma, we have that A k C k ≤ C 1 , ∀k ≥ 1. From (H) and λ 1 = 1 2 , u 1 = u 0 − 1 L 1 F (u 0 ), and thus: C 1 = 1 L 1 F (u 1 ) 2 − 1 L 1 F (u 1 ), F (u 0 ) . Let u * be an arbitrary solution to (MI) (and thus also to (MVI)). As F (u 1 ) − F (u 0 ), u 1 − u 0 ≥ 1 L 1 F (u 1 ) − F (u 0 ) 2 and u 1 = u 0 − 1 L 1 F (u 0 ), it follows that F (u 1 ) 2 ≤ F (u 0 ), F (u 1 ) , and, thus C 1 ≤ 0. Further, as A k > 0, we also have C k ≤ 0, and, hence: F (u k ) 2 ≤ L k λ k 1 − λ k F (u k ), u 0 − u k = L k λ k 1 − λ k F (u k ), u 0 − u * + L k λ k 1 − λ k F (u k ), u * − u k ≤ L k λ k 1 − λ k F (u k ), u 0 − u * ≤ L k λ k 1 − λ k F (u k ) · u 0 − u * , where the last line is by u * being a solution to (MVI) and by the Cauchy-Schwarz inequality. The conclusion of the lemma now follows by dividing both sides of F (u k ) 2 ≤ L k λ k 1−λ k F (u k ) · u 0 − u * by F (u k ) and observing that the statement holds for an arbitrary solution u * to (MI), and thus, it also holds for the one that minimizes the distance to u 0 . Lemma 2.2. Let C k be defined as in Eq. (2.1). Let {A i } i≥1 be defined recursively as A 1 = 1 and A k+1 = A k λ k (1−λ k )λ k+1 for k ≥ 1. Assume that {λ i } i≥1 is chosen so that λ 1 = 1 2 and for k ≥ 1 : λ k+1 1−2λ k+1 ≥ λ k L k (1−λ k )L k+1 . Finally, assume that L k ∈ (0, ∞) and F (u k ) − F (u k−1 ), u k − u k−1 ≥ 1 L k F (u k ) − F (u k−1 ) 2 , ∀k. Then, (∀k ≥ 1) : A k+1 C k+1 ≤ A k C k . Proof. By the assumption of the lemma, 1 L k+1 F (u k+1 ) − F (u k ) 2 ≤ F (u k+1 ) − F (u k ), u k+1 − u k , which, after expanding the left-hand side, can be equivalently written as: 1 L k+1 F (u k+1 ) 2 ≤ F (u k+1 ), u k+1 − u k + 2 L k+1 F (u k ) − F (u k ), u k+1 − u k + 1 L k+1 F (u k ) . From (H), we have that u k+1 − u k = λ k+1 1−λ k+1 (u 0 − u k+1 ) − 2 L k+1 F (u k ) and u k+1 − u k = λ k+1 (u 0 − u k ) − 2(1−λ k+1 ) L k+1 F (u k ). Hence: 1 L k+1 F (u k+1 ) 2 ≤ λ k+1 1 − λ k+1 F (u k+1 ), u 0 − u k+1 − λ k+1 F (u k ), u 0 − u k + 1 − 2λ k+1 L k+1 F (u k ) 2 . Rearranging the last inequality and multiplying both sides by A k+1 , we have: A k+1 1 L k+1 F (u k+1 ) 2 − λ k+1 1 − λ k+1 F (u k+1 ), u 0 − u k+1 ≤ A k+1 (1 − 2λ k+1 ) L k+1 F (u k ) 2 − A k+1 λ k+1 F (u k ), u 0 − u k . The left-hand side of the last inequality if precisely A k+1 C k+1 . The right-hand side is ≤ A k C k , by the choice of sequences {A i } i≥1 , {λ i } i≥1 . A.2 Operator Mapping Proposition 2.4. Let F be an 1 L -cocoercive operator and let G η be defined as in Eq. (1.1), where η ≥ L. Then G η is 1 2η -cocoercive. Proof. As Id − Π U is 1-cocoercive (by Fact 1.3), we have, ∀u, v ∈ E: Π U u − 1 η F (u) − Π U v − 1 η F (v) , u − 1 η F (u) − v − 1 η F (v) = 1 η (G η (v) − G η (u)) + u − v, u − v − 1 η (F (u) − F (v)) ≥ 1 η (G η (v) − G η (u)) + u − v 2 . Hence: 1 η 2 G η (v) − G η (u) 2 ≤ 1 η G η (u) − G η (v), u − v + 1 η 2 G η (u) − G η (v), F (u) − F (v) − 1 η F (u) − F (v), u − v . (A.1) As η ≥ L and F is 1 L -cocoercive, 1 η F (u) − F (v), u − v ≥ 1 η 2 F (u) − F (v) 2 . It remains to apply Young's inequality, which implies G η (u) − G η (v), F (u) − F (v) ≤ 1 2 G η (u) − G η (v) 2 + 1 2 F (u) − F (v) 2 . Algorithm 4: EG Without the Knowledge of L Input: a 0 , u 0 ∈ U , m, ǫ. If not provided at the input or > 1/m, set a 0 = 1/m. ; 1ū 0 = Π U (u 0 − a k F (u 0 )); 2 k = 0, δ 0 = a 0 mǫ 5 √ 2 ; 3 while ū k − u k > δ k do 4 k = k + 1, a k = a k−1 ; 5ū k = Π U (u k − a k F (u k )); 6 u k+1 = argmin u∈U a k F (ū k ), u + a k m 2 u −ū k 2 + 1 2 u − u k 2 ; 7 while a k F (ū k ) − F (u k ),ū k − u k+1 > 1 4 u k+1 −ū k 2 + 1 4 ū k − u k 2 do 8 a k = min a k 2 , ū k −u k F (ū k )−F (u k ) ; 9ū k = Π U (u k − a k F (u k )); 10 u k+1 = argmin u∈U a k F (ū k ), u + a k m 2 u −ū k 2 + 1 2 u − u k 2 ; end 11 δ k = a k mǫ 5 √ 2 ; end return u k A.3 Approximating the Resolvent Let us start by proving the convergence of a version of the Extragradient method of Korpelevich [1977] that does not require the knowledge of the Lipschitz constant L (but does require knowledge of the strong monotonicity parameter m; when computing the resolvent we have m = 1). The algorithm is summarized in Algorithm 4. Observe that the update step for u k from Lines 6 and 10 can be written in the form of a projection onto U ; we chose to write it in the current form as it is more convenient for the analysis. We now bound the convergence of Algorithm 4. Lemma A.1. Let a 0 > 0 and let F be m-strongly monotone and L-Lipschitz. Then, Algorithm 4 outputs a point u k with u k − u * ≤ ǫ after at most k = O L m log( L u 0 −u * mǫ ) oracle queries to F, where u * solves (SVI). Proof. Define A k = k i=0 a i . To prove the lemma, we will use the following gap (or merit) functions: f k = 1 A k k i=0 a i F (ū i ),ū i − u * − m 2 ū i − u * 2 . As F is strongly monotone, f k ≥ 0, ∀k. By convention, we take f −1 = 0 and A −1 = 0, so that A k f k − A k−1 f k−1 = a k F (ū k ),ū k − u * − m 2 ū k − u * 2 . Let us now bound A k f k − A k−1 f k−1 , and observe that A k f k − A k−1 f k−1 ≥ 0. First, write A k f k − A k−1 f k−1 = a k F (ū k ),ū k − u * − m 2 ū k − u * 2 = a k F (ū k ), u k+1 − u * + a k F (u k ),ū k − u k+1 + a k F (ū k ) − F (u k ),ū k − u k+1 − a k m 2 ū k − u * 2 . (A.2) By the first-order optimality of u k+1 in its definition, we have, ∀u : a k F (ū k ) + a k m(u k+1 −ū k ) + u k+1 − u k , u − u k+1 ≥ 0, and, thus: a k F (ū k ), u k+1 − u ≤ a k m u k+1 −ū k , u − u k+1 + u k+1 − u k , u − u k+1 . By the standard three-point identity (which can also be verified directly): u k+1 − u k , u − u k+1 = 1 2 u − u k 2 − 1 2 u − u k+1 2 − 1 2 u k − u k+1 2 . Thus, setting u = u * : a k F (ū k ), u k+1 − u * = a k m u k+1 −ū k , u * − u k+1 + 1 2 u * − u k 2 − 1 2 u * − u k+1 2 − 1 2 u k − u k+1 2 . Observe also that: u k+1 −ū k , u * − u k+1 = 1 2 u * −ū k 2 − 1 2 u k+1 −ū k 2 − u * − u k+1 2 . Thus, we have: a k F (ū k ), u k+1 − u * = 1 2 u * − u k 2 − 1 + a k m 2 u * − u k+1 2 − 1 2 u k − u k+1 2 + a k m 2 u * −ū k 2 − a k m 2 u k+1 −ū k 2 . (A.3) By similar arguments: a k F (u k ),ū k − u k+1 = 1 2 u k+1 − u k 2 − 1 2 u k+1 −ū k 2 − 1 2 ū k − u k 2 . (A.4) Combining Eq. (A.2)-(A.4): A k f k − A k−1 f k−1 = 1 2 u * − u k 2 − 1 + a k m 2 u * − u k+1 2 + a k F (ū k ) − F (u k ),ū k − u k+1 − 1 + a k m 2 u k+1 −ū k 2 − 1 2 ū k − u k 2 . By the condition of the while loop in Line 7 of Algorithm 4, and because A k f k − A k−1 f k−1 ≥ 0, 1 + a k m 2 u * − u k+1 2 + 1 + 2a k m 4 u k+1 −ū k 2 + 1 4 ū k − u k 2 ≤ 1 2 u * − u k 2 . (A.5) The condition of the while loop in Line 7 of Algorithm 4 is satisfied for any a k ≤ 1 2L , as a k F (ū k ) − F (u k ),ū k − u k+1 ≤ a k L ū k − u k · ū k − u k+1 ≤ a k L 2 ū k − u k 2 + ū k − u k+1 2 , where we have used the Cauchy-Schwarz inequality, the fact that F is L-Lipschitz, and the Young inequality. Thus, in any iteration, a k > 1 4L , and the total number of times the while loop from Line 7 is entered is at most log 2 (4L/a 0 ). From Eq. (A.5), u * − u k+1 2 ≤ 1 1+m/(4L) u * − u k 2 ≤ (1 − m 8L ) u * − u k 2 . Thus, for any δ > 0, u * − u k ≤ δ for k ≥ 16L m log( u * −u 0 δ ). Consequently, from Eq. (A.5), ū k − u k ≤ √ 2δ whenever u * − u k ≤ δ. In particular, for δ = a k mǫ 5 √ 2 ≥ mǫ 20 √ 2L , ū k − u k ≤ √ 2δ = a k m 5 ǫ after at most k = 16L m log( 20 √ 2L u * −u 0 mǫ (outer loop) iterations. It remains to show that when ū k − u k ≤ δ, u k − u * ≤ ǫ, and so Algorithm 4 terminates. Observe that u k −ū k = a k G 1/a k (u k ), where G 1/a k is the operator mapping defined in Eq. (2.2). Thus, using Lemma 2.5 and noting that a k ≤ 1/L loc = ū k −u k F (ū k )−F (u k ) , if ū k − u k ≤ a k m 5 ǫ, we have F (ū k ),ū k − u * ≤ 2m 5 ǫ ū k − u * . On the other hand, as F is m-strongly monotone, we also have F (ū k ),ū k − u * ≥ m 2 ū k − u * 2 . Hence, ū k − u * ≤ 4ǫ 5 . Finally, applying the triangle inequality and as a + k ≤ 1/m : u k − u * ≤ u k −ū k + ū k − u * ≤ ǫ 5 + 4ǫ 5 = ǫ. Note that we have already bounded the total number of inner and outer loop iterations. Observing that each inner iteration makes 2 oracle queries to F and each outer iteration makes 2 oracle queries to F outside of the inner iteration, the bound on the total number of oracle queries to F follows. Lemma 2.8. Letū * k = J F +I U (u k ), where u k ∈ U and F is L-Lipschitz. Then, there exists a parameterfree algorithm that queries F at most O((L + 1) log( (L+1) u k −ū * k ǫ ) ) times and outputs a pointū k such that ū k −ū * k ≤ ǫ. Proof. Observe first thatū * k solves (SVI) for operatorF (u) = F (u) + u − u k over the set U . This follows from the definition of the resolvent, which implies: u * k + F (ū * k ) + ∂I U (ū * k ) ∋ u k . Equivalently: 0 ∈F (ū * k ) + ∂I U (ū * k ). The rest of the proof follows by applying Lemma A.1 toF , which is (L + 1)-Lipschitz and 1-strongly monotone. A.4 Inexact Halpern Iteration We start by first proving the following auxiliary result. Proposition A.2. Given an initial point u 0 ∈ U , let u k evolve according to Eq. (2.3), where λ k = 1 k+1 . Then, (∀k ≥ 1) : u k − u * ≤ u 0 − u * + 1 k + 1 k i=1 i e i−1 , where u * is such that P (u * ) = 0. Proof. Let T = Id−P. Then T (u * ) = u * . By Fact 1.2, T is nonexpansive. Observe that we can equivalently write Eq. (2.3) as u k = λ k u 0 + (1 − λ k )T (u k−1 ) + (1 − λ k )e k−1 . Thus, using that u * = T (u * ): u k − u * = λ k (u 0 − u * ) + (1 − λ k )(T (u k−1 ) − T (u * )) + (1 − λ k )e k−1 ≤ λ k u 0 − u * + (1 − λ k ) u k−1 − u * + (1 − λ k ) e k−1 , where we have used the triangle inequality and nonexpansivity of T. The result follows by recursively applying the last inequality and observing that k j=i (1 − λ j ) = i k+1 . Using this proposition, we can now prove the following lemma. Lemma 2.9. Let C k be defined as in Eq. (2.1) with P as the 1 2 -cocoercive operator, and let L k = 2, λ k = 1 k+1 , and A k = k(k+1) 2 , ∀k ≥ 1. If the iterates u k evolve according to (2.3) for an arbitrary initial point u 0 ∈ U , then: (∀k ≥ 1) : A k+1 C k+1 ≤ A k C k + A k+1 e k , (1 − λ k+1 )P (u k ) − P (u k+1 ) . Further, if, ∀k ≥ 1, e k−1 ≤ ǫ 4k(k+1) , then P (u K ) ≤ ǫ after at most K = 4 u 0 −u * ǫ iterations. Proof. By the same arguments as in the proof of Lemma 2.1: 1 2 P (u k+1 ) 2 ≤ P (u k+1 ), u k+1 − u k + P (u k ) − P (u k ), u k+1 − u k + 1 2 P (u k ) . From (2.3) and the definition ofP , we have that u k+1 − u k = λ k+1 1 − λ k+1 (u 0 − u k+1 ) − P (u k ) − e k , and u k+1 − u k = λ k+1 (u 0 − u k ) − (1 − λ k+1 )P (u k ) − (1 − λ k+1 )e k . Hence: 1 2 P (u k+1 ) 2 ≤ λ k+1 1 − λ k+1 P (u k+1 ), u 0 − u k+1 − λ k+1 P (u k ), u 0 − u k + 1 − 2λ k+1 2 P (u k ) 2 + e k , (1 − λ k+1 )P (u k ) − P (u k+1 ) . Plugging λ k+1 = 1 k+2 in the last inequality and using the definition of C k and the choice of A k from the statement of the lemma completes the proof of the first part. Using the same arguments as in the proof of Lemma 2.2, we can conclude from A k+1 C k+1 ≤ A k C k + A k+1 e k , (1 − λ k+1 )P (u k ) − P (u k+1 ) , ∀k ≥ 1 that: P (u k ) 2 2 ≤ 1 k P (u k ) u 0 − u * + 1 A k k i=1 A i e i−1 , (1 − λ i )P (u i−1 ) − P (u i ) = 1 k P (u k ) u 0 − u * + 1 k(k + 1) k i=1 i(i + 1) e i−1 , i i + 1 P (u i−1 ) − P (u i ) . (A.6) Let us now bound each e i−1 , i i+1 P (u i−1 ) − P (u i ) term. Recall that P (u * ) = 0 and P is 2-Lipschitz (as discussed in Section 1.2, this follows from P being 1 2 -cocoercive). Thus, we have: e i−1 , i i + 1 P (u i−1 ) − P (u i ) = e i−1 , i i + 1 (P (u i−1 ) − P (u * )) − (P (u i ) − P (u * )) ≤ 2 e i−1 i i + 1 u i−1 − u * + u i − u * ≤ 2 e i−1 i + 2 i + 1 u 0 − u * + i i + 1 e i−1 + 2 i + 1 i−1 j=1 j e j−1 , where we have used Proposition A.2 in the last inequality. In particular, if e i−1 ≤ ǫ 4i(i+1) , then, ∀i ≥ 1: e i−1 , i i + 1 P (u i−1 ) − P (u i ) ≤ ǫ 2i(i + 1) i + 2 i + 1 u 0 − u * + ǫ/2 . Combining with Eq. (A.6): P (u k ) 2 2 ≤ 1 k P (u k ) u 0 − u * + ǫ 2k ( u 0 − u * + ǫ/2). (A.7) Observe that if u 0 − u * ≤ ǫ/2, as P is 2-Lipschitz and P (u * ) = 0, we would have P (u 0 ) ≤ ǫ, and the statement of the second part of the lemma would hold trivially. Assume from now on that u 0 − u * > ǫ/2. Suppose that P (u k ) > ǫ and k ≥ 4 u 0 −u * ǫ . Then, dividing both sides of Eq. (A.7) by P (u k ) /2 and using that P (u k ) > ǫ and u 0 − u * > ǫ/2, we get: P (u k ) < 2 u 0 − u * (1 + 1/2) k + 2 · ǫ/4 k < 3ǫ 4 + ǫ 4 ≤ ǫ, contradicting the assumption that P (u k ) > ǫ and completing the proof. A.5 Strongly Monotone Lipschitz Operators Theorem 2.12. Given F that is L-Lipschitz and m-strongly monotone, consider running the following algorithm A, starting withū 0 ∈ U : (A) : At iteration k, invoke Algorithm 3 with error parameter ǫ k = 7 16 P (u k−1 ) . Then, A outputs a point u k ∈ U with P (u k ) ≤ ǫ after at most log 2 ( u 0 − u * /ǫ) iterations, where w.l.o.g. ǫ ≤ 1 2 . The total number of oracle queries to F until this happens is O (L + L m ) log( u 0 − u * /ǫ) log(L + L m ) . Proof. The first part of the theorem is immediate, as each call to Algorithm 3 ensures, due to Theorem 2.10, that P (u k ) ≤ 7P (u k−1 ) 16 ≤ 7 P (u k−1 ) 16 + ǫ k 8 ≤ P (u k−1 ) 2 , and P (u 0 ) ≤ 2 u 0 − u * as P is 2-Lipschitz (because it is 1 2 -cocoercive) and P (u * ) = 0. It remains to bound the number of calls to F for each call to Algorithm 3. Using Theorem 2.10 and P (u k ) = Θ( P (u k ) ), each call to Algorithm 3 takes O( L u k−1 −u * P (u k−1 ) log( L u k−1 −u * P (u k−1 ) )) calls to F. De- noteū * k−1 = J F +∂I U (u k−1 ) = u k−1 − P (u k−1 ) . Using Proposition 2.7: F (ū * k−1 ),ū * k−1 − u * ≤ P (u k−1 ) ū * k−1 − u * . On the other hand, as F is m-strongly monotone and u * is an (MVI) solution, m ū * k−1 − u * 2 ≤ F (ū * k−1 ),ū * k−1 − u * . Hence: ū * k−1 −u * ≤ 1 m P (u k−1 ) . It remains to use the triangle inequality and P (u k−1 ) = u k−1 −ū * k−1 to obtain: u k−1 − u * ≤ 1 + 1 m P (u k−1 ) , (A.8) which completes the proof. A.6 Lower Bounds We make use of the lower bound from Ouyang and Xu [2019] and the algorithmic reductions between the problems considered in previous sections to derive (near-tight) lower bounds for all of the problems considered in this paper. The lower bounds are for deterministic algorithms working in a (first-order) oracle model. For convexconcave saddle-point problems with the objective Φ(x, y) and closed convex feasible set X × Y, any such algorithm A can be described as follows: in each iteration k, A queries a pair of points (x k ,ȳ k ) ∈ X × Y to obtain (∇ x Φ(x k ,ȳ k ), ∇ y Φ(x k ,ȳ k )), and outputs a candidate solution pair (x k , y k ) ∈ X × Y. Both the query points pair (x k ,ȳ k ) and the candidate solution pair (x k , y k ) can only depend on (i) global problem parameters (such as the Lipschitz constant of Φ's gradients or the feasible sets X , Y) and (ii) oracle queries and answers up to iteration k : {x i ,ȳ i , ∇ x Φ(x i ,ȳ i ), ∇ y Φ(x i ,ȳ i )} k−1 i=0 . We start by summarizing the result from [Ouyang and Xu, 2019, Theorem 9]. Theorem A.3. For any deterministic algorithm working in the first-order oracle model described above and any L, R X , R Y > 0, there exists a problem instance with a convex-concave function Φ(x, y) : X × Y → R whose gradients are L-Lipschitz, such that ∀k = O(d) : max y∈y Φ(x k , y) − min x∈X Φ(x, y k ) = Ω L(R X 2 + R X R Y ) k , where (x k , y k ) ∈ X × Y is the algorithm output after k iterations and R X , R Y denote the diameters of the feasible sets X , Y, respectively, and where both X , Y, are closed and convex. The assumption of the theorem that k = O(d) means that the lower bound applies in the high-dimensional regime d = Ω( L(R X 2 +R X R Y ) ǫ ), which is standard and generally unavoidable. In the setting of VIs, we consider a related model in which an algorithm has oracle access to F and refer to it as the operator oracle model. Similarly as for the saddle-point problems, we consider deterministic algorithms that on a given problem instance described by (F, U ) operate as follows: in each iteration k the algorithm queries a pointū k ∈ U , receives F (ū k ), and outputs a solution candidate u k ∈ U . Both u k and u k can only depend on (i) global problem parameters (such as the feasible set U and the Lipschitz parameter of F ), and (ii) oracle queries and answers up to iteration k : {ū i , F (ū i )} k−1 i=0 . Note that all methods described in this paper and most of the commonly used methods for solving VIs, such as, e.g., the mirror-prox method of Nemirovski [2004] and dual extrapolation method of Nesterov [2007], work in this oracle model. Proof. Proof of (a): Suppose that this claim was not true. Then we would be able to solve any instance with L-Lipschitz F and U with diameter bounded by D and obtain u k with max u∈U F (u k ), u k − u ≤ ǫ in o( LD 2 ǫ ) iterations, assuming the appropriate high-dimensional regime. In particular, given any fixed convexconcave Φ(x, y) with L-Lipschitz gradients and feasible sets X , Y whose diameter is D/2, let u = [ x y ], F (u) = [ ∇xΦ(x,y) −∇yΦ(x,y) ], U = X × Y. Then, it is not hard to verify that F is monotone and L-Lipschitz (see, e.g., Nemirovski [2004], Facchinei and Pang [2003]) and the diameter of U is D. Thus, by assumption, we would be able to construct a point u k = [ x k y k ] for which max u∈U F (u k ), u k − u ≤ ǫ in o( LD 2 ǫ ) iterations. But then, because Φ is convex-concave, we would also have, for any x ∈ X , y ∈ Y: Φ(x k , y k ) − Φ(x, y k ) = max y∈Y Φ(x k , y k ) − Φ(x k , y k ) + Φ(x k , y k ) − min x∈X Φ(x, y k ) ≤ ∇ y Φ(x k , y k ), y − y k + ∇ x Φ(x k , y k ), x k − y k = F (u k ), u k − u . In particular, we would get: max y∈Y Φ(x k , y k ) − min x∈X Φ(x, y k ) ≤ max u∈U F (u k ), u k − u ≤ ǫ. Because we obtained this bound for an arbitrary L-Lipschitz convex-concave Φ and arbitrary feasible sets X , Y with diameters D/2, Theorem A.3 leads to a contradiction. Proof of (b): If (b) was not true, then we would be able to obtain a point u k with max u∈{U ∩Bu k } F (u k ), u k − u = o(ǫ/D) in k = LD 2 ǫ iterations. But the same point would satisfy max u∈U F (u k ), u k − u = o(ǫ), which is a contradiction, due to (a). Proof of (c): We prove the claim for L = 2. This is w.l.o.g., due to the standard rescaling argument: if F is 1 L -cocoercive, thenF = F/(2L) is 1 2 -cocoercive. Further, if, for some u k ∈ U , max u∈{U ∩Bu k } F (u k ), u k − u = Ω(ǫ), then max u∈{U ∩Bu k } F (u k ), u k − u = Ω(Lǫ). Suppose that the claim was not true for a 1 2 -cocoercive operator F. Then for any M -Lipschitz monotone operator G, we would be able to use the strategy from Section 2.2 to obtain a point u k with max u∈{U ∩Bu k } G(u k ), u k − u = o(ǫ) in k = M D ǫ iterations. This is a contradiction, due to (b). Proof of (d): Suppose that the claim was not true, i.e., that there existed an algorithm that, for any m, L > 0, could output u k with max u∈{U ∩Bu k } F (u k ), u k − u = ǫ/2 in k = o(L/m) iterations, for any m-strongly monotone and L-Lipschitz operator. Then for any L-Lipschitz monotone operator F , we could apply that algorithm toF (·) = F (·) + ǫ 2D (· − u 0 ) to obtain a point u k with max u∈{U ∩Bu k } F (u k ), u k − u = ǫ/2 in k = o(LD/ǫ) iterations. But then we would also have: max u∈{U ∩Bu k } F (u k ), u k − u = max u∈{U ∩Bu k } F (u k ) − ǫ 2D (u k − u 0 ), u k − u ≤ ǫ, which is a contradiction, due to (b). [2011], Mokhtari et al. [2019], Gidel et al. [2019] if L is known, or Stonyakin et al. [2018], if L is not known. For completeness, we provide a simple modification to the Extragradient algorithm of Korpelevich Theorem 2. 10 . 10Let F be a monotone and L-Lipschitz operator and let u 0 ∈ U be an arbitrary initial point. For any ǫ > 0, Algorithm 3 outputs a point with P (u k ) ≤ ǫ after at most 8 u * −u 0 ǫ iterations, where each iteration can be implemented with O((L + 1) log( (L+1) u 0 −u * ǫ ) oracle queries to F. Hence, the total number of oracle queries to F is: Lemma 3. 1 . 1For any deterministic algorithm working in the operator oracle model described above and any L, D > 0, there exists a VI described by an L-Lipschitz-continuous operator F and a closed convex feasible set U with diameter D such that:(a) For all ǫ > 0 such that k = LD 2 ǫ = O(d), max u∈U F (u k ), u k − u = Ω(ǫ); (b) For all ǫ > 0 such that k = LD ǫ = O(d), max u∈{U ∩Bu k } F (u k ), u k − u = Ω(ǫ); (c) If F is 1 L -cocoercive, then for all ǫ > 0 such that k = LD ǫ log(D/ǫ) = O(d), it holds that max u∈{U ∩Bu k } F (u k ), u k − u = Ω(ǫ) (d) If F is m-strongly monotone, then for all ǫ > 0 such that k = L m = O(d), it holds that max u∈{U ∩Bu k } F (u k ), u k − u = Ω(ǫ). Ulrich Kohlenbach. On quantitative versions of theorems due to fe browder and r. wittmann. http://www.optimization-online.org/DB_FILE/2017/11/6336.pdf. Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In Proc. ICLR'18, 2018. Aryan Mokhtari, Asuman Ozdaglar, and Sarath Pattathil. A unified analysis of extra-gradient and optimistic gradient methods for saddle point problems: Proximal point approach. arXiv preprint arXiv:1901.08511, 2019.Advances in Mathematics, 226(3):2764-2795, 2011. Daniel Körnlein. Quantitative results for halpern iterations of nonexpansive mappings. Journal of Mathe- matical Analysis and Applications, 428(2):1161-1172, 2015. GM Korpelevich. Extragradient method for finding saddle points and other problems. Matekon, 13(4): 35-49, 1977. Laurentiu Leustean. Rates of asymptotic regularity for halpern iterations of nonexpansive mappings. Journal of Universal Computer Science, 13(11):1680-1691, 2007. Felix Lieder. On the convergence rate of the Halpern-iteration, 2017. Hongzhou Lin, Julien Mairal, and Zaid Harchaoui. Catalyst acceleration for first-order convex optimization: From theory to practice. The Journal of Machine Learning Research, 18(1):7854-7907, 2017. Qihang Lin, Mingrui Liu, Hassan Rafique, and Tianbao Yang. Solving weakly-convex-weakly-concave saddle-point problems as weakly-monotone variational inequality. arXiv preprint arXiv:1810.10207, 2018. Arkadi Nemirovski. Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM Journal on Optimization, 15(1):229-251, 2004. Yurii Nesterov. Dual extrapolation and its applications to solving variational inequalities and related prob- lems. Mathematical Programming, 109(2-3):319-344, 2007. Yurii Nesterov. How to make the gradients small. Optima. Mathematical Optimization Society Newsletter, (88):10-11, 2012. Yurii Nesterov. Lectures on convex optimization, volume 137. Springer, 2018. Yurii Nesterov and Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete & Continuous Dynamical Systems-A, 31(4):1383-1396, 2011. Yuyuan Ouyang and Yangyang Xu. Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems. Mathematical Programming, Aug 2019. Ernest K Ryu and Stephen Boyd. Primer on monotone operator methods. Applied and Computational Mathematics, 15(1):3-43, 2016. Ernest K Ryu, Kun Yuan, and Wotao Yin. ODE analysis of stochastic gradient methods with optimism and anchoring for minimax problems and GANs. arXiv preprint arXiv:1905.10899, 2019. Fedor Stonyakin, Alexander Gasnikov, Pavel Dvurechensky, Mohammad Alkousa, and Alexander Titov. Generalized mirror prox for monotone variational inequalities: Universality and inexact oracle. arXiv preprint arXiv:1806.05140, 2018. Rainer Wittmann. Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik, 58 (5):486-491, 1992. Hong-Kun Xu. Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 66(1):240-256, 2002. A Omitted Proofs A.1 Unconstrained Setting with a Cocoercive Operator Lemma 2.1. Let C k be defined as in Eq. (2.1) and let u * be the solution to (MI) that minimizes u 0 AcknowledgementsWe thank Prof. Ulrich Kohlenbach for useful comments and pointers to the literature. We also thank Howard Heaton for pointing out a typo in the proof of Lemma 2.1 in a previous version of this paper. Towards principled methods for training generative adversarial networks. Martin Arjovsky, Leon Bottou, Proc. ICLR'17. ICLR'17Martin Arjovsky and Leon Bottou. Towards principled methods for training generative adversarial networks. In Proc. ICLR'17, 2017. . Martin Arjovsky, Soumith Chintala, Léon Bottou, Gan Wasserstein, arXiv:1701.07875arXiv preprintMartin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein GAN. arXiv preprint arXiv:1701.07875, 2017. 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Stochastic model-based minimization of weakly convex functions. SIAM Journal on Optimization, 29(1):207-239, 2019. Performance of first-order methods for smooth convex minimization: a novel approach. Yoel Drori, Marc Teboulle, Mathematical Programming. 1451-2Yoel Drori and Marc Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451-482, 2014. Finite-dimensional variational inequalities and complementarity problems. Francisco Facchinei, Jong-Shi Pang, Springer Science & Business MediaFrancisco Facchinei and Jong-Shi Pang. Finite-dimensional variational inequalities and complementarity problems. Springer Science & Business Media, 2003. Accelerated gradient methods for nonconvex nonlinear and stochastic programming. Saeed Ghadimi, Guanghui Lan, Mathematical Programming. 1561-2Saeed Ghadimi and Guanghui Lan. Accelerated gradient methods for nonconvex nonlinear and stochastic programming. 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Performance of cesium iodide (CsI) thin film photocathode is reported in the present work. Optical absorbance CsI have been analyzed in the spectral range of 190 nm to 900 nm. The optical energy band gap of CsI films are calculated using Tauc plot from absorbance data. The values refractive index was estimated using envelope plot of transmittance data, proposed by Swanepoel. Absolute quantum efficiency (QE) measurement of CsI have been carried out in the wavelength range of 150 nm to 200 nm. Crystallographic nature and surface morphology of CsI investigated by the means of X-ray diffraction (XRD), transmission electron microscopy (TEM) and atomic force microscopy (AFM) are also presented. In addition, elemental composition result of CsI thin film is gained by energy dispersive X-ray analysis (EDAX) is also reported in the present work.

10.1016/j.nima.2015.02.059

1409.5731

11927b1111d7b88542c03ada6c18f6896e11790f

Optical and structural properties of CsI thin film photocathode 19 Sep 2014 R Triloki Department of Physics High Energy Physics laboratory Banaras Hindu University 221005VaranasiINDIA B K Rai Department of Physics High Energy Physics laboratory Banaras Hindu University 221005VaranasiINDIA Singh Department of Physics High Energy Physics laboratory Banaras Hindu University 221005VaranasiINDIA Optical and structural properties of CsI thin film photocathode 19 Sep 2014Cesium iodidequantum efficiencyXRDTEMAFMEDAXabsorbancetransmittanceenergy band gaprefractive indextexture coefficientcrystallite size and grain size Performance of cesium iodide (CsI) thin film photocathode is reported in the present work. Optical absorbance CsI have been analyzed in the spectral range of 190 nm to 900 nm. The optical energy band gap of CsI films are calculated using Tauc plot from absorbance data. The values refractive index was estimated using envelope plot of transmittance data, proposed by Swanepoel. Absolute quantum efficiency (QE) measurement of CsI have been carried out in the wavelength range of 150 nm to 200 nm. Crystallographic nature and surface morphology of CsI investigated by the means of X-ray diffraction (XRD), transmission electron microscopy (TEM) and atomic force microscopy (AFM) are also presented. In addition, elemental composition result of CsI thin film is gained by energy dispersive X-ray analysis (EDAX) is also reported in the present work. Introduction Photocathode devices in the soft X-ray and ultraviolet (UV) wavelength range are very important in the particle physics experiments for particle identification [1,2,3,4]. In soft X-ray and UV wavelength regions alkali halide photocathodes are known to be very efficient photo converters. Cesium Iodide (CsI) is one of the most efficient among them, because CsI photocathode is relatively stable under short exposure to air and has the highest QE among alkali halide photocathodes [5]. Therefore it is widely used in many UV-detecting devices [6,7]. These devices consist films of thicknesses varying from few nanometer (nm) to micrometer (µm), depending upon the mode of operation and application of photocathode. It is very much important to know the absorbance, transmittance and refractive index as a function of wavelength to predict the photoemissive behavior of a photocathode device. Knowledge of these optical constants is also necessary to determine the optical energy band gap of the film. In the present work, optical absorbance of CsI thin films were measured and analyzed. Optical transmittance and energy band gap were estimated from optical absorbance data of CsI thin films. Dispersive behavior of 500 nm thick CsI film was studied by using refractive index. The value of refractive index has been determined by using Swanepoel's method. Photoemission properties of 500 nm thick CsI photocathode is shown in the spectral range of 150 nm to 200 nm. Structural, morphological and elemental composition analysis of CsI film are also reported in the present work. Experimental Details CsI thin films evaporation was carried out into a high vacuum stainless steel chamber. Evaporation chamber was pumped with a turbo-molecular pump (Model: TMU 521 P from Pfeiffer vacuum) having a pumping speed 510 Ltr/Sec for N 2 gas. Prior to CsI evaporation residual atmosphere of the chamber was monitored through a residual gas analyzer (RGA) (model: SRS RGA 300), under a high vacuum environment (3 × 10 −7 Torr). It is observed that large amount of water molecules have been evacuated from the evaporation chamber after 08 hours of pumping (see Figure 1) and main constituent of residual gases (at partial pressure 3 × 10 −7 ) were N 2 (58.2%), H 2 (11.0%), H 2 O(16.2%), O 2 (9.9%) and CO 2 (1.4%). After 08 hours of pumping, the majority of residual gas was N 2 , which does not affect CsI photocathode during the film preparation. Small amount of CsI crystals (Alfa Aesar, 5N purity) were placed in a tantalum (Ta) boat inside the vacuum chamber and carefully heated to allow out gassing from the outer surface of the CsI crystals. After proper out gassing and melting, CsI thin films deposited from Ta boat on quartz, Aluminium (Al) and formvar coated copper grid substrates. For uniform deposition, distance between Ta boat and substrate was kept about 20 cm. The films were deposited at a typical rate of 1 nm to 2 nm per second. The thickness of the films was controlled by a quartz crystal thickness monitor (sycon STM 100 thickness/rate monitor). After the sample preparation, vacuum chamber was purged with dry nitrogen gas, in order to avoid the interaction of water vapor present in humid air to the prepared sample. Immediately after the chamber opening under constant flow of nitrogen gas, CsI films were placed into a vacuum desiccator and further moved to characterization setup. The schematic diagram of experimental setup for QE measurement is shown in Figure 2. The experimental setup for quantum efficiency (QE) measurement, includes a high vacuum stainless steel (S.S.) chamber (pumped with a TMP to 10 −5 Torr), coupled to a vacuum ultra violet (VUV) monochromator (MC)( model -234/302 VUV monochromator, McPherson). The experimental setup is equipped with a 30 Watt (W) magnesium fluoride (MgF2) windowed deuterium (D 2 ) lamp (model 632 deuterium light source, having entire spectral range from 113 nm to 380 nm). The QE was measured in a reflective mode, under vacuum. A positive voltage was applied from high voltage power supply (CAEN -N471A) to a mesh electrode placed at a distance of ∼ 3 mm from the photocathode surface. The photocurrent, induced by monochromatic UV photons to photocathode, was recorded from a picoamme- ter (Keithley -6485). The absolute QE value, which is the ratio of number of emitted photoelectrons (N e ) to number of incident photons (N p ) i.e. QE = N e /N p , derived from the ratio of the current measured from the photocathode to the current measured from a calibrated photomultiplier (Cal. PMT). This was done by alternatively directing the UV beam to both Cal. PMT interchangeable via a photocathode. This PMT was calibrated against a NIST vacuum-photodiode [8]. The stability of D 2 lamp was monitored throughout the measurements by a second reference photomultiplier (Ref. PMT) (Hamamatsu PMT, Model: 658), of the same type, and the measured photocurrent values were corrected correspondingly. Optical properties measurement of CsI thin film photocathodes were carried out using Perkin Elmer λ25 UV/Vis spectrophotometer in the wavelength range of 190 nm to 900 nm. Structural properties was studied by using X'Pert PRO PANalytical X-Ray diffractometer (XRD), based on Brag-Brentano para-focusing geometry, operated at 30 kV and 40 mA with Cukα(λ = 1.54056 Å) radiation. CsI film deposited on formvar coated copper (Cu) grid was moved to transmission electron microscopy (TEM) lab where morphological and structural features of the CsI thin film was studied by FEI Technai 20 G 2 TEM operated at 200 kV accelerating voltage. Elemental composition of CsI photocathode was studied by means of energy dispersive X-ray spectroscopy (EDAX) technique. Optical properties of CsI thin films Optical absorbance UV/Vis absorption of CsI films, deposited on quartz (Qz) substrate, (due to its transparency in the spectral region of 190 nm to 900 nm), performed in spectral range 190 nm to 900 nm as shown in Figure 3. It is observed that the absorbance of CsI films varies in between 0 to 2 for the thicknesses 10 nm to 100 nm, while for thicknesses more than 100 nm absorbance varies in between 0 to 3.5 (shown in inset of Figure 3). Two strong absorption peaks was observed in the UV-region at a wavelength smaller than 225 nm for thinner CsI films. Similar optical absorbance results are reported in previous literature [9,10,11] for thinner and thicker CsI films. The absorption coefficient (α) was also calculated from the absorption spectrum using the relation: α = 1 t ln 1 T (1) where t is the thickness of the film and T is the transmittance of the film. Absorption coefficient (α), estimated using equation (1), lies in between 0.02 to 0.04. Optical transmittance Optical transmittance of CsI films are derived from absorbance data in the wavelength region of 190 nm to 900 nm. Transmittance of CsI films is shown in Figure 4, are in good agreement with the previous reported work [12,13]. Transmittance of CsI films have been derived from the absorbance data using the relation: T = exp(−A)(2) Where A is the absorbance. Several transmittance peaks are observed in the wavelength region 190 nm to 900 nm, as already shown in reference [12,13]. CsI films of thicknesses more than 100 nm, was found to be opaque in the spectral region 190 nm to 225 nm, having transmittance is about only 2-3% (see inset of Figure 4). While CsI film of thicknesses below 50 nm, found to be semitransparent in the spectral region 190 nm to 225 nm, where transmittance varies from 20% to 40%. A sharp increase in transmittance was observed near a wavelength λ ≈ 225 nm indicates crystalline nature of CsI films. Thinner and thicker CsI films are found to be transparent in the spectral region 225 nm to 900 nm, having more than 80% transmittance. The surface quality and homogeneity of CsI films are analyzed from the existence of interference fringes (oscillatory nature in transmittance spectra) in the transmittance spectra. In transparent spectral region (λ > 225) nm, for thinner and thicker CsI films shows distinct characteristics, which impute to inhomogeneities in the films. In transparent spectral region CsI film of thicknesses smaller than 100 nm does not shows any interference fringes pattern in the transmission spectrum, which reveals that the CsI layer does not appear to be continuous, exhibiting small surface area coverage. While CsI film of thicknesses 100 nm and more shows interference fringes patterns, which indicates existence of continuous and homogeneous CsI layers, exhibiting large surface area coverage (see Figure 4). It is also observed fro Figure 4, that oscillatory nature in transmittance spectra of CsI films increases with an increase in the thickness of the film. Oscillatory nature in transmittance spectra of CsI films indicates that homogeneity, continuity surface area coverage, of CsI films increases with an increase in the thickness of the film. 500 nm thick CsI film found to be more homogeneous and continuous, than the thinner CsI films. Transmission spectrum for 500 nm thick CsI film (shown in Figure 5) depict a sharp fall in transmission near the fundamental absorption, which is an identification for the good crystallinity of film [14,15,16,17]. The oscillatory nature of the transmission spectrum observed for 500 nm thick CsI film is attributed to the interference of light transmitted through the thin film and the substrate. Optical energy band gap for 500 nm thick CsI film The energy band gap of the photocathode is one of the key parameter determining the range of its most efficient operation, in particular the sensitivity cutoff. In addition to proper band gap energy of a good photocathode material should allow an efficient electron transport to the emission surface and should have low or negative work function/electron affinity. The absorption in the UV region is attributed to energy band gap absorption of CsI thin film. An obvious increase in the absorption of wavelength less than ∼225 nm (see inset of Figure 6)can be assigned to the intrinsic band gap absorption of CsI film due to the electron transmission from the valence band to conduction band. The absorption band gap (E g ) has been calculated by using the Tauc relation [18,19,20]. (αhν) n = A(hν − E g )(3) where A is the edge width parameter, h is the Planck's constant, ν is the frequency of vibration, hν is the photon energy,α is the absorption coefficient, E g is the band gap and n is either 2 for direct band transitions or 1/2 for indirect band transitions [21]. The direct band gap energy estimated from a Tauc plot of (αhν) 2 versus photon energy hν according to the K. M. Model is shown in Figure 6. The value of photon energy (hν) extrapolated to α = 0 gives an absorption edge which corresponds to a band gap E g . The extrapolation gives band gap E g ≈ 5.4eV corresponds to absorption peak of 500 nm thick CsI film. The energy band gap determined from Tauc relation can be compared with energy band gap E g = 5.9eV derived from experimental QE dependence on wavelength [22] for heat-enhanced CsI thick film photocathode. Figure 6: Variation of (αhν) 2 vs. photon energy hν and absorbance as a function of wavelength (inset) for 500 nm thick CsI film. Determination of refractive index The optical properties of 500 nm thick CsI film can be evaluated from transmittance data using the method proposed by Swanepoel [23,24]. The applicability of this method is limited to thin film deposited on transparent substrate much thicker than the CsI film. The application of this method entails, as a first step, the calculation of the maximum T Max and minimum T min transmittance envelope curves by parabolic interpolation to the experimentally determined positions of peaks and valleys (shown in inset of Figure 5). From maximum of transmittance (T Max ) and minimum of transmittance (T min ), value of refractive index (n λ ) is determined by using the expression proposed by Swanepoel [23] is given below: n = N + N 2 − n 2 s(4) In the weak and medium absorption regions, the value of N is given by N = 2n s T Max − T min T Max T min + n 2 s + 1 2(5) with n s being the refractive index of the substrate. In general, n s is determined by the maximum of the transmission in the transparent region T Max [25] using the relation: n s = 1 T Max +       1 T 2 Max − 1      (6) It is observed that the value of refractive index decreases with increasing wavelength as shown in Figure 7. Table 1 shows the values at the extremes of the spectrum of λ, T Max and T min obtained from envelope plot of Figure 5 (see zoomed view, inset image). The values of refractive index n λ calculated from equation (4) is shown in table 1. The variation of the refractive index n λ with the wavelengths is shown in Figure 7. We observe a sharp fall in refractive index at lower wavelength side and a gradual destruction in refractive index corresponds to higher wavelength side. This variation in refractive index indicates normal dispersive behavior of 500 nm thick CsI film. Table 1: Values of λ, T Max and T min for the transmittance spectrum of CsI ( Figure 6) and the value of refractive index n λ to the corresponding wavelength λ: Photoemission properties CsI film Photoemission properties of 500 nm thick thermally evaporated CsI film is studied in wavelength region of 150 nm to 200 nm, with a scan step size of 2 nm. Absolute QE, which is the ratio of emitted photoelectrons to incident photons, is determined by illuminating the CsI surfaces with photon flux of a given frequency and the resulting photocurrent is measured by a picoammeter (Keithley-6485). Because the observables are current, it is necessary to relate these to QE, using the relation: QE(%) = I pc I pm × QE pm × G pmt(7) Where, I pm is the PMT current, I pc is the photocathode current, QE pm is known QE of calibrated PMT and G pm is gain of PMT. It is clearly observed that from the plot of Figure 8, maximum QE obtained is ∼40% at wavelength 150 nm. The QE was found to decrease with an increase in wavelength of incident photon. experimentally determined QE is in good agreement with the most of literature data for the CsI photocathode [26,27]. Structural properties of CsI film Crystallographic analysis The crystal structure and orientation of the CsI thin film was investigated by X-ray diffraction (XRD) patterns. Figure 9 shows the typical X-ray diffraction patterns of 500 nm thick CsI film deposited on Al substrate. XRD patterns indicate that the CsI film is purely and (220) crystallographic planes respectively. As CsI is deposited on Al substrate, so XRD pattern also contains four Al peaks as shown in Figure 9. The crystallographic orientations eventually obtained allow to attribute a body centered cubic (bcc) structure to the CsI. The lattice constant of crystalline CsI film is calculated using the analytical relation (for cubic crystal system): a = d × (h 2 + k 2 + l 2 )(8) The lattice constant for crystalline CsI thin film is found to be about a = 4.66Å, which is in good agreement with lattice constant reported in International Center for Diffraction Data (ICDD, File number -060311). The crystallite size is calculated using a well known Scherrer's equation [28,29]. D = kλ βcosθ(9) where D is the size of crystallite, k(=0.9) is the crystal constant, λ(= 1.5406Å) is the wavelength of X-ray used, β is the broadening of diffraction line measured at half of its maximum intensity in radians and θ is the angle of diffraction. The crystallite size obtained for most intense (110) crystallographic plane of CsI thin film is about 55 nm, which matches very well with previous reported articles [30,31] Table 2 shows the values of Bragg's angle (2θ), inter planar spacing(d), full width at half maximum(FWHM), texture coefficient (TC) and crystallite size(D), corresponds to their lattice (hkl) planes for 500 nm thick CsI film. The XRD pattern of CsI film shows a highly intense peak at 2θ = 27.06 indicating a strong preferred orientation along the (110) plane. The texture coefficient (TC) of CsI determined from XRD data represents the texture of a particular plane. The deviation in TC from unity implies the preferred growth. The texture coefficient factor can be calculated for each crystallite orien- tation was using the following equation [32]: TC(hkl) = I(hkl)/I 0 (hkl) N −1 n I(hkl)/I 0 (hkl) × 100%(10) where TC(hkl) is the texture coefficient, I(hkl) is the measured relative intensity of a plane (hkl) and I 0 (hkl) is the standard intensity of the plane (hkl) taken from the ICDD (card no: 060311) data and 'n' is the number of reflections taken into account. By using the above equation, the preferred orientation of the lattice plane can be understood. TC(hkl) is expected to be unity for films with randomly oriented crystallites, while higher values indicate the abundance of grains oriented in a given (hkl) direction. The variation of TC(hkl) and crystallite size for the peaks of the CsI film is presented in Table 2. TC(hkl) and average crystallite size of different planes of CsI film are shown in Figure 10. Average crystallite size corresponds to the various crystal plane of CsI is found to be about 47 nm which varies from 43 nm to 55 nm. It is clear from the plot that the average pertaining to (110) reflection, is higher than that of the other planes. This indicates that the preferential orientation of (110) plane in CsI film. Morphological analysis of CsI film A transmission electron microscope (TEM) is a powerful microscope that produces a high-resolution, black and white image from the interaction that takes place between samples and energetic electrons in the vacuum chamber. TEM measurement of 500 nm CsI film deposited on formvar coated copper (Cu) grid was carried out. In order to observe the surface morphology of CsI film, few regions were scanned by TEM, one of them is shown in Figure 11 (top panel). Figure 11(top panel) shows that CsI film surfaces have homogeneous and continuous grain like morphology, with more than 95% surface area was covered by CsI grains. It is observed that 500 nm thick CsI film have grains of various sizes, ranges from 110 nm to 860 nm. Average grain size of CsI film is found to be about 300 nm, as shown in histogram of Figure 11 (bottom panel). The electron diffraction pattern shown in inset of Figure 11(top panel)) indicate that the 500 nm thick CsI film is crystalline in nature and having single crystal like domains. A cubic structure with lattice constant a = 4.66 Å was found to have good match between experimental and calculated values of inter-planar distances. Elemental composition analysis of CsI film The energy dispersive X-ray (EDAX) is used to detect elements present in considerable amount (quantitative determination of bulk element composition) of thermally evaporated CsI thin film. The EDAX analysis of CsI film deposited on formvar coated Cu grid, carried out on TECHNAI 20 G 2 TEM system operated at 200 kV accelerating voltage. EDAX spectrum (shown in Figure 12) represents the different elements present in the CsI film. The data are shown with no smoothing, filtering or processing of any kind. The EDAX spectrum shows a clear peaks corresponding to the 55 Cs K (30.97 KeV) line, 55 Cs L line (4.23 KeV), 53 I K line (28.51 KeV) and 53 I L line (3.94 KeV). EDAX spectrum of CsI film exhibits the cesium(Cs) and iodine(I) signals peak of the elements presented with an approximately atomic ratio of Cs and I of 51:49 (thus the Cs:I ratio is fount to be ∼1:1), which is consistent with the stoichiometry of CsI. The observed 29 Cu K line (8 KeV), 29 Cu L line (1.5 KeV) and 6 C K line (1 KeV) peaks in the EDAX spectrum seen to be originated from the copper grid that was used for monitoring the sample in the TEM machine. No either peaks were observed over the entire 0 Kev to 35 KeV detection windows. Conclusions Photoemissive properties of thermally evaporated 500 nm thick CsI film was investigated by the means of VUV monochromator. Photoemission measurement was performed in the wavelength range of 150 nm to 200 nm and the maximum quantum efficiency achieved is ∼ 40% at the wavelength λ = 150 nm. Optical properties measurement has been performed on Perkin Elmer λ 25 UV-Vis spectrometer. Optical absorption of CsI thin film deposited on quartz substrate was performed in the spectral range 190 nm to 900 nm. Two strong absorption peaks is observed in the UV wavelength region, one is at 207 nm and another is at 222 nm. Transmittance of CsI film has been derived from absorbance data and it is found that CsI film having only about 3% transmittance in the spectral range of 190 to 225 nm, which reveals that 500 nm thick CsI film is opaque in the spectral range 190 to 225 nm. However in the spectral range of 225 nm 900 nm CsI film having more than 80% transmittance, which shows CsI film is transparent in the spectral range of 225 nm 900 nm. Appearance of interference fringes pattern in transparent spectral region indicates, existence of continuous and homogeneous grain like morphology with maximum surface area coverage for 500 nm thick CsI film. Optical energy band gap has been calculated from absorbance data using the Tauc plot of K. M. model and found to be about 5.4 eV. The values of refractive index calculated from envelop plot of optical transmittance data, varies from 1.82 to 1.30 in the spectral range of 275 nm to 900 nm. This variation of refractive index indicates, dispersive behavior of CsI film. Surface morphology, bulk structure as well as crystallographic nature of CsI was studied by the means of TEM and XRD techniques. TEM results reveals that the CsI film have homogeneous and continuous grain like morphology, with more than 95% surface area coverage by CsI grains. Average grain size (composed of many coherent domains) of CsI thin film, obtained from TEM micro structure is about 300 nm. Diffraction pattern of obtained from XRD and TEM measurement reveals that CsI thin film is purely crystalline in nature and having body centered cubic (bcc) structure. The value of lattice constant obtained is about a = 4.66 Å. The coherent scattering domain size (crystallite size) calculated using Scherrer's method is found to be about 55 nm. EDAX result indicates that, CsI film having mainly Cs and I elements. The atomic ratio of Cs and I is found to 1:1, which is consistent with the stoichiometry of CsI. Figure 1 : 1Residual gas composition inside the vacuum chamber: After 30 minutes of pumping (left panel) and after 08 hours of pumping (right panel). Figure 2 : 2A schematic view of the experimental set-up of the 234/302 VUV Monochromator equipped with deuterium lamp, PMT and collimating optics. CsI as a function of wavelengthTransmittance of CsI as a function of wavelength Figure 3 : 3Optical absorption of CsI thin films in the wavelength range of 190 nm to 900 nm and zoomed view of absorbance in UV spectral region (inset) Figure 4 : 4Optical transmittance of CsI thin film in the wavelength range 190 nm to 900 nm. Figure 5 : 5Transmission spectrum of CsI thin film (solid line), including the maximum (T Max ) and minimum (T min ) transmittance envelope curves (dashed and doted lines). Figure 7 : 7Refractive index as a function of wavelength for 500 nm thick CsI film deposited on quartz substrate. Figure 8 : 8Absolute quantum efficiency as a function of wavelength for 500 nm thick CsI photocathode deposited on Al disc. crystalline in nature. XRD pattern of CsI deposited on Al substrate, contains an intense peak at Bragg's angle 2θ = 27.06, assigned to (110) crystallographic plane and three other XRD peaks of CsI found at Bragg's angle 2θ = 38.87, 48.30and56.49 corresponds to (200), Figure 9 : 9X-ray (XRD) pattern of 500 nm thick CsI film deposited on Al substrate. The diffraction pattern is taken in a Bragg Brentano parafocusing geometry. Figure 10 : 10Average crystallite size and texture coefficient of CsI thin film calculated from X-ray diffraction pattern. Figure 11 : 11Transmission electron microscopy image and diffraction pattern (inset) of 500 nm CsI thin film (Top) and Grain size distribution of 500 nm CsI thin film (Bottom). Figure 12 : 12Chemical composition of CsI film was determined by EDAX. The spectra suggests that grains have chemical composition of Cs and I having Cs:I ratio is 1:1. Table 2 : 2Crystallographic properties of 500 nm thick CsI film obtained from XRD:(hkl) 2θ[deg] d[Å] FWHM I(hkl)/I 0 (hkl) TC[%] D(nm) (110) 27.06 3.295 0.1476 1.00 1.59 55 (200) 38.87 2.315 0.1968 0.37 0.59 43 (211) 48.30 1.884 0.1968 0.45 0.72 44 (220) 56.49 1.628 0.2460 0.69 1.10 46 AcknowledgmentsThis work was partly supported by the Department of Science and Technology (DST), the Council of Scientific and Industrial Research (CSIR) and by Indian Space Research Organization (ISRO), Govt. of India. Triloki acknowledges the financial support obtained from UGC under research fellowship scheme for meritorious students (RFSMS) program. Richa Rai acknowledges the financial support obtained from UGC under research fellowship scheme in central universities. Presented at RICH2010. A Breskin, A. Breskin "Presented at RICH2010". . C Lu, Nuclear Instruments and Methods in Physics Research A. 366C. Lu et al., Nuclear Instruments and Methods in Physics Re- search A 366 (1995) 345-354. . 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Today there are a plethora of many-body techniques for calculating nuclear wave functions and matrix elements. I review the status of that reliable workhorse, the interacting shell model, a.k.a. configurationinteraction methods, a.k.a. Hamiltonian diagonalization, and survey its advantages and disadvantages. With modern supercomputers one can tackle dimensions up to about 20 billion! I discuss how we got there and where we might go in the near future.

1809.07869

19f411cf05c8fb8650268c4e504f747e4a2b9895

Current Status of Very-Large-Basis Hamiltonian Diagonalizations for Nuclear Physics 25 Sep 2018 September 27, 2018 May 28-June 3, 2018 Calvin W Johnson Department of Physics San Diego State University 92182-1233CAUSA Current Status of Very-Large-Basis Hamiltonian Diagonalizations for Nuclear Physics 25 Sep 2018 September 27, 2018 May 28-June 3, 2018PRESENTED AT Thirteenth Conference on the Intersections of Particle and Nuclear Physics Indian Wells, CA, Today there are a plethora of many-body techniques for calculating nuclear wave functions and matrix elements. I review the status of that reliable workhorse, the interacting shell model, a.k.a. configurationinteraction methods, a.k.a. Hamiltonian diagonalization, and survey its advantages and disadvantages. With modern supercomputers one can tackle dimensions up to about 20 billion! I discuss how we got there and where we might go in the near future. 1 Introduction and relevance of nuclear structure Some of my colleagues think nuclear structure theory, in particular the nuclear shell model, is as old-fashioned as the horse-and-buggy. But really it's is the exact opposite. Nuclear structure theory has lots of exciting developments. These developments push the shell model from phenomenology to rigorous first-principle calculations, driven partly by new ideas but above all by the explosion in computing capabilities. While in the early days solving a 25 × 25 matrix was the height of computation [Halbert and French, 1957], today we find extremal eigenvalues of matrices exceeding dimensions of 2 × 10 10 [Forssén et al., 2018]. Aside from the intrinsic physics interest of nuclei, careful microscopic calculations are needed for many applications. Detection of known and unknown particles, from neutrinos [Suzuki et al., 2006] to dark matter [Anand et al., 2014], as well as experiments testing fundamental symmetries, such as neutrinoless double-β decay [Horoi and Brown, 2013] and nonconservation of parity and time-reversal symmetries [Haxton and Wieman, 2001], often require knowledge of matrix elements in complex nuclei. For such calculations to be reliable and both precise and accurate, they need to be founded on solid microscopic calculations. Fortunately, in many cases modern nuclear structure theory is rising to the challenge. Key ideas in large-basis diagonalization This paper deals solely with diagonalization of the many-body Hamiltonian in a basis built from shell-model single-particle states, also called the configuration-interaction method or the interacting shell model [Brussard and Glaudemans, 1977, Brown and Wildenthal, 1988, Caurier et al., 2005. The idea is straightforward: expand a state |Ψ in a basis {|α } (assumed to be orthonormal, α|β = δ α,β ), |Ψ = α c α |α ; (1) minimizing Ψ|Ĥ|Ψ / Ψ|Ψ leads to the eigenvalue equation β H α,β c β = Ec α ,(2) where H α,β = α|Ĥ|β is the matrix element of the many-body HamiltonianĤ in this basis. I deal with the question of the choice of basis in section 3 We can broadly classify configuration-interaction (CI) calculations into two categories, phemonenological and ab initio. Phenomenological calcqulations are older, and usually assume a fixed cored and a relatively narrow valence space, such as the 1s 1/2 -0d 3/2 -0d 5/2 space with a fixed 16 O core, or the 1p-0f space with a fixed 40 Ca core [Brussard and Glaudemans, 1977, Brown and Wildenthal, 1988, Caurier et al., 2005. The interactions actually start from some ab initio underlying interaction, and then adjusted to many-body spectra in the target space [Brown and Richter, 2006]. Because of this, it is fair to call them semi -phenomenological. By ab initio I mean a potential fitted to few-body data, such as nucleon-nucleon scattering and the binding energies of the A = 2, 3 and other light systems. These interactions are most commonly built from chiral effective field theory [Entem and Machleidt, 2003], but not always [Wiringa et al., 1995, Shirokov et al., 2016. Despite having essentially the same few-body input, difference choices such as cut-off regulators [Dyhdalo et al., 2016] can strongly influence the final many-body energies. Purely ab initio CI calculations are often called no-core shell model (NCSM) calculations [Navrátil et al., 2009, Barrett et al., 2013, precisely because there is no core: all particles, in principle, are active, and the standard methodology increases the model space until convergence: see section 4 below. In between these two are attempts to derive ab initio effective interactions, with no adjustable parameters, for phenomenological-like valences spaces for medium and heavy nuclei, via a double projection (Okubo-Lee-Suzuki) method [Dikmen et al., 2015], via coupled clusters [Jansen et al., 2014], and via the in-medium similarity renormalization group [Stroberg et al., 2017]. Because we cast the many-body Schrödinger equation as a matrix equation, the main computational task becomes solving a matrix eigenvalue problem. While some bases are larger than other, as discussed below, almost all CI calculations involve large enough dimensions that it would be foolish to try to find all eigenpairs. Instead, one solves for extremal eigenvalues using Arnoldi-type algorithms, almost always the Lanczos algorithm [Whitehead et al., 1977], although there have been attempts to use other methods [Shao et al., 2018]. The drawback is one needs a large number of M-scheme basis states to build up nuclear correlations. There are more sophisticated bases. J-scheme basis states have fixed total angular momentum J. The most widely used J-scheme codes are OXBASH [Brown et al., 1985] and its successor NuShellX [Brown and Rae, 2014]. As such, the J-scheme basis has smaller dimensions than the M-scheme. One can go even further, to so-called symmetry-adapted bases, based upon groups such as SU (3) [Dytrych et al., 2013] or Sp(3,R) [McCoy et al., 2018]. When judiciously truncated in the choice of irreps (subspaces defined by the Casimir operators of the group), such calculations can be even smaller in dimension. Dimensions alone do not measure the computational burden. From Eq. (2) the real computational burden is in the nonzero matrix elements of the Hamiltonian. Mscheme bases are very sparse, as small as ∼ 10 −6 , while J-scheme bases, smaller in dimensions, can have more nonzero matrix elements, and symmetry-adapted bases yet more [Dytrych et al., 2016]. Furthermore, J-scheme basis states are generally represented as a linear combination of M-scheme states, and symmetry-adapted states are either a linear combination of M-scheme states or require non-trivial recursion algorithms, making calculation of the nonzero matrix elements a significant burden; by contrast, in the M-scheme matrix elements are so simple they can be recomputed efficiently on-the-fly, dramatically reducing the memory load, albeit at a price of a more complicated algorithm Nowacki, 1999, Johnson et al., 2018]. There is no 'best' basis, only the recognition of trade-offs. In addition to the choice of many-body basis states, there is the question of the underlying single-particle basis. Phenomenological calculations either assume a harmonic oscillator basis or a Woods-Saxon like basis, but in general as matrix elements are primarily tuned to spectra, the single-particle basis is ambiguous. More rigorous ab initio calculations such as the no-core shell model (NCSM) do have definite singeparticle bases, almost always harmonic oscillator which aids in removing spurious center-of-mass motion. Yet harmonic oscillator wave functions have a steep, unphysical fall off. Hence there have been many efforts to introduce better wave functions [Caprio et al., 2012], a question which has proved challenging. The most promising seem to be natural orbitals [Constantinou et al., 2017], orbitals that diagonalize the ground state one-body density matrix. Convergence and extrapolation Phenomenological calculations take place in a fixed set of valence orbits (unfortunately common usage often conflates orbits and shells), with interactions tuned to that valence space, such as the 1s-0d or sd-space [Brown and Richter, 2006]. Ab initio calculations, conversely, imply a result in a unrestricted or infinite space. Because any actual calculation must be done in a finite space, one must investigate the convergence as the space is increased, and in many cases, extrapolate to the infinite limit. In default NCSM calculations [Barrett et al., 2013], one defines the model space by two parameters: the harmonic oscillator frequency Ω, or, more often,hΩ, for the single-particle basis states, and N max , the maximum number of oscillator quanta allowed above the lowest configuration; historically this has also been called NhΩ. Typically one wants to extrapolate to infinite N max andhΩ. One strategy is to use an exponential extrapolation, e.g. fitting energies to a form a + b exp(−cN max ) [Heng et al., 2017]. This is inspired by similar exponential extrapolations in phenomenological shell model calculations where even the finite model space is so large one must truncate and extrapolate [Horoi et al., 1999]. For the NCSM, however, the results are not very robust. Instead, recent work has found more robust extrapolation by combining N max andhΩ in to infrared and ultraviolet parameters, and following the convergence in those parameters [Coon et al., 2012, More et al., 2013, Wendt et al., 2015. This can also be linked to interpreting N max as a finite 'wall' [Furnstahl et al., 2012]. In a way, these extrapolations are brute force, and limited by the capability of modern computers. The basis dimension grows exponentially with the number of orbits / N max and particles, which is why size-extensive methods such as coupled clusters [Hagen et al., 2010] are attractive, but which have their own set of limitations. These limitations inspire alternatives to the standard NCSM prescription: rather than brute force computation in a larger basis, build in smarter bases, such as use of better single orbitals such as natural orbitals [Constantinou et al., 2017], and selected irreducible representations in symmetry-adapted bases which efficiently exploit deformation degrees of freedom [Dytrych et al., 2013, McCoy et al., 2018. These lose, however, the powerful machinery of extrapolation applied to standard NCSM calculations. Finally, rather than being 'smarter' in our physics, one can ride a current trend and hand over insights to the computer, with novel extrapolations using machine learning [Negoita et al., 2018]. The initial results are impressive, and it remains to see how widespread such techniques can be applied. Basis states for configuration interactionHow to construct the basis set {|α }? One choice is to use many simple states. The most common building block are Slater determinants (antisymmeterized products of single-particle states) or more generally the occupation-space representations of Slater determinants using creation and annihilation operators. Furthermore, one often uses an is M-scheme basis, where each Slater determinant has the same fixed total M or J z , that is, the z-component of angular momentum. This is easy because J z is an additive quantum number. Many CI shell model codes use an M-scheme basis, most notably ANTOINE[Caurier and Nowacki, 1999], MFDn[Sternberg et al., 2008], BIGSTICK[Johnson et al., 2013[Johnson et al., , 2018, and KSHELL[Shimizu, 2013]. 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We briefly discuss construction of energy-dependent effective non-hermitian hamiltonians for studying resonances in open disordered systems . PACS number(s): 03.65.Yz, 03.65.Nk, 72.15.Rn

10.1007/s10773-010-0604-y

1011.5932

580e85235d85970713c7c2732cdaac6976b3f1cf

Effective Non-Hermitian Hamiltonians for Studying Resonance Statistics in Open Disordered Systems 26 Nov 2010 Joshua Feinberg 1e-mail:[email protected] Department of Physics Department of Physics University of Haifa at Oranim 36006TivonIsrael Technion-Israel Inst. of Technology 32000HaifaIsrael Effective Non-Hermitian Hamiltonians for Studying Resonance Statistics in Open Disordered Systems 26 Nov 20101resonancesspectral determinantdisordered systemsaverage density of resonances We briefly discuss construction of energy-dependent effective non-hermitian hamiltonians for studying resonances in open disordered systems . PACS number(s): 03.65.Yz, 03.65.Nk, 72.15.Rn Introduction Open systems typically give rise to resonances. A resonance is a long-living quasistationary state, which eventually decays into the continuum. Physically, it may be thought of as a particle, initially trapped inside the system, which eventually escapes to infinity. One common approach to studying resonances is based on the analytic properties of the scattering matrix S(E) in the complex energy plane. Resonances correspond to poles E n = E n − i 2 Γ n(1) of S(E) on the non-physical sheet [1,2]. In an alternative equivalent approach, which we shall follow here, one solves the Schrödinger equation subjected to the boundary condition of purely outgoing wave outside the range of the potential. This boundary condition, which describes a process in which a particle is ejected from the system, renders the problem non-Hermitian. The Schrödinger equation with this boundary condition leads to complex eigenvalues E n which correspond to resonances [1,2]. For a recent lucid discussion of resonances in quantum systems, with particular emphasis on the latter approach, see [3,4]. The outgoing-wave approach leads, in a natural way, to non-Hermitian effective hamiltonians ,whose complex eigenvalues are the resonances of the studied system [5,6,7]. Such effective hamiltonians are very useful for studying resonances in scattering theory, including scattering in chaotic and disordered systems [8,9,10,11,12,13,14]. There are many examples of resonances in atomic and nuclear physics. Recently, there has been considerable interest in resonances which arise in chaotic and disordered systems. See [9] for a recent review. One of the main goals in these studies is computation of the distribution P (Γ) of resonance widths. There is ample amount of work on computing P (Γ) in one-dimensional disordered chains [13,14,15,16,17,18,19]. Numerical results presented in some of these works indicate that P (Γ) ∼ Γ −γ in a large range of values of Γ, where the exponent γ is very close to 1. A more general quantity than P (Γ) is the 1 density of resonances (DOR) ρ(x, y) = n δ(x − ReE n )δ(y − ImE n ).(2) It is widely believed that the averaged DOR in the complex plane contains information about the Anderson transition [9,17,19,22]. This expectation 2 is based on an analogy with Thouless' arguments concerning the sensitivity of eigenstates to the boundary conditions in Hermitian localization theory [20,21]. Indeed, the coupling of the disordered system to the external world plays in our case a role similar to changing the boundary conditions in Thouless' picture. Namely, the width of a typical resonance in the insulating regime should be exponentially small, Γ typ ∼ exp −L/ξ(E) (L being the size of the system), whereas in the metallic regime the typical width is Γ typ ∼ D/L 2 , namely, the inverse Thouless time scale (D is the diffusion coefficient in the disordered metal). Thus, Γ typ , measured in units of level spacing ∆, is analogous to the Thouless conductance. This picture was already pursued numerically in [22]. The continuum limit of the disordered chain was studied in [23]. For simplicity, a chain opened only at one end was studied. The spectral determinant for the problem was derived, and the averaged DOR was expressed in terms of a certain integral over the solution of a certain singular two-dimensional Fokker-Planck equation. (That Fokker-Planck equation determined the probability distribution of the logarithmic derivative of the outgoing wave at the open end of the chain.) The present work was motivated in part by [13,14]. In particular, an analytical approach was developed in [14] for studying resonances, which is based on counting poles of the resolvent of the non-Hermitian tight-binding effective hamiltonian of the open chain. In the case of a semi-infinite disordered chain, coupled to a semi-infinite perfect lead, these authors have derived an exact integral representation for the DOR, valid for arbitrary disorder and chain-lead coupling strength. In the limit of weak chain-lead coupling (in which resonances are typically narrow) they were able to rigorously derive a universal scaling formula for the DOR, valid for any degree of disorder and everywhere inside the unperturbed energy band of the closed chain. The 1/Γ behavior of the DOR follows from that formula. In this paper we shall review and explain how to construct energy dependent non-hermitian hamiltonians for studying resonance statistics in open systems. While many (but by no means all) of the results presented in this paper are known, we believe our presentation offers a somewhat fresh look at these issues. Upon elimination of the leads, one can reformulate the problem in terms of an effective non-hermitian hamiltonian, which depends only on the degrees of freedom of the disordered system. In this effective description, the outgoing-wave boundary condition in the original system is translated into a local non-hermitian, energy dependent boundary condition at the contact points (or more generally, contact regions) of the system and the leads. This paper is organized as follows. In Section 2 we discuss resonances in a generic quantum system coupled to the external world by a single one-dimensional lead ( a single channel lead). We derive a general expression for the DOR in terms of an appropriate diagonal matrix element of the resolvent of the original closed system. From this expression, we derive an integral representation for the averaged DOR of the disordered system. In Section 3 we specialize to the case of an open one dimensional disordered chain, derive the corresponding effective hamiltonian, and obtain its continuum limit. The resulting continuum effective non-hermitian hamiltonian differs from the hermitian one of the closed system by a complex energy dependent boundary condition. The structure revealed in this way is quite generic, and we conclude in Section 4 by mentioning similar continuum effective hamiltonians for higher dimensional systems. Resonances in a System Connected to a Single Perfect Lead In order to keep the discussion as simple as possible, let us consider a quantum system connected to a single perfect semi-infinite one-dimensional lead, which lies along the negative x-axis. This construction is described in Figure 1. We shall model the lead by means of a tight-binding hopping hamiltonian, with nearestneighbor hopping amplitude t. The sites on the lead lie at the points x n = na, n = 0, −1, −2, . . ., a being the lattice spacing. Let us assume that the (closed) quantum system has an N -dimensional state space, and that it is described by an N × N hermitian matrix H ij , H = H † . We further assume that the system lives on some graph with N nodes, and that H ij is the matrix element connecting site i to site j (the link < ij > is directed, of course). Let us now connect the lead's end n = 0 to some site in the system, which with no loss of generality we pick to be site i = 1. The hopping amplitude along the buffer link < 01 > is taken to be t , which need not be equal to t. In particular, t = 0 corresponds to a closed hermitian system. Disorder is modeled by some probability distribution for the matrix H, which means, in general, both random hopping and random site energies on the graph. H ij 1 0 −1 · · · | | | t t t disordered system contact link perfect lead H = H † ψ n = Ae −ikn outgoing wave Re k > 0 Figure 1: A disordered system is opened up and coupled to the external world by a perfect lead, stretched along the negative x-axis. The lead is modeled by a tight-binding hamiltonian, with nearest neighbor hopping amplitude t. The < 01 > link connects the lead and the system, with corresponding hopping amplitude t . The Schrödinger equation is to be solved with outgoing wave boundary condition. In this figure we have set the lattice spacing a = 1. See the text for more details. The Schrödinger equation for this system is therefore − t(ψ n+1 + ψ n−1 ) = zψ n , n ≤ −1 −tψ −1 − t ψ 1 = zψ 0 , n = 0 −t ψ 0 + (H ψ) 1 = zψ 1 n = 1 (H ψ) n = zψ n , n ≥ 2 .(3) Here we lumped the wave-function amplitudes inside the system into an N -dimensional vector ψ, and z is the complex eigenvalue. Imposing outgoing-wave boundary condition in the perfect lead means that ψ n = Ae −ikna , n ≤ 0 ,(4) where the wave-vector k must be restricted to the right half of the fundamental Brillouine zone, namely, 0 ≤ Re ka ≤ π ,(5) as the wave propagates freely to the left, into the lead. (This choice has the obvious continuum limit Re k ≥ 0 describing free propagation to the left.) We shall now eliminate the lead entirely from (3), following the idea presented in [8]. To this end we first substitute (4) in the first equation in (3), from which we find that z = z(k) = −2t cos(ka) .(6) Next, we eliminate ψ 0 = A = t t e ika ψ 1 from the n = 0 equation, and substitute it in the n = 1 equation. In this way we discover that − t 2 t e ika ψ 1 + (H ψ) 1 = z(k)ψ 1 .(7) The remaining equations (H ψ) n = z(k)ψ n (n ≥ 2), as well as the equation (7) for n = 1, can be neatly written as H − t 2 t e ika P ψ = z(k) ψ ,(8) where P = |1 1| is the projector on site 1, to which the lead is connected. Thus, we can ignore the lead from now on, and describe the open system itself by an effective hamiltonian H ef f = H − ηe ika P ,(9) with η = t 2 t .(10) H ef f is a non-hermitian N × N matrix, which also depends on energy, through k, according to (6). This explicit k dependence is of course, the price we had to pay in order to eliminate the lead from the description of our system, and it should actually be expected of an effective description -the effective hamiltonian normally depends on the energy scale one studies. Of course, when t = η = 0, the lead is disconnected, and H ef f = H of the closed system. The picture we have in mind is that starting with the closed system (η = 0), all N energy eigenstates are real and sharply defined. Then we open the system adiabatically, i.e., increase η slowly. As a result, each sharply defined energy eigenstate in the original system should broaden continuously into a resonance, with complex energy z(k, η). Thus, we end up with N complex resonance energies, the solutions of H ef f (k) ψ = −2t cos(ka) ψ. Let us briefly elaborate on the domain in the complex-k plane which corresponds to resonances. We shall follow the discussion in [3,4], which give probabilistic interpretation to the modulus square of the time-dependent resonance eigenfunction Ψ n (t). For this, we shall temporarily include the lead in our discussion. In a resonance (i.e., quasi-stationary) state the amplitude Ψ n (t) = ψ n e −iz(k)t has to grow in magnitude into the lead, where the particle is likely to be after a long time. For the same reason, it also has to decay as function of time, at any finite fixed site. Thus, in addition to (5), we must also demand that both Im k and Im z = 2t sin(Re ka) · sinh(Im ka) be negative. Thus, sin(Re ka) > 0, which holds automatically due to (5). As was originally discussed in [3] (and later extended in [4]), we see that as time goes by, we can maintain the numerical value of the spatial integral of |Ψ n (t)| 2 (the probability) , provided we allow the integration domain to expand at a constant speed to the left, which is nothing but the ballistic velocity of the ejected particle. To summarize, resonances must all lie in the strip 0 ≤ Re ka ≤ π , Im k < 0 (11) in the fourth quadrant of the complex k plane. Similarly, anti-resonances, which describe a situation in which the system absorbs particles from the lead 3 , must all lie in the strip −π ≤ Re ka ≤ 0 , Im k < 0 in the third quadrant of the complex k plane. The Secular Equation Resonances are the roots of the equation det (z − H ef f ) = 0 ,(12) with k lying in the appropriate strip (11). Practically, it is easier to compute the ratio of determinants F (k) = det (z − H ef f ) det (z − H) = det z − H + ηe ika P det (z − H) = det 1 + ηe ika GP ,(13) where G = 1 z − H(14) is the resolvent of H. In (12)- (14) we must of course set z according to (6). Note that (GP ) nm = G n1 δ m1 . Hence, 1 + ηe ika GP is a lower diagonal matrix, and computation of the last determinant in (13) is immediate. We find simply that F (k) = 1 + ηe ika G 11 (z(k)) .(15) Thus, in order to solve for the resonance spectrum of our model, all we require is the G 11 element of the Green's function of the original closed system. The latter is the Green's function of a hermitian hamiltonian, and therefore well-studied. Note that we have not specified the specific nature of the closed system corresponding to H. Our discussion is completely generic! The DOR For a given realization of H, F (k) is a holomorphic function of k, and has zeros at the eigenvalues of H ef f and poles (on the real axis) at the eigenvalues of H. Let k 0 α and k p β be, respectively, the zeros and (purely real) poles of F (k). Thus, F (k) F (k) = α 1 k − k 0 α − β 1 k − k p β .(16) From the identity ∂ ∂k * 1 k − q = πδ (2) (k − q) ,(17) which is nothing but Gauss' Law in 2d electrostatics (for a unit point charge located at position k = q), we thus find 1 π ∂ ∂k * F (k) F (k) = α δ (2) (k − k 0 α ) − β δ (2) (k − k p β ) .(18) Averaging this equation with its complex-conjugate, we finally obtain that ρ(k, k * ) = 1 2π ∂ 2 ∂k∂k * log F (k) 2 = α δ (2) (k − k 0 α ) − β δ (2) (k − k p β ) .(19) Since the poles live entirely on the real axis, going off it and into the fourth quadrant in the complex k-plane, we obtain our desired DOR. Continuing the analogy with 2d electrostatics [24], observe that (19) W (k, k * ) = − log F (k) 2 = − log det(z − H ef f ) 2 det(z − H) 2(20)H =   0 z − H ef f z * − H † ef f 0   .(21) In fact, given a non-hermitian operator, such as H ef f , whose spectrum we wish to study, the method of hermitization [25] instructs us to construct its hermitized form (21), and study its spectrum, which of course lies entirely on the real axis. Thus, for example, the Green's function 1/(ζ − H) is analytic in the complex ζ plane, save for poles (or a cut, upon averaging) along the real axis, where the spectrum is located. Thus, one may bring the power of analytic function theory to bear in analyzing the spectrum, which cannot be done for the non-hermitian H ef f . The Averaged DOR In our closed disordered system G 11 (z) = X(z) + iY (z)(22) is a complex valued random variable, with probability distribution P(X, Y ; z) = δ (X − X(z)) δ (Y − Y (z)) ,(23) which we assume to be known. Thus, from (19) and (23) we immediately obtain an integral representation for the averaged DOR as ρ av (k, k * ) = 1 2π ∂ 2 ∂k∂k * dXdY P(X, Y ; z(k)) log 1 + ηe ika (X + iY ) 2 .(24) The One-Dimensional Disordered Chain and the Continuum Limit of its H ef f We shall now depart from the general discussion and take H to be the tight-binding hamiltonian of a disordered chain with N sites, i.e., the one-dimensional Anderson model. We take the nearest-neighbor hopping amplitudes to be t, as in the lead. The site energies n (n = 1, 2, . . . , N ) are i.i.d. random variables taken from some probability distribution q( ). Thus, the corresponding hermitian matrix H in (3) (and in Fig.1), in the previous section, is given by H mn = −t(δ m,n+1 + δ m+1,n ) + n δ mn , 1 ≤ m, n ≤ N .(25) The resulting Schrödinger equation is therefore − t(ψ n+1 + ψ n−1 ) + n ψ n = zψ n ,(26) with Dirichlet boundary conditions ψ 0 = ψ N +1 = 0 ,(27) corresponding to a closed chain. As can be seen from (9), the effective Schrödinger equation for the open system (with the lead eliminated, of course) is obtained from (26) (or (25)) simply by replacing n bỹ n = n − ηe ika δ n1 .(28) Statistics of resonances in this model was studied in detail in [13,14,15]. to the complex action S = N n=1 a ( n − ζ)|ψ n | 2 − D δ n1 a ψ * 1 ψ 2 − ψ 1 a + t t 2 e ika − 1 a |ψ 1 | 2 −D N n=2 aψ * n δ 2 ψ n a 2 ,(30) where D = ta 2 and ζ = z + 2t = 4t sin 2 ka 2 (31) are, respectively, the diffusion constant (the lattice version of 2 2m ) and the shifted (renormalized) energy, and δ 2 ψ n = ψ n+1 − 2ψ n + ψ n−1(32) is the symmetric second difference (the lattice discretized version of the self-adjoint laplacian). Strictly speaking, we should really apply the variational principle to the real action corresponding to the hermitized form (21). However, in order to keep the discussion as brief as possible, and since all we want to obtain in this section is the continuum limit of H ef f , and not to pursue the averaged DOR in detail, we shall contend ourselves with the complex action S. The Continuum Limit The continuum limit is obtained by sending a → 0 and t → ∞ simultaneously, while keeping D = ta 2 = 2 2m and k finite. Furthermore, t → ∞ as well, such that the ratio t t 2 = e λa(33) with λ finite. In this limit we also obtain the familiar relation ζ = Dk 2 . In the limit, the lattice amplitudes tend to the continuous wave function, ψ n = ψ(na) → ψ(x) and the site energies tend to the potential n → V (x) . Obviously, δ 2 ψn a 2 → ∂ 2 x ψ(x) and ψ 2 −ψ 1 a → ψ (a+). Finally, of course, δ n1 a → δ(x − a)S = L 0+ dx (V (x) − ζ) |ψ(x)| 2 − Dψ * (x)∂ 2 x ψ(x) − Dδ(x − a) ψ * (x)∂ x ψ(x) + (λ + ik)|ψ(x)| 2 = L 0+ dx ψ * (x) H cont ef f − ζ ψ(x) ,(35) with the boundary conditions (34) understood. We can immediately read off the continuum effective effective hamiltonian from (35), namely, H cont ef f = Dp 2 + V (x) − Dδ(x − a) (ip + λ + ik)(36) with p = −i∂ x as usual. Note that we have explicitly left the infinitesimal lattice spacing a in (35) ∂ x ψ(0+) + (λ + ik)ψ(0) = 0 ,(37) which depends on energy, through k. Since in the lead, x < 0 we have the outgoing wave ψ(x) = ψ(0)e −ikx , the derivative ∂ x ψ(x) jumps: ψ (0+) − ψ(0−) = −λψ(0). This jump is the result of the singular contact potential term −λDδ(x) in (36). Note from (36) that λ → −∞ penalizes for having ψ(0) = 0. Thus, this limit corresponds to Dirichlet boundary conditions, namely, t = 0 and a closed chain, as can be seen also from (33). More precisely, for a very small but finite, one integrates the Schrödinger equation −D∂ 2 x ψ(x) + V (x)ψ(x) = ζψ(x) , subjected to ψ(L) = 0, from the right end of the system all the way to x = a, where the large coefficient of the boundary layer interaction takes over, and fixes ψ (a−) = −(λ + ik)ψ(a). The wave function has then to relax to zero at x = 0 across the thin boundary layer, with tremendous slope. This segment of the wave function is an artifact, which we cut and throw, and replace by the resonance boundary condition (37). Concluding Remarks Concerning Higher Dimensional Systems The structure revealed by analyzing the continuum limit of the one dimensional case is quite generic. The effective non-hermitian hamiltonian is generically given by the original differential expression (in the coordinate representation), supplemented by an appropriate energy dependent complex boundary condition. In conclusion, let us mention briefly two simple 3d examples in the continuum, which correspond to having infinitely many weak channels connecting the system to the environment. These results are straightforward: • Disordered half-space coupled uniformly to the environment through a contact plane. Let's take the disordered system to live in the z > 0 half-space, and let it communicate with its environment through the xy plane. For a given complex energy z = 2 Q 2 2m there is a continuum of resonances indexed by the components of the wave-vector k ⊥ , perpendicular to the z axis, which are real. They correspond to the direction in which the particle is ejected from the system. Let q = k 3 be the on-shell complex component of momentum in the z-direction, such that q 2 = Q 2 − k 2 ⊥ . Then, the resonance amplitude immediately outside the system, must satisfy the outgoing boundary condition (∂ z + iq sign Im q)ψ k ⊥ (z = 0 − ) = 0 . We can then obtain ∂ z ψ k ⊥ (z = 0 + ) right inside the system by considering the contact potential −Dλδ(z), in complete analogy with (37). Since the boundary condition (38) is rotationally symmetric with respect to the z-axis, we expect the averaged DOR to inherit this symmetry as well. • A disordered ball of radius a coupled uniformly to the environment through its surface. In this case, we should consider resonances with definite angular momentum quantum numbers l, m in the outside world. For a given complex energy z = 2 Q 2 2m , the corresponding outgoing wave amplitude ψ lm (r) = A lm h l (Qr) must be proportional to a Hankel function. Thus, it must trivially satisfy ψ lm (a+) = Q h l (u) h l (u) ψ lm (a) ,(39) where u = Qa. Again, the radial derivative immediately inside the ball may be obtained by taking into account the jump in the radial derivative due to a uniform radial-shell contact potential. Due to spherical symmetry, the boundary condition (39) is independent of m. Consequently, the averaged DOR should inherit this property as well. acknowledgements I wish to thank Boris Shapiro for many valuable discussion on resonances in disordered systems. This work was supported in part by the Israel Science Foundation (ISF). on the RHS. Moreover, note that the real quantity det(z − H ef f ) 2 in (20) is proportional to the determinant of the 2N × 2N hermitian operator and n a → dx, and (27) tend to the continuum Dirichlet boundary conditions ψ(0) = ψ(L) = 0 (34) with L = N a (where of course N → ∞). Plugging all these limiting quantities in (30), we obtain the continuum limit of S as and (36) as a mnemonic. In fact, δ(x − a) in these expressions really stands for a thin boundary layer around the left end of the chain, with a very large coefficient, which penalizes for having ∂ x ψ(x) + (λ + ik)ψ(x) = 0 in the immediate vicinity of x = 0+. The continuum Schrödinger equation derived by applying the variational principle to (35) generates in this way the continuum resonance boundary condition In order to avoid cluttering of our formulas, we do not use the resonance width Γ n in (1) as an argument of ρ, but rather y = −Γ n /2.2 I learned this argument about the expected scaling behavior of the DOR from B. Shapiro. Here the wave should propagate freely in the lead to the right, towards the system. Hence −π ≤ Re ka ≤ 0. Furthermore, Im k < 0, since at t = 0 the it is overwhelmingly probable to find the particle in the lead, while at the same time we must also have Im z > 0, since the probability to find the particle at n = 0 must grow. L D Landau, E M Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Course of theoretical physics. OxfordPergamon3L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic The- ory, Course of theoretical physics , vol. 3 (Pergamon, Oxford, 1977). 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B61 (2000), R2444. . M Weiss, J A Mendez-Bermudez, T Kottos, Phys. Rev. 7345103M. Weiss, J. A. Mendez-Bermudez, and T. Kottos, Phys. Rev. B73 (2006), 045103. . J T Edwards, D J J Thouless ; D, Thouless, J. Phys. C. 593Phys. Rep.J. T. Edwards and D. J. Thouless, J. Phys. C 5 (1972), 807; D. J. Thouless, Phys. Rep. 13 (1974), 93. . E Abrahams, P W Anderson, D C Licciardello, T V Ramakrishnan, Phys. Rev. Lett. 42673E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan, Phys. Rev. Lett.42 (1979), 673. T Kottos, M Weiss, ; M Weiss, J A Mendez-Bermudez, T Kottos, cond-mat/0509195Resonance Width Distribution for High-Dimensional Random Media. 89T. Kottos and M. Weiss, Phys. Rev. Lett.89 (2002), 056401; M. Weiss, J. A. Mendez-Bermudez and T. Kottos, Resonance Width Distribution for High- Dimensional Random Media, cond-mat/0509195. . J Feinberg, Pramana. 73565J. Feinberg, Pramana 73 (2009), 565. . F Haake, F Izrailev, N Lehmann, D Saher, H.-J Sommers, Phys. Rev. Lett. 359H.-J. Sommers, A. Crisanti, H. Sompolinski and Y. Stein881895Z. Phys.F. Haake, F. Izrailev, N. Lehmann, D. Saher and H.-J. Sommers, Z. Phys. B88 (1992), 359; H.-J. Sommers, A. Crisanti, H. Sompolinski and Y. Stein, Phys. Rev. Lett. 60 (1988), 1895. . J Feinberg, A Zee, 643; ibid. B504Nucl. Phys. 501579J. Feinberg and A. Zee, Nucl. Phys. 501 (1997), 643; ibid. B504 (1997), 579.

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have periodicity and pattern similar to stable and unstable periodic orbits already existent for the unperturbed oscillator. These features indicate that the reported replicate periodic windows are associated with chaos control of the considered oscillators.

10.1016/j.chaos.2011.08.002

1105.1802

ee077802b11b0129496f1b556ae7a41cbfe3c3f4

Replicate Periodic Windows in the Parameter Space of Driven Oscillators 9 May 2011 E S Medeiros Instituto de Física Universidade de São Paulo São PauloBrasil S L T De Souza Universidade Federal de São João del-Rei Campus Alto Paraopeba Minas Gerais Brazil R O Medrano-T Instituto de Física Universidade de São Paulo São PauloBrasil Departamento de Ciências Exatas e da Terra Universidade Federal de São Paulo Diadema, São PauloBrasil I L Caldas Instituto de Física Universidade de São Paulo São PauloBrasil Replicate Periodic Windows in the Parameter Space of Driven Oscillators 9 May 2011numbers: 0545-a0260Cb0545Gg0545Pq Keywords: Driven OscillatorsControlling ChaosParameter SpaceNonfeedback Method In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have periodicity and pattern similar to stable and unstable periodic orbits already existent for the unperturbed oscillator. These features indicate that the reported replicate periodic windows are associated with chaos control of the considered oscillators. I. INTRODUCTION The parameters of deterministic dynamical systems play an important role to specify the transitions between chaotic and periodic behavior. This parameter influence on the attractor transition can be represented in the parameter space [1][2][3][4][5]. In particular, for both discrete-time [1,6] and continuous dynamical systems [7], it is known that the set of parameters for which a system exhibit periodic behavior are periodic windows immersed in chaotic regions, in the form of shrimps, in the bi-dimensional parameter space. In the past two decades, periodic windows have been numerically obtained for a large number of applied dynamical systems like lasers [8,9], electronic circuits [9][10][11], mechanical oscillators [12], and also in population dynamics [13]. Periodic windows have also been identified in experiments with electronic circuits [14][15][16]. Additionally, the bi-dimensional parameter diagram is an important tool to analyze the outcome of techniques used to control chaos for allowing a global overview of periodic and chaotic behavior [17]. In literature, different approaches have been proposed to control chaos. Essentially, two main methods have been successfully employed in several dynamical systems, namely the feedback and the nonfeedback methods. The feedback methods are applied to maintain the trajectory in a desired unstable periodic orbit embedded in the chaotic attractor [18]. In contrast, the nonfeedback methods eliminate the chaotic behavior by a slightly modification in the system dynamics [19]. The suppression of chaos by applying an external weak perturbation is an interesting application of a nonfeedback method. Suppression of chaos by adding a weak harmonic perturbation has been numerically and theoretically reported specially for systems with an original harmonical driven [20]. For example, a weak harmonic perturbation has been used to suppress chaos in the forced Duffing oscillator [21], and a similar perturbation has been applied to control chaos in a Josephson junction oscillator [22]. Considering that in [20][21][22] the controlling of these two well-known systems were essentially accomplished for limited ranges of system parameters, a further parameter space analysis considering a weak harmonic perturbation is required. The main motivation of this article is to determine the alterations in the parameter space due to weak harmonic perturbations applied to continuous-time dynamical systems. To investigate that we analyze how such perturbations modify the attractors and the parameter space of the harmonically driven Duffing and Josephson oscillators. For the first time in the literature, we observe that this harmonic perturbation replicates the periodic windows in the parameter space of these systems. Moreover, we find evidences that new periodic orbits, whose parameters are in the new periodic windows, are similar to unstable periodic orbits embedded in the unperturbed chaotic attractor and periodic orbits already existing. We discover that the reported replicate periodic windows are associated with chaos control of the considered oscillators. This letter is organized as follows: In Section II, we obtain the parameter space of the unperturbed Duffing oscillator and we establish the suitable parameters to implement the weak perturbation. In Section III, for the Duffing oscillator, we investigate the periodicity, shape and Lyapunov exponents of the perturbed periodic and of the unperturbed chaotic orbit. In Section IV, we present, for the Josephson oscillator, another example of replicate periodic windows caused by a weak periodic perturbation. Finally, in Section 5 we summarize our main conclusions. II. PARAMETER SPACE STRUCTURE OF THE DUFFING OSCILLATOR The Duffing oscillator is a well-known model to describe oscillations of a mass obeying a fourth order symmetric potential [23]. This system has a large number of applications, specially to model physical systems, and has been extensively studied in theoretical, numerical and experimental approaches [24][25][26]. Here, we consider a simple version of the Duffing system, which describe oscillations of a single-well potential. The time evolution of this system is determined by solution of the following dimensionless equation: x + cẋ + x 3 = β cos(ωt). (1) Here, the parameter c is the amplitude of the system damping, β is the forcing amplitude, and ω is the natural system frequency settled at ω = 1.0. We numerically obtain solutions of the Duffing equation by using a fourthorder Runge-Kutta method with fixed step h = 0.001. Additionally, to investigate transitions between chaotic and periodic attractors of the Duffing oscillator, the largest non zero Lyapunov exponent is obtained [27] for each point of a bi-dimensional grid of the system parameters (parameter space). In Figure 1, we show the bi-dimensional parameter space (c × β) obtained for the Duffing oscillator (also shown in [25]). The Lyapunov exponent features of each point in the grid are represented by assigning different colors. In Figure 1, blue color represents parameter sets for which the attractors are chaotic (positive Lyapunov exponent), while grey-scale represents parameter sets for which the attractors are periodic (negative Lyapunov exponent). In black we represent points of the superstable lines corresponding to periodic attractors for which the largest non zero Lyapunov exponents is a minimun inside a periodic window. We observe, in Figure 1, that the parameter sets for which the Duffing oscillator behaves periodically are in aggregated periodic windows (gray scale area). Those periodical structures, also known as shrimps, have been well described in literature [1][2][3]. In Figure 1(b), the red circle and the black plus symbol indicate examples of parameter sets whose orbits are, respectively, periodic and chaotic; these two orbits will be further considered in Section III. To show the different behavior of the two attractors associated with the two parameter sets marked in Figure 1(b), we obtain a stroboscopic map by collecting the velocity and the displacement at (Time-2π/ω). In Figure 2(a), we show the periodic orbit correspondent to the parameters represented in red inside the periodic window amplified in Figure 1(b). Similarly, in Figure 2(b), we show the chaotic attractor for parameters represented by a plus symbol in Figure 1 (b). We call the attention, in Figure 2, that both periodic and chaotic attractors are in the same phase space region. Next, in Section III, we analyze how these two orbits are modified by a weak periodic perturbation. III. INTRODUCING A WEAK PERTURBATION IN THE DUFFING OSCILLATOR We introduce a weak harmonic perturbation by adding a second harmonic term in the original Duffing driven. The perturbation amplitude is taken as a control parameter of the system. On the other hand, the perturbation frequency is settled in an integer ratio of the original system frequency. Similar procedures have been already reported in literature [20][21][22]. So, the time evolution of this perturbed system is determined by solution of the following dimensionless equation: x + cẋ + x 3 = β cos(ωt) + αsin(Ωt). ( Here, α is the weak perturbation amplitude (α << β) and Ω is the perturbation frequency fixed at Ω = 2ω. Other rational multiples of ω could be used to Ω [17,21]. In Figure 3, we present the parameter space of the perturbed Duffing oscillator for two different perturbation amplitudes α = 0.04 and α = 0.08. By comparing the parameter spaces of Figure 1(b) and Figures 3(a,b), we note the existence of replicated periodic windows. In particular, the central periodic window of Figure 1(b) is duplicated in Figure 3. Moreover, the new periodic windows appear slightly displaced in Figures 3(a) and 3(b). Thus, in Figure 3(b), for the perturbation amplitude α = 0.08, the black plus marked point is inside the replicated periodic window. In other words, the chaotic attractor showed in Figure 2(a) has been changed to a periodic one. Next, we discuss the alteration produced by the weak perturbation on the chaotic orbit shown in Figure 2(b). For this purpose, in Figure 4(a), we show the stroboscopic phase space of the new periodic orbit (in black plus symbols), that substitutes the former chaotic orbit shown in Figure 2(b). Additionally, to compare similarities between perturbed and unperturbed orbits, in Figure 4(a), we present (in red points) the unperturbed periodic orbit whose parameters are indicated in red in Figure 1(b). In Figure 4(b), we show the convergence of the largest non zero Lyapunov exponent of the two orbits of Figure 4(a). We recognize, in Figure 4(a), that the new perturbed periodic orbit and the previous periodic orbit have the same periodicity and similar pattern. Moreover, the largest non zero Lyapunov exponent of both orbits also have a similar convergence. We investigate the relation between the new perturbed periodic orbit and a possible unstable periodic orbit embedded in the chaotic sea. Thus, for the parameters marked by a plus symbol in Figure 1(b), we integrate the unperturbed equation for a short integration time, starting with initial conditions settled in the unperturbed periodic orbit with parameters marked by red circle in Figure 1(b). In Figure 5, the blue triangle symbol denotes the chaotic orbit in the short integration time, the red circle symbol denotes unperturbed existent periodic orbit, and the black plus symbol denotes the perturbed periodic orbit. In Figure 5, we observe that for a short integration time, the chaotic orbit remains in the neighbor of the perturbed and the unperturbed periodic orbit. Figure 1(b)). The analysis of results shown in Figures 4 and 5 indicate that the periodic window replication described in this work gives rise to new periodic windows whose parameters correspond to both stable and unstable periodic orbits with the same periodicity and pattern found in previous existing unperturbed oscillator. IV. WEAK PERIODIC PERTURBATION IN THE JOSEPHSON JUNCTION OSCILLATOR The superconducting Josephson junction is usually modeled by an electronic circuit equation [28]. Considerable efforts has been devoted to understand the onset of chaos in this system and its critical parameters for this transition [29][30][31]. The system is described by a pendulum-like dimensionless equation [22]: φ + Gφ + sin φ = I + A sin(ωt).(3) Here, the parameter G gives the amplitude of the system damping, I is the direct current component of the circuit, ω is the natural system frequency settled at ω = 0.25, while A is the alternating current component of the circuit and the system forcing amplitude. In Figure 6, we obtain the bi-dimensional parameter space (G × A) of the Josephson oscillator given by Equation 3. In this figure, blue points correspond to positive Lyapunov exponent (chaotic behavior), while grey-scale points correspond to negative Lyapunov exponent (periodic behavior). The black region corresponds to the superstable lines. We verify, in Figure 6, the existence of aggregated periodic windows in the parameter space of the Josephson oscillator. As in Figure 1(b), the red circle and the black plus symbol indicate in Figure 6 In Figures 7(a) and 7(b) we present, respectively, the periodic and the chaotic orbits for the two parameter sets indicated in Figure 6(b), to be further analyzed in this section. For the Josephson oscillator, similarly to the perturbed Duffing oscillator (Considered in Section 3), a harmonic term is added to the original one: φ + Gφ + sin φ = I + A sin(ωt) + α sin(Ωt). where, α is the perturbation amplitude and Ω is the perturbation frequency fixed at Ω = ω/2. In Figure 8 we ticed in Figure 3 for the Duffing oscillator, in this case the existence of replicate periodic windows in the modified parameter space is also observed. Moreover, comparing Figures 6 and 8, we see that the parameter set indicated by a plus symbol in these two figures represent a chaotic attractor in Figure 6(b), for the unperturbed oscillator, and a periodic one, in Figure 8(b), for the per-turbed oscillator. Next, to show how these orbits are modified by the considered perturbation with amplitude α = 0.01, we show in Figure 9 the stroboscopic phase space of the perturbed Josephson oscillator with three orbits. Thus, in Figure 9, the black plus symbol represents the controlled periodic orbit indicated by a black plus symbol in the parameter space shown in Figure 8(b), with the same parameters of the former chaotic orbit marked with the black plus symbol in Figure 6(a). The blue triangles indicate the chaotic orbit (whose parameter set is indicated in Figure 6(b)) under a short integration time for the initial condition fixed on the periodic attractor. The red circle symbol denotes unperturbed periodic orbit. Thus, the results shown in Figures 6 -9 confirm the evidence, found for the Duffing oscillator, that the new periodic orbits are similar to previous stable and unstable periodic orbits of the unperturbed oscillator. V. CONCLUSIONS We investigate the control of chaos for two driven oscillators by a weak harmonic forcing. To identify periodic and chaotic regions in the bi-dimensional parameter space, we compute the largest non zero Lyapunov exponents for the attractors in the considered parameter ranges. We identify shrimp-shaped periodic windows immersed into a chaotic region. The parameter space is much modified whenever a weak amplitude forcing is applied. New similar periodic windows arise in the neighborhood of the original windows. We verify that periodic orbits are similar (and with the same periodicity) for parameters inside the original and the new periodic windows. These results are similar to those obtained for the driven impact oscillator [17]. By analyzing stroboscopic maps of unperturbed and perturbed attractors we find evidences that the new reported periodic windows are formed by parameters for which the observed new periodic attractors are similar to preexisting stable and unstable periodic orbits. Therefore, we conjecture that the replicate periodic windows reported in this work are associated with chaos control and reproduce further other periodic orbits of the considered oscillators. FIG. 1 :FIG. 2 : 12(Color online) (a) Bi-dimensional (c × β) parameter space of the unperturbed Duffing oscillator, for ω = 1.0. (b) Magnification of the squared area. The red circle indicates periodic behavior at β = 10.6780 and c = 0.2670, the black plus symbol indicates chaotic behavior at β = 10.7020 and c = 0.2670. (Color online) Stroboscopic phase space of the Duffing oscillator. (a) Periodic orbit correspondent to the parameters β = 10.6780 and c = 0.2670. (red circle in Figure 1(b)). (b) Chaotic attractor correspondent to the parameters β = 10.7020 and c = 0.2670 (black plus symbol in FIG. 3 :FIG. 4 : 34(b) examples of parameters for which the correspondent orbits are, respectively, periodic and chaotic. (Color online) Bi-dimensional (c × β) parameter space of the perturbed Duffing oscillator for ω = 1.0 and Ω = 2.0. The perturbation amplitude is α = 0.04 in (a) and α = 0.08 in (b). (Color online) (a) The red circle symbol denotes the periodic orbit for β = 10.6780, c = 0.2670 and α = 0.0, the same orbit of Figure 2(b). The black plus symbol denotes the controlled orbit for β = 10.7020, c = 0.2670, Ω = 2.0, and α = 0.08. (b) The red line indicates the largest Lyapunov exponent of the unperturbed periodic orbit. The black line indicates the largest Lyapunov exponent of the controlled orbit. FIG. 5 : 5(Color online) The blue triangle indicates the chaotic orbit under a short integration time for β = 10.7020, c = 0.2670 and α = 0.0. The black plus symbol indicates the controlled periodic orbit for β = 10.7020, c = 0.2670, α = 0.08 and Ω = 2.0. The red circle symbol indicates the unperturbed periodic orbit for β = 10.6780, c = 0.2670 and α = 0.0. FIG. 6 :FIG. 7 : 67show the parameter space of the perturbed Josephson equation. Similarly to what was no-(Color online) (a) Bi-dimensional (G × A) parameter space of the unperturbed Josephson oscillator, the natural system frequency is settled for ω = 0.25, the direct current component is settled for I = 0.905. (b) Magnification of the squared area. The red circle indicates the parameters A = 0.5289 and G = 0.4403, the black plus symbol indicates the parameters A = 0.5488 and G = 0.4416. Stroboscopic phase space of Josephson oscillator. (a) Periodic orbit correspondent to the parameters A = 0.5488 and G = 0.4416. (red circle inFigure 6(b)). (b) Chaotic attractor correspondent to the parameters A = 0.5488 and G = 0.4416 (black plus symbol inFigure 6(b)). FIG. 8 : 8(Color online) Bi-dimensional (G × A) parameter space of the perturbed Josephson oscillator, the natural system frequency and the perturbation frequency are settled, respectively, for ω = 0.25 and Ω = 0.125. (a) The perturbation amplitude is settled for α = 0.005. (b) The perturbation amplitude is settled for α = 0.01. online) The blue triangle indicates the chaotic orbit under a short integration time for A = 0.5488, G = 0.4416 and ω = 0.25. The black plus symbol indicates the controlled periodic orbit for A = 0.5488, G = 0.4416, ω = 0.25 and Ω = 0.125. The red circle symbol indicates the previous periodic orbit for A = 0.5488, G = 0.4416 and ω = 0.25. VI. ACKNOWLEDGEMENTSThis work was made possible by partial financial support from the following Brazilian government agencies: FAPESP, CNPq, and Capes. . J A C Gallas, Phys. Rev. Lett. 702714J. A. C. Gallas, Phys. Rev. Lett. 70 (1993) 2714. . J A C Gallas, Phys. A. 202196J. A. C. Gallas, Phys. A 202 (1994) 196. . 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We review work on the formation of gravitino condensates via the super-Higgs effect in the early Universe. This is a scenario for both inflating the early universe and breaking local supersymmetry (supergravity), entirely independent of any coupling to external matter. The goldstino mode associated with the breaking of (global) supersymmetry is "eaten" by the gravitino field, which becomes massive (via its own vacuum condensation) and breaks the local supersymmetry (supergravity) dynamically. The most natural association of gravitino condensates with inflation proceeds in an indirect way, via a Starobinsky-inflation-type phase. The higher-order curvature corrections of the (quantum) effective action of gravitino condensates induced by integrating out massive gravitino degrees of freedom in a curved space-time background, in the brokensupergravity phase, are responsible for inducing a scalar mode which inflates the Universe. The scenario is in agreement with Planck data phenomenology in a natural and phenomenologically-relevant range of parameters, namely Grand-Unified-Theory values for the supersymmetry breaking energy scale and dynamically-induced gravitino mass.

10.1051/epjconf/20159503023

1412.6437

f301e5f97ec81a0f5070e261e449eddc79936bfb

Gravitino Condensates in the Early Universe and Inflation Nick E Mavromatos Department of Physics Theoretical Particle Physics and Cosmology Group King's College London Strand LondonWC2R 2LSUK Physics Department-Theory Division CERN 1211Geneve 23CHSwitzerland Gravitino Condensates in the Early Universe and Inflation EPJ Web of Conferences will be set by the publisher DOI: will be set by the publisher We review work on the formation of gravitino condensates via the super-Higgs effect in the early Universe. This is a scenario for both inflating the early universe and breaking local supersymmetry (supergravity), entirely independent of any coupling to external matter. The goldstino mode associated with the breaking of (global) supersymmetry is "eaten" by the gravitino field, which becomes massive (via its own vacuum condensation) and breaks the local supersymmetry (supergravity) dynamically. The most natural association of gravitino condensates with inflation proceeds in an indirect way, via a Starobinsky-inflation-type phase. The higher-order curvature corrections of the (quantum) effective action of gravitino condensates induced by integrating out massive gravitino degrees of freedom in a curved space-time background, in the brokensupergravity phase, are responsible for inducing a scalar mode which inflates the Universe. The scenario is in agreement with Planck data phenomenology in a natural and phenomenologically-relevant range of parameters, namely Grand-Unified-Theory values for the supersymmetry breaking energy scale and dynamically-induced gravitino mass. Introduction and Summary The inflationary paradigm is at present a successful one, offering an elegant solution to the so-called horizon and flatness problems of the standard Big Bang cosmology, whilst simultaneously seeding both the large-scale structure of the universe and temperature anisotropies of the CMB via quantum fluctuations occurring during the inflationary epoch. The precise microscopic mechanism of inflation is however unknown at present. The data favour, or -from a rather more conservative viewpoint -are in agreement with, a scalar field or fields with canonical kinetic terms slowly rolling down an almost flat potential in the context of Einstein gravity, generating in the process 50 -60 e-folds of inflation, along with adiabatic, nearly scale invariant primordial density perturbations [1,2]. From the best fit value of the running spectral index n s ∼ 0.96 for the gravitational perturbations in the slow-roll parametrisation, found by Planck [1], and the usual relations among the slow-roll inflationary parameters [2] n s = 1 − 6 + 2η , r = 16 , we then find r 0.11 given the non-observation of primordial gravitational wave-like (transverse and traceless) perturbations by Planck or WMAP collaborations (that is the absence of B-mode polarisations). This observational fact implies that the energy scale E I of inflation is much smaller than the Planck scale m P , lying in the ballpark of the Grand Unified Theory (GUT) scale [1,2] GeV, with M Pl = 2.4 × 10 18 GeV the reduced Planck mass, H I the Hubble scale during inflation and r the tensor-to-scalar perturbation ratio [2].The upper bound on r < 0.11 placed by the Planck Collaboration [1] implies H I = 1.05 r 0.20 1/2 × 10 14 GeV ≤ 0.78 × 10 14 GeV , that is an upper bound four order of magnitudes smaller than the reduced Planck mass. An important issue at present is the extent to which this inflationary process is tied to physics at the Grand Unification (GUT) scale, and in particular, to a possible supersymmetric phase transition occurring in the early universe. Links of supersymmetry to inflation may be arguably expected from the fact that supersymmetry provides a rather natural reason [3] for the smallness (compared to Planck scale) value of the inflationary Hubble scale (2). If supersymmetry is realised in nature however, it is certainly broken. It is known that simple realisations of global supersymmetry (SUSY) breaking, such as in the Wess-Zumino model [4], can provide, when embedded in gravitational environments, slow-roll models for inflation consistent with the Planck data [1]. Rigorous embeddings of global SUSY to local supersymmetry (SUGRA) have also been considered and explored in the literature over the years in connection with various scenarios for inflation [5], such as hybrid [6], chaotic [7], no-scale SUGRA/Starobinsky-like [8]. In the latter case inflation is linked to higher curvature terms in the gravitational action (such as R 2 terms), as in the original Starobinsky model [9], and others [10,11]. Such models have been compared against the recently available data, with the conclusion that Planck data [1] compatibility is straightforward. However, In March 2014, the scientific community has been stunned by claims from the BI-CEP2 collaboration [12] on the measurement for the first time of B-mode polarization in the cosmic microwave background radiation, which was interpreted as evidence for gravitational waves at the time of the last scattering, with a tensor-to-scalar ratio r = 0.16 +0.06 −0.05 after dust subtraction. The BICEP2 data are consistent with a scalar spectral index n s 0.96 and no appreciable running, in agreement with the Planck data [1], but the Hubble parameter during slow-roll inflation H I indicated by the BICEP2 data is larger than then upper limit (2), imposed by Planck, that is one has now H Bicep2 I ∼ 0.94×10 14 GeV for r = 0.16. If confirmed by subsequent experiments, such a large value of r would exclude Starobinsky-type inflationary potentials and mostly favour φ 2 tachyonic models for inflation [2]. However, there is currently an active debate as to whether the BICEP2 signal truly represents primordial gravitational waves, or is polluted by Galactic foregrounds and gravitationally-lensed E-modes [13]. According to such works, the BICEP2 signal could be compatible with a cosmology with r 0.1 if there is a dust polarization effect as large as presently allowed by Planck [1] and other data. These remarks have recently been reinforced by data on the foreground dust in the BICEP2 region released by the Planck collaboration [14], which point to significant foreground pollution that would affect the interpretation of the BICEP2 B-mode polarization data. One therefore needs to wait for the results of the planned joint analysis by the Planck and BICEP2 teams, before any definite conclusions are drawn on this important issue. Until this is resolved, it is advisable to keep an open mind about the possible range of r values that models of inflation could yield. Therefore it seems premature to abandon Starobinsky-like models as potential candidates for realistic models of inflation compatible with the data. In this talk we shall present a rather minimal inflationary scenario which is associated indirectly with a Starobinsky type inflation. The approach is documented in a series of previous publications [15][16][17], and is based on the possibility of dynamically breaking SUGRA solely by means of exploiting the four-gravitino interactions that characterise (any) supergravity action, via the fermionic torsion parts of the spin connection. The primary example, where the calculations of the effective potential were detailed, was that of N = 1, D = 4 simple SUGRA without matter [18,19]. The dynamical breaking process may be concretely realised by means of a phase transition from the supersymmetric phase where the bilinear ψ µ ψ µ representing the effective scalar degree of freedom has zero vacuum expectation value, to one where σ ≡ ψ µ ψ µ 0. The quantum excitations about this condensate vacuum are then identified with a gravitino condensate scalar field. Since this must be an energetically favourable process to occur, it then follows that the effective potential experienced by the gravitino condensate must be locally concave about the origin. The corresponding one-loop effective potential of the gravitino condensate scalar field, obtained after integrating out fermionic (gravitino) and bosonic (graviton) degrees of freedom therefore has the characteristic form of a Coleman-Weinberg double well potential, offering the possibility of hilltoptype inflation, with the condensate field playing the role of the inflaton [15,16], which would be the simplest scenario. However, in order to guarantee a slow-roll inflation one needs unnaturally large values of the gravitino-condensate wave function renormalisation. This prompted us to discuss a second, rather indirect way, by means of which the gravitino condensate is associated with inflation [17]. This is realised via the higher (in particular quadratic-order) curvature corrections of the (quantum) effective action of the gravitino condensate field, obtained after integration of graviton and gravitino degrees of freedom in the massive gravitino phase. These corrections induce a Starobinsky-type inflation [9], which occurs for quite natural values of the parameters of the N = 1 SUGRA model and its variants, as we shall review below. The structure of the talk is as follows: In Section 2 we review the formalism and physical concepts underlying dynamical breaking of SUGRA and the associated super-Higgs effect, within the context of simple four dimensional N = 1 models, including superconformal extensions thereof (with broken conformal symmetry) that are necessitated for phenomenological reasons, as explained in the text. In Section 3 we discuss the simplest possible scenario for hilltop inflation, where gravitino condensate fields near the origin of the effective potential play the role of the inflation field. Unfortunately, for (phenomenologically desirable) supersymemtry breaking scales that are near or below the GUT scale, the model is compatible with slow roll for very large (unnatural) values of the condensate wave function renormalisation. This prompts us to discuss in Section 4 alternative scenarios for inflation of Starobinsky type that may occur in the massive gravitino phase, near the non-trivial minimum of the effective potential. In such scenarios, which are compatible with the Planck results, the role of the inflaton field is played by the scalar mode that describes the effects of scalar-curvature-square terms that characterise the gravitational sector of the effective action in the broken SUGRA phase, after integrating out the massive gravitinos. Finally, conclusions and outlook are presented in section 5. S SG = d 4 x e 1 2κ 2 R (e) − ψ µ γ µνρ D ν ψ ρ + L torsion ,(3)κ 2 = 8πG γ µνρ = 1 2 {γ µ , γ νρ } , γ νρ = 1 2 γ ν , γ ρ , where R(e) and D ν ψ ρ ≡ ∂ ν ψ ρ + 1 4 ω νab (e) γ ab ψ ρ are defined via the torsion-free connection and, given the gauge condition γ · ψ = 0 ,(4) one can write L torsion = − 1 16 ψ ρ γ µ ψ ν ψ ρ γ µ ψ ν + 2ψ ρ γ ν ψ µ × 2κ 2 ,(5) arising from the fermionic torsion parts of the spin connection 1 . Extending the action off-shell requires the addition of auxiliary fields to balance the graviton and gravitino degrees of freedom. These fields however are non-propagating and may only contribute to topic at hand through the development of scalar vacuum expectation values, which would ultimately be resummed into the cosmological constant. Making further use of the gauge condition (4) in concert with the Fierz identities (as detailed in [16]), we may write L torsion = λ S ψ ρ ψ ρ 2 + λ PS ψ ρ γ 5 ψ ρ 2 + λ PV ψ ρ γ 5 γ µ ψ ρ 2(6) where the couplings λ S , λ PS and λ PV express the freedom we have to rewrite each quadrilinear in terms of the others via Fierz transformation. This freedom in turn leads to a known ambiguity in the context of mean field theory [23], which we addressed in [16], where we refer the reader for details. Following the original ideas of dynamical symmetry breaking by Nambu and Jona-Lasinio [24], we wish to linearise these four-fermion interactions via suitable auxiliary fields, e.g. 1 4 ψ ρ ψ ρ 2 ∼ σ ψ ρ ψ ρ − σ 2 ,(7) where the equivalence (at the level of the action) follows as a consequence of the subsequent Euler-Lagrange equation for the auxiliary scalar σ. Our task is then to look for a non-zero vacuum expectation value σ which would serve as an effective mass for the gravitino. To induce the super-Higgs effect [21] we also couple in the Goldstino associated to global supersymmetry breaking via the addition of 1 We note in passing that such four-fermion interactions are characteristic of any Einstein-Cartan theory of fermions in curved space-time [22]. In fact, in a standard spin-1/2 fermion-gravity theory, the torsion-induced four fermion interactions assume a repulsive axial (pseudovector)-current-current form − ψγ µ γ 5 ψ ψγ µ γ 5 ψ . As we demonstrate in the Appendix of the second article in ref. [16], a corresponding repulsive axial-current-current term for the gravitino torsion terms can also be obtained by appropriately utilising Fierz identities in analogy with the Einstein-Cartan theory. One thus would naively conclude that dynamical breaking of supergravity may not be possible. However, in all such theories, the Fierz identities among the fermions, including gravitinos, make the actual coefficient of such terms ambiguous. Only the non-perturbative physics can settle the value of the four-fermion terms [23], and hence dynamical breaking of symmetry via scalar gravitino condensates is a realistic possibility. L λ = f 2 det δ µν + i 2 f 2 λγ µ ∂ ν λ γ·ψ=0 = f 2 + . . . ,(8) where λ is the Goldstino, f expresses the scale of global supersymmetry breaking, and . . . represents higher order terms which may be neglected in our weak-field expansion of the determinant. It is worth emphasising at this point the universality of (8); any model containing a Goldstino may be related to L λ via a non-linear transformation [20], and thus the generality of our approach is preserved. Upon the specific gauge choice (4) for the gravitino field and an appropriate redefinition, one may eliminate any presence of the Goldstino field from the final effective action describing the dynamical breaking of local supersymmetry, except the cosmological constant term f 2 in (8), which serves as a reminder of the pertinent scale of supersymmetry breaking. The non-trivial energy scale this introduces, along with the disappearance (through field redefinitions) of the Goldstino field from the physical spectrum and the concomitant development of a gravitino mass, characterises the super-Higgs effect. We may then identify in the broken phase an effective action S = 1 2κ 2 d 4 x e (R (e) − 2Λ) − ψ µ γ µνρ D ν ψ ρ + m dyn ψ µ ψ µ ,(9) where Λ is renormalised cosmological constant, to be contrasted with the (negative) tree level cosmological constant Λ 0 ≡ κ 2 σ 2 − f 2 ,(10) and m dyn ∝ σ is a dynamically generated gravitino mass, the origin of which will be explained presently. It is worth stressing at this point that Λ 0 must be negative due to the incompatibility of supergravity with de Sitter vacua; if SUGRA is broken at tree level, then of course no further dynamical breaking may take place. For phenomenological reasons which have been outlined in detail in refs. [15,16], and we shall discuss below, we adopt an extension of N = 1 SUGRA which incorporates local supersymmetry in the Jordan frame, enabled by an associated dilaton superfield [10]. The scalar component ϕ of the latter can be either a fundamental space-time scalar mode of the gravitational multiplet, i.e. the trace of the graviton (as happens, for instance, in supergravity models that appear in the low-energy limit of string theories), or a composite scalar field constructed out of matter multiplets. In the latter case these could include the standard model fields and their superpartners that characterise the Nextto-Minimal Supersymmetric Standard Model, which can be consistently incorporated in such Jordan frame extensions of SUGRA [10]. Upon appropriate breaking of conformal symmetry, induced by specific dilaton potentials (which we do not discuss here), one may assume that the dilaton field acquires a non-trivial vacuum expectation value ϕ 0. One consequence of this is then that in the broken conformal symmetry phase, the resulting supergravity sector, upon passing (via appropriate field redefinitions) to the Einstein frame is described by an action of the form (3), but with the coupling of the gravitino four-fermion interaction terms being replaced bỹ κ ≡ e − ϕ κ ,(11) while the Einstein term in the action carries the standard gravitational coupling 1/2κ 2 . Expanding the graviton field about a de Sitter background [25] (under the assumption that it is a solution of the one-loop effective equations) with renormalised cosmological constant Λ > 0, and integrating out both bosonic and fermionic quantum fluctuations to one loop yields the following effective potential for the gravitino condensate field σ in the flat space-time limit Λ → 0, as detailed in ref. [16], V eff = V (0) B + V (1) B + V (1) F = − Λ 0 κ 2 + V (1) B + V (1) F , Λ 0 ≡ κ 2 σ 2 − f 2 ,(12)EPJ Web of Conferences where V (1) B = 45κ 4 512π 2 f 2 − σ 2 2         3 − 2 ln         3κ 2 f 2 − σ 2 2µ 2                 ,(13) and V (1) F =κ 4 σ 430976π indicate the contributions to the effective potential from bosonic and fermionic fields respectively, and µ is an inverse renormalisation group (RG) scale. The effective potential (12) is depicted in fig. 1. We may firstly note that as we flow from UV to IR (i.e. in the direction of increasing µ), we obtain the correct double-well shape required for the super-Higgs effect, and secondly that tuning f allows us to shift V eff and thus attain the correct vacuum structure (i.e. non-trivial minima σ c such that V eff (σ c ) = 0). Moreover, the shape of the effective potential changes, as one varies the New Frontiers in Physics 2014 (renormalisation) scale µ from ultraviolet to infrared values (i.e. flowing in the direction of increasing µ), in such a way that the broken symmetry phase (double-well shaped potential) is reached in the IR. This indicates that the dynamical generation of a gravitino mass is actually an IR phenomenon, in accordance with the rather general features of dynamical mass in field theory. In the broken phase, the mass of the gravitino condensate is then given by m 2 σ ≡ V eff (σ c ) ,(15) where σ c is the minimum of V eff and a prime denotes a functional derivative with respect to the gravitino-condensate field. As observed from (13), the bosonic contributions to the effective potential contain logarithmic terms which would contribute imaginary terms, leading to instabilities, unless σ 2 c < f 2 .(16) From (12) it is straightforward to see that this condition is equivalent to the negativity of the treelevel cosmological constant Λ 0 , which is entirely sensible; if Λ 0 > 0 then SUGRA is broken at tree level (given the incompatibility of supersymmetry with de Sitter vacua) and there can be no dynamical breaking. As such, we must then tune f for a given value of µ to find self consistent minima σ c satisfying (16), thereby ensuring a real V eff . In fact, here lies the importance of the super-Higgs effect, and thus of a non-zero positive f 2 > σ 2 c > 0, in allowing dynamical breaking of local supersymmetry 2 . As discussed in refs. [15,16], phenomenologically realistic situations, where one avoids transplanckian gravitino masses, for supersymmetry breaking scales f at most of order of the Grand Unification (GUT) scale 10 15−16 GeV, as expected from arguments related to the stability of the electroweak vacuum, can occur only for largeκ couplings, typically of orderκ ∼ 10 3 − 10 4 κ. Given the relation (11) this corresponds to dilaton vev of O (−10), where the negative sign may be familiar in the context of dilaton-influenced cosmological scenarios [27]. If we consider for concreteness the caseκ = 10 3 κ, which is a value dictated by the inflationary phenomenology of the model [15], we may find solutions with a vanishing one-loop effective potential at the non-trivial minima corresponding to: κ 2 σ c 3.5 ,κ 2 f 3.7 ,κ µ 4.0 ,(17) which leads to a global supersymmetry breaking scale f 4.7 × 10 15 GeV ,(18) and dynamical gravitino mass m dyn 2.0 × 10 16 GeV . At the non-trivial minima we findκ 4 V (1) F −1.4,κ 4 V (1) B 5.9×10 −13 , with tree-level cosmological constantκ 2 Λ 0 −1.4. We thus observe that fermion contributions to the effective potential are much stronger than the corresponding bosonic contributions for the cases of large couplingsκ κ. These values are phenomenologically realistic, thereby pointing towards the viability (from the point of view of producing realistic results of relevance to phenomenology) of the scenarios of dynamical breaking of local supersymmetry in conformal supergravity models 3 . On the other hand, in standard SUGRA scenarios, whereκ = κ, one finds, as already mentioned, transplanckian values for the dynamically generated gravitino mass [16]: m dyn 2.0 × 10 19 GeV, and a global supersymmetry breaking scale f 4.7 × 10 18 GeV, far too high to make phenomenological sense. Connection with Slow-Roll Hill-top Inflation In order to discuss the possible connection with inflation, we need to calculate one more important ingredient; the wave-function renormalisation. In principle, this should be calculated in a curved de Sitter space-time, which characterises the (unbroken) phase of SUGRA, when the condensate field is near the trivial maximum of the effective potential (12). This is a complicated task. However, it turns out that, since, according to the data [1,2], the de Sitter phase Hubble parameter in phenomenologically relevant inflationary models is expected to be several orders of magnitude smaller than the Planck scale, m P (2), the space-time curvature during inflation is not too large, and thus a flat space-time estimate of the wave function renormalisation may suffice. The effective Lagrangian describing the gravitino bound state, with a non-trivial wave-function renormalization, is L eff = Zκ 2 2 ∂ µ σ∂ µ σ − V eff (σ) ,(20) where the rescaling σ =σ/κ √ Z leads to the canonically normalised Lagrangiañ L eff = 1 2 ∂ µσ ∂ µσ −Ṽ eff (σ) ,(21) and the coupling constants in the potentialṼ eff are defined as V (n) eff (0) ≡ V (n) eff (0) Z n/2 .(22) The latter normalisations ultimately yield the slow roll parameters = 1 Z M 2 Pl 2 V eff V eff 2 , η = 1 Z M 2 Pl V eff V eff , ξ = 1 Z 2 M 4 Pl V eff V eff V 2 eff .(23) 3 A comment concerning SUGRA models in the Jordan frame with such large values for their frame functions is in order here. In our approach, the dilaton ϕ could be a genuine (dimensionless) dilation scalar field ϕ = 2φ arising in the gravitational multiplet of string theory, whose low-energy limit may be identified with some form of SUGRA action. In our normalization the string coupling would be g s ≡ e φ =κ −1/2 . In such a case, a value ofκ = e − ϕ κ = O(10 3−4 ) would imply a large negative v.e.v. of the (four-dimensional) dilaton field of order φ = −O(5) < 0, and thus a weak string coupling squared g s = O(10 −2 ), which may not be far from values attained in phenomenologically realistic string cosmologies [27]. On the other hand, in the Jordanframe SUGRA models of [10], the frame function reads Φ ≡ e −ϕ = 1 − 1 3 S S + u,d H i H † i − 1 2 χ − H 0 u H 0 d + H + u H − d + h. c. , in the notation of [28] for the various matter super fields of the next-to-minimal supersymmetric standard model that can be embedded in such supergravities. The quantity χ is a constant parameter. At energy scales much lower than GUT, it is expected that the various fields take on subplanckian values, in which case the frame function is almost one, and henceκ κ for such models today. To ensureκ κ, and thus large values of the frame function, Φ 1, as required in our analysis, one needs to invoke trasnplanckian values for some of the fields, H 0 u,d , and large values of χ, which may indeed characterize the inflationary phase of such theories. A similar situation occurs for the values of the higgs field (playing the role of the inflaton) in the non-supersymmetric Higgs inflation models [29]. New Frontiers in Physics 2014 That large values of Z 1 are necessarily linked to slow-roll hilltop inflation in this case is to be expected from the fact that the effective potential (12) can be approximated near the origin (i.e. for small field values of the condensateσ → 0) as: V eff f 2 − (Zκ 2 ) −1σ2 ,σ → 0 ,(24) for a canonically normalised condensate fieldσ. To ensure that the slow-roll parameter |η| < 1 (23), then, we must have Z M 4 Pl f 2 .(25) Since the (observed) running spectra index is of order η s 0.96 [1], we must further impose that |η| < 10 −2 . As already mentioned, we assume that we can use the flat space-time results for the wave-function renormalisation obtained in [16], to obtain a correct order of magnitude estimate that is valid in the curved space-times during the inflationary period [15]. We first note, that in the broken phase, with the phenomenologically acceptable values of the gravitino mass m dyn and supersymmetry breaking scales f the function Z is of order one, which is consistent with the exit from the slow-roll inflationary phase. Near the origin of the potential (12), the wave-function renornmalization is [16] Z − 1 2π 2 ln ω 2 µ , ωμ ≡μ/C off , g ≡ λ S C 2 off /2π 2 .(26) whereμ is a transmutation mass scale and C off is a flat-space-time cutoff, which may be taken to be the Planck scale, for low energy theories. For appropriate values of g this corresponds to the limit of large Z 1 and small ωμ, both of which are phenomenologically desirable 4 . Phenomenologically realistic models of broken SUSY have f < 10 16 GeV = 10 −2 M Pl (cf. (18)), hence we must have Z 10 10 , implying very small, practically vanishing, transmutation mass scales. A typical case, compatible with the phenomenologically acceptable values (18) and (19) is given in figure 2, from which we observe that agreement with Planck results is achieved for values of the wave function renormalisation of order Z ∼ O 10 16 for the phenomenologically relevant values of the couplingsκ/κ ∼ 10 3 . This corresponds to practically zero transmutation mass scales ofμ → 0. It may be interesting to notice that increasingκ/κ higher still has the effect of scaling V eff whilst leaving the shape of the potential qualitatively unchanged, allowing smaller and smaller values of f and m dyn . Whilst this decrease in f tends to naturally increase the slow-roll parameter η, by virtue of (25) this scenario may still be rendered compatible with slow-roll inflation if Z is scaled accordingly to counteract this. As such, Planck compatible inflation as demonstrated in figure 2 can be achieved for any value ofκ/κ. The Planck-compatible result is (0.959, 0.04) ≤ {n s , r} ≤ (0.964, 0.03) for 50 and 60 e-folds, respectively, corresponding to f ∼ 5 e ϕ × 10 18 GeV. This is the case for any value of the (negative) dilaton vev ϕ , however, as already mentioned, for realistic supersymmetry breaking phenomenology one should really fix f around or below the GUT scale. One way to interpret this result is the following. Near the origin of the potential one is in the unbroken phase, and hence the gravitino condensate has not yet fully formed, or rather is beginning to form, corresponding to a very small value of the gravitino mass. This small value grows in actual time, until the condensate sits in the minimum of the potential after rolling downhill, at which point the gravitino mass is stabilised at phenomenologically acceptable value, e.g. of order the GUT scale. The duration of the whole process is that of the slow-roll inflation period, and exit from this phase occurs near the non-trivial minimum of the potential (12). [1] for n s and r with the gravitino-condensate hill-top inflation indicated explicitly (dark green). The latter model leads to higher r than Starobinsky-type R 2 inflation (orange), although requires a very high value of the gravitino-condensate wave-function renormalisation, of order larger than O(10 16 ). One may object to the huge value of the wave function renormalisation (25) during the slow-roll inflationary phase as unnatural 5 . There are alternative scenarios of slow-roll inflation linked to this model which do not require such large Z, which we shall now come to discuss. These are associated with another type of inflation that may occur in the broken SUGRA phase, where, in contrast to the hill-top inflationary scenario discussed so far, the gravitino condensate field lies near its value that minimises the potential (12). In this scenario, the inflaton field is not the gravitino condensate, but it is linked to the scalar mode that parametrises a R 2 -Starobinsky-like [9] inflation that is associated with the effective gravitational action obtained after integrating out the massive gravitino-condensate degrees of freedom. This scenario was discussed in detail in ref. [17], and we now proceed to review it briefly. Starobinsky-type inflation in the broken SUGRA phase Starobinsky inflation is a model for obtaining a de Sitter (inflationary) cosmological solution to gravitational equations arising from a (four space-time-dimensional) action that includes higher curvature terms. Specifically, an action of the type in which the quadratic curvature corrections consist only of scalar curvature terms [9] 5 One may be tempted to discuss, within the context of our minimal model, an alternative scenario, according to which global SUSY breaks at a transplanckian scale f 1 (in Planck units). In this case, the SUSY matter partners would completely decouple from the low-energy spectrum, and hence there would be no experimental evidence for SUSY. On the other hand, local SUSY (SUGRA) would ensure inflation via the gravitino condensation mechanism described in this work, while the induced transplanckian dynamical mass for the gravitino, would remove any possibility of observing it as well. From (25) we can then conclude that slow-roll inflation could be achieved for natural values of the wave-function renormalisation Z < O(10), but in this case the stability of the electroweak vacuum would be delinked from any SUSY arguments. One could also try to relax the slow-roll assumption but this opens up a whole new game, where comparison with data may be complicated, and we do not consider it here. S = 1 2 κ 2 d 4 x √ −g R + β R 2 , β = 8 π 3 M 2 ,(27) where κ 2 = 8πG, and G = 1/m 2 P is Newton's (gravitational) constant in four space-time dimensions, with m P the Planck mass, and M is a constant of mass dimension one, characteristic of the model. The important feature of this model is that inflationary dynamics are driven purely by the gravitational sector, through the R 2 terms, and that the scale of inflation is linked to M. From a microscopic point of view, the scalar curvature-squared terms in (27) are viewed as the result of quantum fluctuations (at one-loop level) of conformal (massless or high energy) matter fields of various spins, which have been integrated out in the relevant path integral in a curved background space-time [31]. The quantum mechanics of this model, proceeding by means of tunnelling of the Universe from a state of "nothing" to the inflationary phase of ref. [9] has been discussed in detail in ref. [32]. The above considerations necessitate truncation to one-loop quantum order and to curvature-square (fourderivative) terms, which implies that there must be a region of validity for curvature invariants such that O R 2 /m 4 p 1. This is of course a condition satisfied in phenomenologically realistic scenarios of inflation [1,2], for which the inflationary Hubble scale H I satisfies (2) (the reader should recall that R = 12H 2 I in the inflationary phase). Although the inflation in this model is not driven by rolling scalar fields, nevertheless the model (27) (and for that matter, any other model where the Einstein-Hilbert space-time Lagrangian density is replaced by an arbitrary function f (R) of the scalar curvature) is conformally equivalent to that of an ordinary Einstein-gravity coupled to a scalar field with a potential that drives inflation [33]. To see this, one firstly linearises the R 2 terms in (27) by means of an auxiliary (Lagrange-multiplier) field α(x), before rescaling the metric by a conformal transformation and redefining the scalar field (so that the final theory acquires canonically-normalised Einstein and scalar-field terms): g µν → g E µν = (1 + 2 βα(x)) g µν ,α (x) → ρ(x) ≡ 3 2 ln (1 + 2 βα (x)) .(28) These steps may be understood schematically via d 4 x √ −g R + β R 2 → d 4 x −g E R E + g E µ ν ∂ µ ρ ∂ ν ρ − V ρ )) ,(29) where the arrows have the meaning that the corresponding actions appear in the appropriate path integrals, with the potential V(ρ) given by: V(ρ) = 1 − e − √ 2 3 ρ 2 4 β = 3M 2 1 − e − √ 2 3 ρ 2 32 π .(30) The potential is plotted in fig. 3. We observe that it is sufficiently flat for large values of ρ (compared to the Planck scale) to produce phenomenologically acceptable inflation, with the scalar field ρ playing the role of the inflaton. In fact, the Starobinsky model fits the Planck data on inflation [1] well. The agreement of the model of ref. [9] with the Planck data has triggered an enormous interest in the current literature in revisiting the model from various points of view, such as its connection with no-scale supergravity [8] and (super)conformal versions of supergravity and related areas [7,11]. In the latter works however the Starobinsky scalar field is fundamental, arising from the appropriate scalar component of some chiral superfield that appears in the superpotentials of the model. Although of great value, illuminating a connection between supergravity models and inflationary physics, and especially for explaining the low-scale of inflation compared to the Planck scale, it can be argued that these works contradict the original spirit of the Starobinsky model (27). Therein, higher curvature corrections are viewed as arising from quantum fluctuations of matter fields in a curved space-time background, such that inflation is driven by the pure gravity sector in the absence of fundamental scalars. Figure 3. The effective potential (30) of the collective scalar field ρ that describes the one-loop quantum fluctuations of matter fields, leading to the higher-order scalar curvature corrections in the Starobinski model for inflation (27). The potential is sufficiently flat to ensure slow-roll conditions for inflation are satisfied, in agreement with the Planck data, for appropriate values of the scale 1/β ∝ M 2 (which sets the overall scale of inflation in the model). . Generic shape of the one-loop effective potential for the gravitino condensate field σ c in dynamically broken (conformal) Supergravity models in the presence of a non-trivial de Sitter background with cosmological constant Λ > 0 [17]. The Starobinsky inflationary phase is associated with fluctuations of the condensate and gravitational field modes near the non-trivial minimum of the potential, where the condensate σ c 0, and the potential assumes the value Λ > 0, consistent with supersymmetry breaking. The dashed green lines denote "forbidden" areas of the condensate field values, violating the condition (16), for which imaginary parts appear in the effective potential, thereby destabilising the broken symmetry phase. ⇢ In this section we consider an extension of the analysis of ref. [16] to the case where the de Sitter parameter Λ is perturbatively small compared to m 2 P , but not zero, so that truncation of the series to order Λ 2 suffices. This is in the spirit of the original Starobinsky model [9], with the rôle of matter fulfilled by the now-massive gravitino field. Specifically, we are interested in the behaviour of the effective potential near the non-trivial minimum, where σ c is a non-zero constant. It is important to notice at this point that, in contrast to the original Starobinsky model [9], where the crucial for inflation R 2 terms have been argued to arise from the conformal anomaly in the path integral of massless (conformal) matter in a de Sitter background, and thus their coefficient was arbitrary, in our scenario, such terms arise in the one-loop effective action of the gravitino condensate field, evaluated in a de Sitter background, after integrating out massive gravitino fields, whose mass was generated dynamically. The order of the de Sitter cosmological constant, Λ > 0 that breaks supersymmetry, and the gravitino mass are all evaluated dynamically (self-consistently) in our approach from the minimization of the effective potential. Thus, the resulting R 2 coefficient, which determines the phenomenology of the inflationary phase, is calculable [17]. Moreover, in our analysis, unlike Starobinsky's original work, we will keep the contributions from both graviton (spin-two) and gravitino quantum fluctuations. Specifically, we are interested in the behaviour of the effective potential near the non-trivial minimum, where σ c is a non-zero constant (cf fig. 4). The one-loop effective potential, obtained by integrating out gravitons and (massive) gravitino fields in the scalar channel (after appropriate euclideanisation), may be expressed as a power series in Λ: Γ S cl − 24π 2 Λ 2 α F 0 + α B 0 + α F 1 + α B 1 Λ + α F 2 + α B 2 Λ 2 + . . . ,(31) where S cl denotes the classical action with tree-level cosmological constant Λ 0 : − 1 2κ 2 d 4 x √ g R − 2Λ 0 , Λ 0 ≡ κ 2 σ 2 − f 2 ,(32) with R denoting the fixed S 4 background we expand around ( R = 4Λ, Volume = 24π 2 /Λ 2 ), and the α's indicate the bosonic and fermionic quantum corrections at each order in Λ. The leading order term in Λ is then the effective action found in [16] in the limit Λ → 0, Γ Λ→0 − 24π 2 Λ 2 − Λ 0 κ 2 + α F 0 + α B 0 ≡ 24π 2 Λ 2 Λ 1 κ 2 ,(33) and the remaining quantum corrections then, proportional to Λ and Λ 2 may be identified respectively with Einstein-Hilbert R-type and Starobinsky R 2 -type terms in an effective action (34) of the form Γ − 1 2κ 2 d 4 x √ g R − 2Λ 1 + α 1 R + α 2 R 2 ,(34) where we have combined terms of order Λ 2 into curvature scalar square terms. For general backgrounds such terms would correspond to invariants of the form R µνρσ R µνρσ , R µν R µν and R 2 , which for a de Sitter background all combine to yield R 2 terms. The coefficients α 1 and α 2 absorb the non-polynomial (logarithmic) in Λ contributions, so that we may then identify (34) with (31) via α 1 = κ 2 2 α F 1 + α B 1 , α 2 = κ 2 8 α F 2 + α B 2 .(35) To identify the conditions for phenomenologically acceptable Starobinsky inflation around the non-trivial minima of the broken SUGRA phase of our model, we impose first the cancellation of the "classical" Einstein-Hilbert space term R by the "cosmological constant" term Λ 1 , i.e. that R = 4 Λ = 2 Λ 1 . This condition should be understood as a necessary one characterising our background in order to produce phenomenologically-acceptable Starobinsky inflation in the broken SUGRA phase following the first inflationary stage, as discussed in ref. [15]. This may naturally be understood as a generalisation of the relation R = 2Λ 1 = 0, imposed in ref. [16] as a self-consistency condition for the dynamical generation of a gravitino mass. From thid it follows that the cosmological constant Λ satisfies the four-dimensional Einstein equations in the non-trivial minimum, and in fact coincides with the value of the one-loop effective potential of the gravitino condensate at this minimum. As we discussed in [16], this non-vanishing positive value of the effective potential is consistent with the generic features of dynamical breaking of supersymmetry [34]. In terms of the Starobinsky inflationary potential (30), the value Λ > 0 corresponds to the approximately constant value of this potential in the high ϕ-field regime of fig. 3, where Starobinsky-type inflation takes place. Thus we may set Λ ∼ 3 H 2 I , where H I the (approximately) constant Hubble scale during inflation, which is constrained by the current data to satisfy (2). The effective Newton's constant in (34) is then κ 2 eff = κ 2 /α 1 , and from this, we can express the effective Starobinsky scale (27) in terms of κ eff as β eff ≡ α 2 /α 1 . This condition thus makes a direct link between the action (31) with a Starobinsky type action (27). Comparing with (27), we can then identify the Starobinsky inflationary scale in this case as M = 8π 3 α 1 α 2 .(36) We may then determine the coefficients α 1 and α 2 in order to evaluate the scale 1/β of the effective Starobinsky potential given in fig. 3 in this case, and thus the scale of the second inflationary phase. To this end, we use the results of ref. [16], derived via an asymptotic expansion as explained in the appendix therein, to obtain the following forms for the coefficients α F 1 = 0.067κ 2 σ 2 c − 0.021κ 2 σ 2 c ln Λ µ 2 + 0.073κ 2 σ 2 c ln κ 2 σ 2 c µ 2 , α F 2 = 0.029 + 0.014 ln κ 2 σ 2 c µ 2 − 0.029 ln Λ µ 2 ,(37) and α B 1 = −0.083Λ 0 + 0.018 Λ 0 ln Λ 3µ 2 + 0.049 Λ 0 ln − 3Λ 0 µ 2 , α B 2 = 0.020 + 0.021 ln Λ 3µ 2 − 0.014 ln − 6Λ 0 µ 2 ,(38) where Λ 0 has been defined in (32), σ c denotes the gravitino scalar condensate at the non-trivial minimum of the one-loop effective potential (cf. fig. 4), andκ = e − ϕ κ is the conformally-rescaled gravitational constant in the model of ref. [10], defined previously via (11). In the case of standard N = 1 SUGRA, ϕ = 0. We note at this stage that the spin-two parts, arising from integrating out graviton quantum fluctuations, are not dominant in the conformal case [16], providedκ/κ ≥ O(10 3 ), which leads [15] to the agreement of the first inflationary phase of the model with the Planck data [1]. However, if the first phase is succeeded by a Starobinsky phase, it is the latter only that needs to be checked against the data. To this end we search numerically for points in the parameter space such that; the effective equations ∂Γ ∂Λ = 0 , ∂Γ ∂σ = 0 ,(39) New Frontiers in Physics 2014 are satisfied, Λ is small and positive (0 < Λ < 10 −5 M 2 Pl , to ensure the validity of our expansion in Λ) and 10 −6 < M/M Pl < 10 −4 , to match with known phenomenology of Starobinsky inflation [1]. Forκ = κ (i.e. for non-conformal supergravity), we were unable to find any solutions satis- fying these constraints. This of course may not be surprising, given the previously demonstrated non-phenomenological suitability of this simple model [16]. If we considerκ >> κ however, we find that we are able to satisfy the above constraints for a range of values. We present this via the two representative cases below, indicated in fig. 5, where f is the scale of global supersymmetry breaking, and we have set the normalisation scale via κµ = √ 8π. Every point in the graphs of the figures is selected to make the Starobinsky scale of order M ∼ 10 −5 M Pl , so as to be able to achieve phenomenologically acceptable inflation in the massive gravitino phase, consistent with the Planck-satellite data [1]. Exit from the inflationary phase is a complicated issue which we shall not discuss here, aside from the observation that it can be achieved by coherent oscillations of the gravitino condensate field around its minima, or tunnelling processes à la Vilenkin [32]. This is still an open issue, which may be addressed via construction of more detailed supersymmetric models, including coupling of the matter sector to gravity. ity) and inflation. From our analysis above, it seems that in order to ensure phenomenologically relevant (for the stability of the electroweak vacuum) supersymmetry breaking scales and gravitino masses, one needs to apply the above ideas, not to the minimal SUGRA model, but to Jordan-frame extensions thereof, involving a third scalar field (dilaton). In the context of the next to minimal supersymmetric standard model, which Jordan-frame SUGRA models can incorporate, such dilatons may be composite of appropriate matter superfields, involving Higgs (supermultiplets). Details of the microscopic matter model are important to ensure the correct cosmological evolution, in particular satisfaction of the Big-Bang-Nucleosynthesis constraints. A GUT scale gravitino can be made to decay fast enough so as not to disturb the BBN, but this depends on the details of the matter sector of the theory, which we have not discussed so far. We plan to do so in the future. Nevertheless, we believe that the above dynamical breaking of supergravity scenario and the links with Starobinsky inflation are interesting paradigms, which have a chance of leading to realistic phenomenological scenarios compatible with the cosmological and particle physics data. 2 Super-Higgs effect and dynamical breaking of N = 1 SUGRA Our starting point is the N = 1 D = 4 (on-shell) action for 'minimal' Poincaré supergravity in the second order formalism, following the conventions of ref. [18] (with explicit factors of the (dimensionful) gravitational constant κ 2 = 8πG = 1/M 2 Pl , in units = c = 1, where M Pl the reduced Planck mass in four space-time dimensions): Figure 1 . 1Upper panel: The effective potential(12), expressed in units of the couplingκ (11). Lower panel: As above, but showing schematically the effect of tuning the RG scale µ and the supersymmetry breaking scale f , whilst holding, respectively, f and µ fixed. The arrows in the respective axes correspond to the direction of increasing µ and f . Figure 2 . 2Planck data Figure 4 4Figure 4. Generic shape of the one-loop effective potential for the gravitino condensate field σ c in dynamically broken (conformal) Supergravity models in the presence of a non-trivial de Sitter background with cosmological constant Λ > 0 [17]. The Starobinsky inflationary phase is associated with fluctuations of the condensate and gravitational field modes near the non-trivial minimum of the potential, where the condensate σ c 0, and the potential assumes the value Λ > 0, consistent with supersymmetry breaking. The dashed green lines denote "forbidden" areas of the condensate field values, violating the condition (16), for which imaginary parts appear in the effective potential, thereby destabilising the broken symmetry phase. Figure 5 . 5Left panel: Results forκ = 10 3 κ. Right panel: Results forκ = 10 4 κ. It should be mentioned at this point that in refs.[26], the importance of the super-Higgs effect was ignored, which led to the incorrect conclusion that imaginary parts exist necessarily in the one-loop effective potential (in the same class of gauges as the one considered in ref.[16] and here) and hence dynamical breaking of SUGRA was not possible. As we have seen above, such imaginary parts are absent when the condition (16) is satisfied, and thus dynamical breaking of SUGRA occurs. Note that larger than one values of the wave-function renormalisation for the composite gravitino condensate fields do not contradict unitarity. A similar situation is encountered in composite Higgs symmetry breaking models in field theory[30]. ConclusionsIn this talk we considered a minimal inflationary scenario, by means of which a gravitino condensate in supergravity models is held responsible for breaking local supersymmetry dynamically and inducing inflation in an indirect way by means of a Starobinsky-type inflation in the massive gravitino phase. Although Inflation of hilltop type via gravitino condensate, where the inflaton is the gravitino condensate field itself, appears at first sight the simplest scenario, nevertheless to ensure slow roll in such a case would require unnaturally high values of the condensate wave-function renormalization, unless the supersymmetry-breaking scale assumed transplanckian values. It is in this sense that the Starobinsky-type scenario for inflation, which is associated with the scalar mode that collectively parametrizes the effects of the quadratic-curvature contributions to the effective action of the gravitino condensate, after integrating out graviton and massive gravitino degrees of freedom, appears quite natural. It involves parameters that assume values of a natural and phenomenologically relevant order of magnitude, specifically global supersymmetry scale and gravitino masses of the order of GUT mass scales or less.Such a scenario is a truly minimal scenario for natural inflation, in the sense that it involves two scalar primordial composite modes, to achieve dynamical breaking of a gauge symmetry (supergrav- ]; for a general survey of Planck results including inflation, see. 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We show that the following double integralremains invariant as one trades the parameters p and q for p ′ = 1 − p 2 and q ′ = 1 − q 2 respectively. This invariance property is suggested from symmetry considerations in the operating characterstics of a semiconductor Hall-effect device.

10.1007/s11139-017-9915-4

1701.06310

d5cb26e2bd385b84aaf7fee8e71720459341f184

A FUNCTIONAL IDENTITY INVOLVING ELLIPTIC INTEGRALS 24 Jan 2017 M Lawrence Yajun Zhou A FUNCTIONAL IDENTITY INVOLVING ELLIPTIC INTEGRALS 24 Jan 2017Incomplete elliptic integralscomplete elliptic integralsLanden's transformation Subject Classification (AMS 2010): 33E05 (Primary)78A35 (Secondary) We show that the following double integralremains invariant as one trades the parameters p and q for p ′ = 1 − p 2 and q ′ = 1 − q 2 respectively. This invariance property is suggested from symmetry considerations in the operating characterstics of a semiconductor Hall-effect device. Introduction When an electron current flows perpendicular to a magnetic field through a conducting medium, the charges are forced to deviate to one side creating an imbalance which results in a measurable electric potential conveying important information about the material. A device based on this, so-called Hall effect, has been studied in detail by Ausserlechner [1] who has found that its operating features are summed up in the Hall-geometry-factor G(λ f , λ p ) = 1 K ′ 1−p 1+p K 1−f 1+f 1 0 x 0 d y 1−( 1−p 1+p ) 2 (1−y 2 ) √ 1−y 2 √ 1 − x 2 1 − 1 − 1−f 1+f 2 (1 − x 2 ) d x. Here p and f are related to the input and output resistances by λ f = 2K(f )/K ′ (f ) and λ p = K ′ (p)/[2K(p)] , with the complete elliptic integral of the first kind being defined by K( √ t) := π/2 0 d θ 1 − t sin 2 θ ≡ K ′ ( √ 1 − t). Due to the symmetry of the device G(λ f , λ p )/ λ f λ p must be unchanged under the substitution (λ f , λ p ) → (2/λ f , 2/λ p ). This can be recast into the remarkable identity that is invariant under (p, q) → ( 1 − p 2 , 1 − q 2 ), which is our aim to prove in this note. A Double Integral Identity Theorem 1. For parameters p, q ∈ (0, 1), define correspondingly p ′ = 1 − p 2 , q ′ = 1 − q 2 , then we have an integral identity A(p, q) = A(p ′ , q ′ ), where A(p, q) := π 0 d x x 0 d y 1 √ 1 − p cos x √ 1 + q cos y = 4 (1 − p)(1 + q) π/2 0 d θ 1 + 2p 1−p sin 2 θ θ 0 d φ 1 − 2q 1+q sin 2 φ .(1) Before proving the functional equation stated in the theorem above, we need to convert double integrals like A(p, q) into single integrals over the products of elliptic integrals and elementary functions, as described in the lemma below. Lemma 2. For 0 < β < α < 1, the following identity holds: 1 π/2 0 d θ 1 − α sin 2 θ θ 0 d φ 1 − β sin 2 φ = 1 π β 0 K( √ 1 − β)K( √ t) √ 1 − t + √ 1 − α d t √ 1 − t + 1 π 1 β K( √ β)K( √ 1 − t) √ 1 − t + √ 1 − α d t √ 1 − t ,(2) where the integrations are carried out along straight line-segments joining the end points. Proof. In what follows, we write Y λ (X) := X(1 − X)(1 − λX) for X ∈ (0, 1) and λ ∈ (0, 1), with the square root taking positive values. It is clear that the complete elliptic integral K( √ λ), λ ∈ (0, 1) satisfies K( √ λ) = 1 2 1 0 d X Y λ (X) .(3) For 0 < β < α < 1, we have an addition formula of Legendre type [4, Eq. 2.3.26] π Y α (U ) 1 U d u Y β (u) = 1 0 2αK( √ 1 − β) 1 − αU V V d V Y α (V ) + 1 0 2αK( √ β) 1 − (1 − αU )V V d V Y 1−α (V ) − 1 1−α 1−β d X Y 1−β (X) 1 1−(1−β)X α d V Y α (V ) αV 1 − αU V .(4) Integrating over U ∈ (0, 1), we obtain π 1 0 d U Y α (U ) 1 U d u Y β (u) = 4πK( √ α)K( √ β) − π 1 0 d U Y α (U ) U 0 d u Y β (u) = − 2K( √ 1 − β) 1 0 log(1 − αV ) d V Y α (V ) + 2K( √ β) 1 0 log 1−(1−α)V 1−V d V Y 1−α (V ) + 1 1−α 1−β d X Y 1−β (X) 1 1−(1−β)X α d V Y α (V ) log(1 − αV ).(5) 1 The constraint 0 < β < α < 1 is needed in the derivation of (2), the validity of which extends to α = 2p/(p − 1) < 0, β = 2q/(1 + q) ∈ (0, 1), by virtue of analytic continuation. Here, the first two single integrals over V can be evaluated in closed form [ 1 0 log(1 − αV ) d V Y α (V ) = K( √ α) log(1 − α),(6)1 0 log 1−(1−α)V 1−V d V Y 1−α (V ) = πK( √ α) + K( √ 1 − α) log(1 − α),(7)1 1−α 1−β d X Y 1−β (X) 1 1−(1−β)X α d V Y α (V ) log(1 − αV ) = 2K( √ 1 − β) π 1 0 (1 − βU ) d U Y β (U ) 1 0 d W W (1 − W ) log(1 − αW − β(1 − W )) 1 − [αW + β(1 − W )]U − 2K( √ β) π 1 0 [1 − (1 − β)U ] d U Y 1−β (U ) 1 0 d W W (1 − W ) log(1 − αW − β(1 − W )) 1 − [1 − αW − β(1 − W )]U .(8)Substituting W = (1 − βU )V /(1 − βU V ) such that W 1 − W = (1 − βU )V 1 − V ,(9) we obtain 1 0 (1 − βU ) d U Y β (U ) 1 0 d W W (1 − W ) log(1 − αW − β(1 − W )) 1 − [αW + β(1 − W )]U = 1 0 d U U (1 − U ) 1 0 d V V (1 − V ) log 1 − α + (α−β)(1−V ) 1−βUV 1 − αU V ,(10) where log 1 − α + (α−β)(1−V ) 1−βUV − log(1 − αV ) 1 − αU V = β 0 1 1 − tU V − 1 − α (1 − t)(1 − V ) + (1 − α)(1 − tU )V d t t − α(11) allows us to integrate over V and U in a sequel on the right-hand side, leading to 1 0 (1 − βU ) d U Y β (U ) 1 0 d W W (1 − W ) log(1 − αW − β(1 − W )) 1 − [αW + β(1 − W )]U = 2π β 0 K( √ t) t − α 1 − 1 − α 1 − t d t + K( √ α) 2 log(1 − α) .(12) Here, in the last step, we have evaluated 1 0 d U U (1 − U ) 1 0 d V V (1 − V ) log(1 − αV ) 1 − αU V = π 1 0 log(1 − αV ) d V Y α (V ) = πK( √ α) log(1 − α)(13) with the aid of (6). Likewise, starting with a variable substitution W = [1 − (1 − β)U ]V /[1 − (1 − β)U V ] such that W 1 − W = [1 − (1 − β)U ]V 1 − V ,(14) we may compute 1 0 [1 − (1 − β)U ] d U Y 1−β (U ) 1 0 d W W (1 − W ) log(1 − αW − β(1 − W )) 1 − [1 − αW − β(1 − W )]U = 2π β 1 K( √ 1 − t) t − α 1 − 1 − α 1 − t d t − πK( √ α) 2 + K( √ 1 − α) 2 log(1 − α) .(15) Thus, the claimed identity is verified. Exploiting the integral identity in the lemma above, together with some modular transformations of elliptic integrals, we will prove Theorem 1. Proof of Theorem 1. We recall that the Legendre function of the first kind of degree −1/4 is defined by P −1/4 (1 − 2t) := 2 F 1 1 4 , 3 4 1 t = 1 √ 2π 1 0 u(1 − tu) 1 − u −1/4 d u 1 − u , t ∈ C [1, +∞).(16) The following relations between P −1/4 and the complete elliptic integral K are recorded in Ramanujan's notebook [2, Chap. 33, Theorems 9.1 and 9.2]: K 2q 1 + q = π 2 √ 1 + qP −1/4 (1 − 2q 2 ),(17)K 1 − q 1 + q = π 2 1 + q 2 P −1/4 (2q 2 − 1),(18) which are provable by standard transformations of the respective hypergeometric functions, provided that q ∈ (0, 1). With the information listed in the last paragraph, we see that A(p, q) = 2q/(1+q) 0 √ 2P −1/4 (2q 2 − 1)K( √ t) √ 1 − t √ 1 − p + √ 1 + p d t √ 1 − t + 1 2q/(1+q) 2P −1/4 (1 − 2q 2 )K( √ 1 − t) √ 1 − t √ 1 − p + √ 1 + p d t √ 1 − t .(19) On one hand, with t = 4 √ s/(1+ √ s) 2 and Landen's transformation [3, item 163.02] K( √ s) = 1 1 + √ s K 2 4 √ s 1 + √ s , 0 < s < 1,(20) we have 2q/(1+q) 0 K( √ t) √ 1 − t √ 1 − p + √ 1 + p d t √ 1 − t = 2 (1− √ 1−q 2 )/(1+ √ 1−q 2 ) 0 K( √ s) (1 − √ s) √ 1 − p + (1 + √ s) √ 1 + p d s √ s .(21) On the other hand, it is clear from a substitution t = 1 − s that 1 2q/(1+q) K( √ 1 − t) √ 1 − t √ 1 − p + √ 1 + p d t √ 1 − t = (1−q)/(1+q) 0 K( √ s) √ s √ 1 − p + √ 1 + p d s √ s = (1−q)/(1+q) 0 √ 2K( √ s) (1 − √ s) 1 − √ 1 − p 2 + (1 + √ s) 1 + √ 1 − p 2 d s √ s .(22) Here, the last equality results from a pair of elementary identities for p ∈ (0, 1): 1 + √ 1 − p 2 2 ± 1 − √ 1 − p 2 2 = √ 1 ± p,(23) which are readily verified by squaring both sides. Therefore, with p ′ = 1 − p 2 , q ′ = 1 − q 2 , we have A(p, q) = (1−q ′ )/(1+q ′ ) 0 2 √ 2P −1/4 (1 − 2q ′2 )K( √ s) (1 − √ s) √ 1 − p + (1 + √ s) √ 1 + p d s √ s + (1−q)/(1+q) 0 2 √ 2P −1/4 (1 − 2q 2 )K( √ s) (1 − √ s) √ 1 − p ′ + (1 + √ s) √ 1 + p ′ d s √ s ,(24) which is evidently equal to A(p ′ , q ′ ). q cos y Date: January 25, 2017. AcknowledgementM.L.G. thanks Udo Ausserlechner (Infinion Technologies) and Michael Milgram (Geometrics Unlimited) for insightful correspondence. Financial support of MINECO (Project MTM2014-57129-C2-1-P) and Junta de Castilla y Leon (UIC 0 11) is acknowledged. Closed form expressions for sheet resistance and mobility from Van-der-Pauw measurement on 90 o symmetryic devices with four arbitrary contacts. Udo Ausserlechner, file:/localhost/opt/grobid/grobid-home/tmp/dx.doi.org/10.1016/j.sse2015.11.030Solid-State Electronics. 116Udo Ausserlechner. Closed form expressions for sheet resistance and mobility from Van-der-Pauw measurement on 90 o symmetryic devices with four arbitrary contacts. Solid-State Electronics 116,46-55 (2016) dx.doi.org/10.1016/j.sse2015.11.030 Ramanujan's Notebooks (Part V). C Bruce, Berndt, Springer-VerlagNew York, NYBruce C. Berndt. Ramanujan's Notebooks (Part V). Springer-Verlag, New York, NY, 1998. Handbook of Elliptic Integrals for Engineers and Scientists, volume 67 of Grundlehren der mathematischen Wissenschaften. F Paul, Morris D Byrd, Friedman, SpringerBerlin, Germany2nd editionPaul F. Byrd and Morris D. Friedman. Handbook of Elliptic Integrals for En- gineers and Scientists, volume 67 of Grundlehren der mathematischen Wissen- schaften. Springer, Berlin, Germany, 2nd edition, 1971. Kontsevich-Zagier integrals for automorphic Green's functions. II. Yajun Zhou, 10.1007/s11139-016-9818-9Ramanujan J. to appear, see arXiv:1506.00318v3 [math.NT] for erratum/addendum)Yajun Zhou. Kontsevich-Zagier integrals for automorphic Green's functions. II. Ramanujan J., 2016. doi:10.1007/s11139-016-9818-9 (to appear, see arXiv:1506.00318v3 [math.NT] for erratum/addendum). . Dpto, De Física Teórica, Donostia International Physics Center, P. Manuel de Lardizabal. 947011Facultad de Ciencias, Universidad de ValladolidPaseo Belén. Spain E-mail address: [email protected]. de Física Teórica, Facultad de Ciencias, Universidad de Valladolid, Paseo Belén 9, 47011 Valladolid, Spain; Donostia International Physics Center, P. Manuel de Lardizabal 4,, E-20018 San Sebastián, Spain E-mail address: [email protected] Program in Applied and Computational Mathematics (PACM). Princeton, NJ 08544; BeijingPrinceton University ; Academy of Advanced Interdisciplinary Sciences (AAIS), Peking Universitymail address: [email protected], [email protected] in Applied and Computational Mathematics (PACM), Princeton University, Princeton, NJ 08544; Academy of Advanced Interdisciplinary Sciences (AAIS), Peking University, Beijing 100871, P. R. China E-mail address: [email protected], [email protected]

Static stochastic VRPs aim at modeling real-life VRPs by considering uncertainty on data. In particular, the SS-VRPTW-CR considers stochastic customers with time windows and does not make any assumption on their reveal times, which are stochastic as well. Based on customer request probabilities, we look for an a priori solution composed preventive vehicle routes, minimizing the expected number of unsatisfied customer requests at the end of the day. A route describes a sequence of strategic vehicle relocations, from which nearby requests can be rapidly reached. Instead of reoptimizing online, a so-called recourse strategy defines the way the requests are handled, whenever they appear. In this paper, we describe a new recourse strategy for the SS-VRPTW-CR, improving vehicle routes by skipping useless parts. We show how to compute the expected cost of a priori solutions, in pseudo-polynomial time, for this recourse strategy. We introduce a new meta-heuristic, called Progressive Focus Search (PFS), which may be combined with any local-search based algorithm for solving static stochastic optimization problems. PFS accelerates the search by using approximation factors: from an initial rough simplified problem, the search progressively focuses to the actual problem description. We evaluate our contributions on a new, real-world based, public benchmark. Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!) 3 Application example. Let us consider the problem of managing a team of on-duty doctors, operating at patient home places during nights and week-ends. On-call periods start with all the doctors at a central depot, where each get assigned a taxi cab for efficiency and safety. Patient requests arrive dynamically. We know from historical data the probability that a request appears depending on the region and time of day.Each online request comes with a hard deadline, and a recourse strategy can be used to decide whether the request can be satisfied in time and how to adapt the routes accordingly. If it cannot be handled in time, the request is rejected and entrusted to an (expensive) external service provider. The goal is to minimize the expected number of rejected requests. In such context, involving very short deadlines, relocating idle doctors anticipatively is often critical. Modeled as a SS-VRPTW-CR, it is possible to compute a first-stage solution composed of (sequences of) waiting locations, optimizing the expected quality of service.Contributions.We introduce an improved recourse strategy for the SS-VRPTW-CR that optimizes routes by skipping some useless parts. Closed-form expressions are provided to efficiently compute expected costs for the new recourse strategy. Another contribution is a new meta-heuristic, called Progressive Focus Search (PFS), for solving static stochastic optimization problems. PFS accelerates the solution process by using approximation factors, both reducing the size of the search space and the complexity of the objective function. These factors are progressively decreased during the search process so that, from an initial rough approximation of the problem, the search gradually focuses to the actual problem. We also introduce a new public benchmark for the SS-VRPTW-CR, based on real-word data coming from the city of Lyon. Experimental results on this benchmark show that PFS obtains better results than a classical search. Important insights are brought to light. By comparing with a basic (yet realistic) policy which does not exploit stochastic knowledge, we show that our stochastic models are particularly beneficial when the number of vehicles increases and when time windows are tight.Organization. In Section 1, we review existing studies on VRPs with stochastic customers and clearly position the SS-VRPTW-CR with respect to them. In Section 2, we formally define the general SS-VRPTW-CR. Section 3 presents a new recourse strategy, which we present as both a generalization and an improvement over the recourse strategy previously proposed in Saint-Guillain, Solnon, and Deville (2017). In Section 4, we describe the Progressive Focus Search metaheuristic for static stochastic optimization problems and show how to instantiate it to solve the SS-VRPTW-CR. In Section 5, we describe a new public benchmark for the SS-VRPTW-CR, derived from real-world data, and describe the experimental settings.The experimental results are analyzed in Section 6 for small instances and in Section 7 for larger instances.Finally, further research directions are discussed in Section 8.

1902.03930

46718734b7866bd1c35494e51d1d10378e3f379c

Progressive Focus Search for the Static and Stochastic VRPTW with both Random Customers and Reveal Times Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times 2 8 Feb 2019 Michael Saint-Guillain [email protected] Christine Solnon [email protected] Yves Deville Institut National des Sciences Appliquées de Lyon Université catholique de Louvain Belgium, France Université catholique de Louvain Belgium Progressive Focus Search for the Static and Stochastic VRPTW with both Random Customers and Reveal Times Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times 2 8 Feb 2019Article submitted to ; manuscript no. (Please, provide the manuscript number!)Submitted to manuscript (Please, provide the manuscript number!) Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication. Static stochastic VRPs aim at modeling real-life VRPs by considering uncertainty on data. In particular, the SS-VRPTW-CR considers stochastic customers with time windows and does not make any assumption on their reveal times, which are stochastic as well. Based on customer request probabilities, we look for an a priori solution composed preventive vehicle routes, minimizing the expected number of unsatisfied customer requests at the end of the day. A route describes a sequence of strategic vehicle relocations, from which nearby requests can be rapidly reached. Instead of reoptimizing online, a so-called recourse strategy defines the way the requests are handled, whenever they appear. In this paper, we describe a new recourse strategy for the SS-VRPTW-CR, improving vehicle routes by skipping useless parts. We show how to compute the expected cost of a priori solutions, in pseudo-polynomial time, for this recourse strategy. We introduce a new meta-heuristic, called Progressive Focus Search (PFS), which may be combined with any local-search based algorithm for solving static stochastic optimization problems. PFS accelerates the search by using approximation factors: from an initial rough simplified problem, the search progressively focuses to the actual problem description. We evaluate our contributions on a new, real-world based, public benchmark. Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!) 3 Application example. Let us consider the problem of managing a team of on-duty doctors, operating at patient home places during nights and week-ends. On-call periods start with all the doctors at a central depot, where each get assigned a taxi cab for efficiency and safety. Patient requests arrive dynamically. We know from historical data the probability that a request appears depending on the region and time of day.Each online request comes with a hard deadline, and a recourse strategy can be used to decide whether the request can be satisfied in time and how to adapt the routes accordingly. If it cannot be handled in time, the request is rejected and entrusted to an (expensive) external service provider. The goal is to minimize the expected number of rejected requests. In such context, involving very short deadlines, relocating idle doctors anticipatively is often critical. Modeled as a SS-VRPTW-CR, it is possible to compute a first-stage solution composed of (sequences of) waiting locations, optimizing the expected quality of service.Contributions.We introduce an improved recourse strategy for the SS-VRPTW-CR that optimizes routes by skipping some useless parts. Closed-form expressions are provided to efficiently compute expected costs for the new recourse strategy. Another contribution is a new meta-heuristic, called Progressive Focus Search (PFS), for solving static stochastic optimization problems. PFS accelerates the solution process by using approximation factors, both reducing the size of the search space and the complexity of the objective function. These factors are progressively decreased during the search process so that, from an initial rough approximation of the problem, the search gradually focuses to the actual problem. We also introduce a new public benchmark for the SS-VRPTW-CR, based on real-word data coming from the city of Lyon. Experimental results on this benchmark show that PFS obtains better results than a classical search. Important insights are brought to light. By comparing with a basic (yet realistic) policy which does not exploit stochastic knowledge, we show that our stochastic models are particularly beneficial when the number of vehicles increases and when time windows are tight.Organization. In Section 1, we review existing studies on VRPs with stochastic customers and clearly position the SS-VRPTW-CR with respect to them. In Section 2, we formally define the general SS-VRPTW-CR. Section 3 presents a new recourse strategy, which we present as both a generalization and an improvement over the recourse strategy previously proposed in Saint-Guillain, Solnon, and Deville (2017). In Section 4, we describe the Progressive Focus Search metaheuristic for static stochastic optimization problems and show how to instantiate it to solve the SS-VRPTW-CR. In Section 5, we describe a new public benchmark for the SS-VRPTW-CR, derived from real-world data, and describe the experimental settings.The experimental results are analyzed in Section 6 for small instances and in Section 7 for larger instances.Finally, further research directions are discussed in Section 8. Introduction In the Vehicle Routing Problem with Time Windows, a set of customers must be serviced by a homogeneous fleet of capacitated vehicles, while reconciling each customer's time windows and vehicle travel times, as well as cumulated customers' demands and vehicle capacities. Whereas deterministic VRP(TW)s assume perfect information on input data, in real-world applications some input data may be uncertain when computing a solution. In this paper, we focus on cases where the customer presence is a priori unknown. Furthermore, we assume to be provided with some probabilistic knowledge on the missing data. In many situations, the probability distributions can be obtained from historical data. In order to handle 1 arXiv:1902.03930v1 [cs.AI] 8 Feb 2019 Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times 2 Article submitted to ; manuscript no. (Please, provide the manuscript number!) Figure 1 Recourse strategies for the SS-VRP with stochastic customers and demands (Bertsimas, 1992). The vehicle has a capacity of 3. The first stage solution states the a priori sequence of customer visits. When applying strategy a, the vehicle unloads at the depot after visiting c. In strategy b, absent customers (a, d, f) are skipped. new customers who appear dynamically, the current solution must be adapted as such random events occur. Depending on the operational context, we distinguish two fundamentally different assumptions. If the currently unexecuted part of the solution can be arbitrarily redesigned, then we are facing a Dynamic and Stochastic VRP(TW) (DS-VRP(TW)). In that case, the solution is adapted by re-optimizing the new current problem while fixing the executed partial routes. If the routes can only be adapted by following some predefined scheme, then we are facing a Static and Stochastic VRP(TW) (SS-VRP(TW)). In the SS-VRP(TW), whenever a bit of information is revealed, the current solution is adapted by applying a recourse strategy. Based on the probabilistic information, we seek a first stage (also called a priori) solution that minimizes its a priori cost, plus the expected sum of penalties caused by the recourse strategy. In order for the evaluation function to remain tractable, the recourse strategy must be efficiently computable, hence simple enough to avoid re-optimization. For example, in Bertsimas (1992) the customers are known, whereas their demands are revealed online. Two different assumptions are considered, leading to different recourse strategies, as illustrated in Fig. 1. In strategy a, each demand is assumed to be revealed when the vehicle arrives at the customer place. If the vehicle reaches its maximal capacity, then the first stage solution is adapted by adding a round trip to the depot. In strategy b, each demand is revealed when leaving the previous customer, allowing to skip customers having null demands. In the recent review of Gendreau et al. (2016), the authors argue for new recourse strategies: With the increasing use of ICT, customer information is likely to be revealed on a very frequent basis. In this context, the chronological order in which this information is transmitted no longer matches the planned sequences of customers on the vehicle routes. In particular, the authors consider as paradoxical the fact that the existing literature on SS-VRPs with random Customers (SS-VRP-C) assumes full knowledge on the presence of customers at the beginning of the operational period. In this paper, we focus on the SS-VRPTW with both random Customers and Reveal times (SS-VRPTW-CR) introduced by Saint-Guillain, Solnon, and Deville (2017), in which no assumption is made on the moment at which a request is known. The goal is to compute the first-stage solution that minimizes the expected number of rejected requests, while avoiding assumptions on the moment at which customer requests are revealed. To handle uncertainty on the reveal times, waiting (re)locations are part of first-stage solutions. 5 Unlike the SS-VRPTW-CR, the set of customers is revealed at the beginning of the operations. Heilporn, Cordeau, and Laporte (2011) introduced the Dial-a-Ride Problem (DARP) with stochastic customer delays. Each customer is present at its pickup location with a stochastic delay. A customer is then skipped if it is absent when the vehicle visits the corresponding location, involving the cost of fulfilling the request by an alternative service (e.g., a taxi). In a sense, stochastic delays imply that each request is revealed at some uncertain time during the planning horizon. That study is thus related to our problem, except that in the SS-VRPTW-CR, part of the requests will reveal to never appear. Problem description: the SS-VRPTW-CR This section recalls the definition of the SS-VRPTW-CR, initially provided in Saint-Guillain, Solnon, and Deville (2017). In fact, it contains parts taken from section 3 of the aforementioned paper. We note W 0 = W ∪ {0} and C 0 = C ∪ {0}. The fleet is composed of K vehicles of maximum capacity Q. Let R = C × H be the set of potential requests. An element r = (c, Γ) of R represents a potential request revealed at time Γ at customer vertex c. It is associated the following deterministic attributes: a demand q r ∈ [1, Q], a service duration s r ∈ H and a time window [e r , l r ] with Γ ≤ e r ≤ l r ≤ h. We note p r the probability that r appears on vertex c at time Γ and assume independence between requests. Although our formalism imposes Γ ≥ 1 for all potential requests, in practice a request may be known with probability 1, leading to a deterministic request. Finally, different customers in C can share the same geographical location, making it possible to consider different types of requests in terms of deterministic attributes. To simplify notations, we use Γ r to denote the reveal time of a request r ∈ R and c r for its customer vertex. Furthermore, a request r may be written in place of its own vertex c r . For instance, the distance d v,cr may also be written as d v,r . Table 1 summarizes the main notations. First-stage solution. The first-stage solution is computed offline, before the beginning of the time horizon. It consists of a set of K vehicle routes visiting a subset of the m waiting vertices, together with duration variables denoted by τ indicating how long a vehicle should wait on each vertex. More specifically, we denote by (x, τ ) a first-stage solution to the SS-VRPTW-CR. x = {x 1 , ..., x K } defines a set of K disjoint sequences of waiting vertices of W , each starting and ending with the depot. Each vertex of W occurs at most once in x. We note W x ⊆ W , the set of waiting vertices visited in x. The vector τ associate a waiting time τ w ∈ H with every waiting vertex w ∈ W x . For each sequence x k = w m 1 , ..., w m k , the vehicle is back at the depot by the end of the time horizon: k−1 i=1 d wm i ,wm i+1 + k−1 i=2 τ wm i ≤ h In other words, x defines a solution to a Team Orienteering Problem (TOP, see Chao, Golden, and Wasil (1996)) to which each visited location is assigned a waiting time by τ . Given a first-stage solution (x, τ ), we define on(w) = [on(w), on(w)] for each vertex w ∈ W x such that on(w) (resp. on(w)) is the arrival (resp. departure) time at w. In a sequence w m 1 , ..., w m k in x, we then have on( w m i ) = on(w m i−1 ) + d wm i−1 ,wm i and on(w m i ) = on(w m i ) + τ wm i for i ∈ [2, k] and assume that on(w m 1 ) = 1. Recourse strategy and optimal first-stage solution Given a first stage solution (x, τ ), a recourse strategy states how the requests, which appear dynamically, are handled by the vehicles. In other words, it defines how the second-stage solution is gradually constructed, based on (x, τ ) and depending on these online requests. A more formal description of recourse strategies in the context of the SS-VRPTW-CR is provided in Saint-Guillain, Solnon, and Deville (2017). Let a recourse strategy R. An optimal first-stage solution (x, τ ) to the SS-VRPTW-CR minimizes the expected cost of the second-stage solution: (SS-VRPTW-CR) Minimize x,τ Q R (x, τ ) (1) s.t. (x, τ ) is a first-stage solution.(2) The objective function Q R (x, τ ), which is nonlinear in general, is the expected number of rejected requests, i.e., requests that fail to be visited under recourse strategy R and first-stage solution (x, τ ).Note that Q R (x, τ ) actually represents an expected quality of service, which does not take travel costs into account. In fact, in most practical applications that could be formulated as an SS-VRPTW-CR, quality of service prevails whenever the number of vehicles is fixed, as travel costs are usually negligible compared to the labor cost of the mobilized mobile units. Formulation (1)-(2) states the problem in general terms, hiding two non-trivial issues. Given a recourse strategy R, finding a computationally tractable way to evaluate Q R constitutes the first challenge. We address it in Section 3, based on a new recourse strategy we propose. The second problem naturally concerns the minimization problem, or how to deal with the solution space. This is addressed in Section 4. For completeness, a mathematical formulation of the constraints involved by (2) is provided in Appendix A. 7 A new recourse strategy The strategy we introduce, called R q+ , is a generalization and an improvement of strategy R ∞ introduced in Saint-Guillain, Solnon, and Deville (2017). First, it generalizes R ∞ by taking vehicle capacities into account. Second, R q+ improves R ∞ by saving operational time when possible, by avoiding some pointless round trips from waiting vertices. For the sake of completeness and since R q+ generalizes R ∞ , part of this section includes elements from section 4 of Saint-Guillain, Solnon, and Deville (2017), which are common to both strategies. We emphasize the common points and differences between these two strategies at the end of this section. Description of R q+ Informally, the recourse strategy R q+ accepts a request revealed at time t if the assigned vehicle is able to adapt its first-stage tour to visit the customer, given its set of previously accepted requests. Time window and capacity constraints should be respected, and already accepted requests should not be disturbed. Ideally, whenever a request appears and prior to determine whether it can be accepted, a vehicle should be selected to minimize objective function (1). Furthermore, if several requests appear at the same time unit and amongst the subset of these that are possibly acceptable, some may not contribute optimally to (1). Given a set of accepted requests, the order in which they are handled also plays a critical role. Unfortunately, none of these decisions can be made optimally without reducing to a NP-hard problem. In order for R q+ to remains efficiently computable, they are necessarily made heuristically. The solution proposed in Saint-Guillain, Solnon, and Deville (2017) makes these decisions beforehand. Before the start of the operations and in order to avoid reoptimization, the set R of potential requests is ordered. Each potential request r ∈ R is also preassigned to exactly one planned waiting vertex in W x , and therefore one vehicle, based on geographical considerations. Request ordering The ordering heuristic is independent of the current first-stage solution. Different orders may be considered, provided that the order is total, strict, and ∀r 1 , r 2 ∈ R, if the reveal time of r 1 is smaller than the reveal time of r 2 then r 1 must be smaller than r 2 in the request order. We order R by increasing reveal time first, end of time window second, and lexicographic order to break further ties. 3.1.2. Request assignment according to a first-stage solution Given a first-stage solution (x, τ ), we assign each request of R either to a waiting vertex visited in x or to ⊥ to denote that r is not assigned. We note w : R → W x ∪ {⊥} this assignment. It is computed for each first-stage solution (x, τ ) before the application of the recourse strategy. To compute this assignment, for each request r, we first compute the set W x r of waiting vertices from which satisfying r is possible, if r appears: where t min r,w and t max r,w are defined as follows. Time t min r,v = max{on(w(r)), Γ r , e r − d v,r } is the earliest time at which the vehicle can possibly leave vertex v ∈ C ∪ W in order to satisfy request r. Time t max r,v = min{l r − d v,r , on(s(w(r))) − d v,r − s r − d r,s(w(r)) }, where s(w(r)) is the waiting vertex that directly follows w(r) in the first-stage solution (x, τ ), is the latest time at which a vehicle can leave vertex v to handle rand arrive at s(w(r)) in time. Given the set W x r of feasible waiting vertices for r, we define the waiting vertex w(r) associated with r as follows: W x r = {w ∈ W x : t min r,w ≤ t max r,w } • If W x r = ∅, then w(r) = ⊥ (r is always rejected as it has no feasible waiting vertex); • Otherwise, w(r) is set to the feasible vertex of W x r that has the least number of requests already assigned to it (further ties are broken with respect to vertex number). This heuristic rule aims at evenly distributing potential requests on waiting vertices. Once finished, the request assignment ends up with a partition {π ⊥ , π 1 , ..., π K } of R, where π k is the set of requests assigned to the waiting vertices visited by vehicle k and π ⊥ is the set of unassigned requests (such that w(r) = ⊥). We note π w , the set of requests assigned to a waiting vertex w ∈ W x . 3.1.3. Using R q+ to adapt a first-stage solution at time t At each time step t, the recourse strategy is applied to decide whether to accept or reject the new incoming requests, if any, and determine the appropriate vehicle actions. Let A t−1 be the set of accepted requests up to time t − 1. Note that A t−1 is likely to contain some requests that have been accepted but are not yet satisfied (i.e. not yet visited). Availability time. The decision to accept or reject a request r ∈ π k appearing at time t = Γ r depends on when vehicle k will be available for r. By available, we mean that it has finished serving all its accepted requests that precede r, according to the predefined order on R. This time is denoted by available(r). It is only defined when all the accepted requests, that must be served before r by the same vehicle, are known. If r is the first request of its waiting vertex, the first of π w(r) , then: available(r) = on(w), w = w(r) = w(r). Otherwise, let r − be the request that directly precedes r in π w(r) . As the requests assigned to w(r) are ordered by increasing reveal time, we know all these accepted requests for sure when t ≥ Γ r − . Given current time t ≥ Γ r , function available(r) is defined in R q+ as: available(r) = max available(r − ) + d v(r − ),r − , e r − + s r − + d r − ,v(r) , if r − ∈ A t available(r − ) + d v + (r − ),v(r) , otherwise. If r is the first request of its waiting vertex, the location v(r) from which the vehicle travels towards request r is necessarily the waiting vertex w(r). Otherwise, v(r) depends on whether r reveals by the time the vehicle finishes to satisfy the last accepted request: v(r) =      c r − , if Γ r ≤ max available(r − ) + d v + (r − ),r − , e r − + s r − ∧ r − ∈ A t v(r − ), if r − / ∈ A t w(r) otherwise. 9 Request notifications. A t is the set of requests accepted up to time t. It is initialized with A t−1 as all previously accepted requests must still be accepted at time t. Then incoming requests (i.e. revealed at time t) are considered in increasing order with respect to < R . r is either accepted (added to A t ) or rejected (not added to A t ). A request r is accepted if (i) it is assigned to a waiting location, (ii) the vehicle is available, and (iii) its capacity is not exceeded. Formally, r is added to A t if and only if: w(r) = ⊥ ∧ available(r) ≤ t max r,v(r) ∧ q r + r ∈π k ∩A t q r ≤ Q.(3) Vehicle operations. Once A t has been computed, vehicle operations for time unit t must be decided. Each vehicle operates independently from all other vehicles. If vehicle k is traveling between a waiting vertex and a customer vertex, or if it is serving a request, then its operation remains unchanged. Otherwise, its operations are defined in Algorithm 1. Algorithm 1: Operations of vehicle k, at current time t. Vertex v is the position of vehicle k, w the waiting vertex it is currently assigned to, s(w) the waiting vertex (or the depot) that follows w in x. if w(r next ) = w and r next has already been revealed and accepted then 8 wait until t min r next ,v , travel to r next and satisfy the request; 9 if w(r next ) = w but r next is not known yet (t < Γ r next ) then travel back to waiting location w; 10 if w(r next ) = w then wait until on s(w) − d r,s(w) and travel to s(w); 1 if t = on(s(w)) − d v, 3.1.4. Relation to strategy R ∞ In strategy R ∞ the vehicles handle requests by performing systematic round trips for their current waiting locations. In R q+ , a vehicle travels directly towards a revealed request r from a previously satisfied one r', provided that r appears by the time the service of r gets completed. Furthermore, a vehicle is now allowed to travel directly from a customer vertex c ∈ C to the next planned waiting vertex, without passing by the waiting vertex associated with c. Figure 2 illustrates, informally, the differences between R ∞ and R q+ . Expected cost of second-stage solutions under R q+ Given a recourse strategy R and a first-stage solution (x, τ ) to the SS-VRPTW-CR, a naive approach for computing Q R (x, τ ) would be to follow the strategy described by R in order to confront (x, τ ) with each and every possible scenario ξ ⊆ R. Since there can be up to 2 |R| possible scenarios, this naive approach Recall that we assume that request probabilities are independent of each other; i.e., for any couple of requests r, r ∈ R, the probability p r∧r that both requests will appear is given by p r∧r = p r · p r . Q R (x, τ ) is equal to the expected number of rejected requests, which in turn is equal to the expected number of requests that are found to appear minus the expected number of accepted requests. Under the independence hypothesis, the expected number of revealed requests is given by the sum of all request probabilities, whereas the expected number of accepted requests is equal to the cumulative sum, for every request r, of the probability that it belongs to A h , i.e., Q R (x, τ ) = r∈R p r − r∈R Pr{r ∈ A h } = r∈R p r − Pr{r ∈ A h }(4) In the case of R q+ , the satisfiability of a request r depends on the current time and vehicle load, but also on the vertex from which the vehicle would leave to serve it. The candidate vertices are necessarily either the current waiting location w = w(r) or any vertex hosting one of the previous requests associated with w. Consequently, under R q+ , the probability Pr{r ∈ A h } is decomposed over all the possible time, load, and vertex configurations in which the vehicle can satisfy r: where: Pr{r ∈ A h } =t g v 1 (r, t, q) ≡ Pr{request r appeared ∧ t = max(t min r,v , available(r)) ∧ v(r) = v ∧ the vehicle carries a load of q} Each tuple (v, t, q) in the summation (5), where v is either w(r) or a previously visited customer vertex r , represents a possible configuration for accepting r. The probability to accept r is then equivalent to the probability to fall into one of those states. In particular, note that Pr{t = max(t min r,v , available(r))} represents the probability that, if r is accepted, the vehicle leaves its current position at time t in order to satisfy it. The calculus of g v 1 is further developed in Appendix C. Given n customer vertices, a horizon of length h and vehicle capacity of size Q, the computational complexity of computing the whole expected cost Q R q+ (x, τ ) is in O n 2 h 3 Q , as detailed in the appendix. Space complexity A naive implementation of equation (5) would basically fill up an n 2 × h 3 × Q array. We draw attention to the fact that even a small instance with n = Q = 10 and h = 100 would then lead to a memory consumption of 10 9 floating point numbers. Using a common eight-byte representation requires more than seven gigabytes. Like strategy R ∞ , important savings are obtained by noticing that the computation of g 1 functions for a given request r under R q+ only relies on the previous potential request r − . By computing g 1 while only keeping in memory the expectations of r − (instead of all nh potential requests), the memory requirement is reduced by a factor nh. This however comes at the price of making any incremental computation, based on probabilities belonging to a similar first-stage solution, impossible. Progressive Focus Search for Static and Stochastic Optimization Solving a static stochastic optimization problem, such as the SS-VRPTW-CR, involves finding values for a set of first-stage decision variables that optimize an expected cost with respect to some recourse strategy: min x Q R (x), x ∈ X Solving this kind of problem is always challenging. Besides the exponential size of the (first-stage) solution space X, the nature of the objective function Q R , an expectation, is usually computationally demanding. Because enumerating all possible scenarios is usually impossible in practice, some approaches tend to circumvent this bottleneck by restricting the set of considered scenarios, using for example the sample average approximation method (Ahmed and Shapiro, 2002). In some cases, expectations may be directly computed in (pseudo) polynomial time, by reasoning on the random variables themselves rather than on the scenarios. However, the required computational effort depends on the recourse strategy R and usually remains very demanding, as it is the case for the SS-VRPTW-CR. The Progressive Focus Search (PFS) metaheuristic aims at addressing these issues with two approximation factors, intended to reduce the size of the solution space and the complexity of the objective function. The initial problem P init is simplified into a problem P α,β having simplified objective function and solution space. Parameters α and β define the approximation factors of the objective function and of the solution space, respectively, and P α,β = P init when α = β = 1. Whenever α > 1 or β > 1, the optimal cost of P α,β is an approximation of that of P init . Starting from some initial positive values for α and β, the idea of PFS is to progressively decrease these values using an update policy. The simplified problem P α,β is iteratively optimized for every valuation of (α, β), using the best solution found at the end of one iteration as starting point in the solution space for the next iteration. The definition of the simplified problem P α,β depends on the problem to be solved. In Sections 4.1 and 4.2, we give some general principles concerning α and β and describe how to apply them to the case of the SS-VRPTW-CR. In Section 4.3, we describe the generic PFS metaheuristic. Reducing objective function computational complexity with α We assume the expected cost to be computed by filling matrices in several dimensions. In order to reduce the complexity, some of these dimensions must be scaled down. This is achieved by changing the scale of the input data and the decision variable domains related to the selected dimensions, dividing the values by the scale factor α and rounding to integer if necessary. For example, in the SS-VRPTW-CR the dimensions considered at computing the objective function are: the number of waiting vertices n, the vehicle capacity Q, and the time horizon h. Let h = 18000 be the time horizon in the initial problem, corresponding to five hours in units of one second. If we choose to reduce the time dimension with respect to a scale factor α = 60, then all durations in the input data (travel times, service times, time windows, etc.) are rounded to the nearest multiple of 60. Thus, the time horizon in the simplified problem P α,β = P 60,β is of h 60 = 300, corresponding to a five-hour time horizon in units of one minute. The domains of waiting times decision variables are reduced accordingly, scaled from [1, 18000] in P 1,β to [1, 300] in P 60,β . Similarly, if we choose to reduce the vehicle capacity dimension with respect to a scale factor α = 1000, and if the vehicle capacity in the initial problem is Q = 500000, e.g. 500 kg in steps of 1 g, then all demands must be rounded to multiples of 1000. The capacity in P 1000,β becomes Q 1000 = 500, thus 500 kg in units of 1 kg. When scaling dimensions of different nature, such as time and capacity, different scale factors should be considered, leading to a vector α. Experiments have shown us that the closer α is to 1, the more accurate the approximation of the actual objective function is. Progressively reducing α during the search process allows us to quickly compute rough approximations at the beginning of the search process, when candidate solutions are usually far from being optimal, and spend more time computing more accurate approximations at the end of the search process, when candidate solutions get closer to optimality. Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!) 13 4.2. Simplifying the solution space size with β When applying a scaling factor α, for consistency reasons the nature of the scaled input data may impose to the domains of some decision variables to be reduced accordingly. Yet the solution space can further simplified by reducing the domains of (part of) the remaining decision variables, or even by further reducing the same decision variables. Let Dom(v) be the initial set of values that may be assigned to v, that is, the domain of a decision variable v. Domain reduction is not necessarily done for all decision variables, but only for a selected subset of them, denoted as V β . The simplified problem is obtained by selecting |Dom(v)|/β values and only considering these candidate values when searching for solutions, for each decision variable v ∈ V β . Ideally, the selection of this subset of values should be done in such a way that the selected values are evenly distributed within the initial domain Dom(v). We note Dom α,β (v), the domain of a decision variable v in the simplified problem P α,β . For example, in the SS-VRPTW-CR a subset of decision variables defines the waiting times on the visited waiting vertices: τ w defines the waiting time on w, with Dom(τ w ) = [1, h]. If the temporal dimension is not scaled with respect to α, or if α = 1, then Dom α,β (τ w ) is reduced to a subset of [1, h] that contains h/β values. To ensure that these values are evenly distributed in [1, h], we may keep multiples of β. However, if the temporal dimension is scaled with respect to α, the selected values must thereafter be scaled. Another subset of decision variables in the SS-VRPTW-CR defines the waiting vertices to be visited by the vehicles. The initial decision variable domains are then equivalent to W . Reducing the domains of these decision variables can be achieved by restricting to a subset of W that contains |W |/β waiting vertices. To ensure that these values are evenly distributed in the space, we may use geographical clustering techniques. Progressively decreasing the value of β allows us to progressively move from diversification to intensification: at the beginning of the search process, there are fewer candidate values for the decision variables of V β . The solution method is therefore able to move quickly towards more fruitful regions of the search space. For minimization (resp. maximization) problems, we can easily show that the optimal solution of a simplified problem P α,β is an upper (resp. lower) bound of the optimal solution of the problem P α,1 ; this is a direct consequence of the fact that every candidate solution of P α,β is also a candidate solution of P α,1 . PFS algorithm PFS requires the following input parameters: • An initial problem P init ; • Initial values (α 0 , β 0 ) for α and β, as well as final values (α min , β min ); • An update policy U that returns the new values α i+1 and β i+1 given α i and β i ; • A computation time policy T such that T (α, β) returns the time allocated for optimizing P α,β ; • A solution algorithm Θ such that, given a problem P , an initial solution s, and a time limit δ, Θ(P, s, δ) returns a possibly improved solution s for P . Article submitted to ; manuscript no. (Please, provide the manuscript number!) PFS is described in Algorithm 2. At each iteration i, the simplified problem P α i ,β i is built (line 3), and the current solution s is updated accordingly (line 3): every value assigned to a decision variable which is concerned by the scale factor α is updated with respect to the new scale α i , and if a value assigned to a decision variable does not belong to the current domain associated with α i and β i , then it is replaced with the closest available value. Note that the updated solution may not be a feasible solution of P α i ,β i (because of value replacements and rounding operations on input data). Therefore the optimizer Θ must support starting with infeasible solutions. Algorithm Θ is then used to improve s with respect to the simplified problem P α i ,β i within a CPU time limit defined by the computation time policy T (line 4). Finally, new values for α and β are computed, according to the update policy U (line 5). This iterative optimization process stops when α i−1 = α min and β i−1 = β min , i.e., when the last optimization of s with Θ has been done with respect to the targeted level of accuracy defined by (α min , β min ). To ensure termination, we assume that the update policy U eventually returns (α min , β min ) after a finite number of calls. Finally, if the final value of α is larger than 1, so that s is a scaled solution, then s is scaled down to become a solution of the initial problem P 1,1 (line 8). Benchmark and Experimental Plan In this section, we introduce the new benchmark as well as the experimental concepts and tools used for experimentations reported in Sections 6 and 7. A benchmark derived from real-world data We derive our test instances from the benchmark described in Melgarejo, Laborie, and Solnon (2015) for the Time-Dependent TSP with Time Windows (TD-TSPTW). This benchmark has been created using real accurate delivery and travel time data obtained from the city of Lyon, France. It is available at http:// becool.info.ucl.ac.be/resources/ss-vrptw-cr-optimod-lyon, as well as the solution files and detailed result tables of the experiments conducted in the following sections. Algorithm 2: Progressive Focus Search (PFS) 1 Initialize i to 0 and construct an initial solution s to problem P init ; 2 repeat 3 Build problem P α i ,β i and update the current solution s to P α i ,β i ; 4 s ← Θ(P α i ,β i , s, T (α i , β i )); 5 (α i+1 , β i+1 ) ← U(α i , β i ); 6 Increment i 7 until α i−1 = α min ∧ β i−1 = β min ; 8 if α min > 1 then Update the current solution s to P 1,1 ; 9 return s; Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!) 15 The benchmark contains two different kinds of instances: instances with separated waiting locations and instances without separated waiting locations. Each instance with separated waiting locations is denoted by nc-mw-i, where n ∈ {10, 20, 50} is the number of customer vertices, m ∈ {5, 10, 30, 50} is the number of waiting vertices, and x ∈ [1, 15] is the random seed. Each instance without separated waiting locations is denoted by nc+w-i. In these instances, every customer vertex is also a waiting vertex, C = W . Instances sharing the same number of customers n and the same random seed x (e.g. 50c-30w-1, 50c-50w-1 and 50c+w-1) always share exactly the same set of customer vertices C. In all instances, the duration of an operational day is eight hours and the time horizon is h = 480, which corresponds to one-minute time steps. To each potential request r = (c r , Γ r ) is assigned a time window [Γ r , Γ r + ∆ − 1], where ∆ is taken uniformly from {5, 10, 15, 20}. Note that the time window always starts with the reveal time Γ r . This aims at simulating operational contexts similar to the practical application example described in introduction, the on-demand health care service at home, requiring immediate responses within small time windows. See the e-companion EC.1 for more details on the complete process used to generate instances. Compared approaches and experimental settings Experiments have been done on a cluster composed of 64-bit AMD Opteron 1.4-GHz cores. The code is developed in C++11 with GCC4.9, using -O3 optimization flag. The current source code of our library for (SS-)VRPs is available from the online repository: bitbucket.org/mstguillain/vrplib. We consider both recourse strategies R q+ and R q , a generalized version of R ∞ for capacitated vehicles. We compare their respective contribution and applicability, then we combine them to take the best of each, using several variations of PFS. An exact method allows us to measure optimality gaps, in order to assess the quality of the solutions found by PFS. In order to evaluate the interest of exploiting stochastic knowledge, that is by modeling the problem as a SS-VRPTW-CR, the solutions are also compared with a wait-and-serve policy which does not anticipate, i.e. in which vehicles are never relocated. 5.2.1. A capacitated version of R ∞ Recourse strategy R q+ is designed to be able to cope with vehicle maximal capacity constraints. In order to compare both strategies R ∞ and R q+ , and since part of our experiments involve limited vehicle capacities, an adapted version of R ∞ is required. We call this generalization R q . Vehicles behave under R q exactly as under R ∞ , but are limited by their capacity. Its request acceptance rule follows the condition in (3), except that the definition of available(r) and t max r,v(r) are those stated in Saint-Guillain, Solnon, and Deville (2017) for R ∞ , and that v(r) = w(r). In Appendix B, we explain how to efficiently compute Q R q (x, τ ). We also show how the resulting equations naturally reduce to the ones proposed in Bertsimas (1992), when particularized to the special case of the SS-VRP-C. We found that, given n customer vertices, a horizon of length h and vehicle capacity of size Q, computing Q R q (x, τ ) is of complexity O nh 2 Q . This is significantly lower than under R q+ , which requires O n 2 h 3 Q operations in the worst case. However, such a lower complexity naturally comes at the Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times 16 Article submitted to ; manuscript no. (Please, provide the manuscript number!) price of a significantly higher expected cost in average, motivating the need for an adequate trade-off. We empirically address this question in Section 6. Progressive Focus Search. We have considered different update and computation time policies U and T in our experiments. In this section, we only describe the optimizer Θ and the approximation factors α and β used when conducting experiments with PFS. Local Search Optimizer The optimizer Θ is the local search (LS) introduced in Saint-Guillain, Solnon, and Deville (2017) to solve the SS-VRPTW-CR. Starting from an initial randomly generated first-stage solution, LS iteratively modifies it by using a set of 9 neighborhood operators: four classical ones for the VRP, i.e., relocate, swap, inverted 2-opt, and cross-exchange (see Kindervater andSavelsbergh (1997), Taillard et al. (1997)), and five new operators dedicated to waiting vertices: insertion/deletion of a randomly chosen waiting vertex in/from W x , increase/decrease of the waiting time τ w of a randomly chosen vertex w ∈ W x , and transfer of a random waiting duration from one waiting vertex to another. After each modification of the first-stage solution, its expected cost is updated using the appropriate equations, depending on whether strategy R q or R q+ is considered. The acceptance criterion follows the Simulated Annealing (SA) metaheuristic of Kirkpatrick, Gelatt, and Vecchi (1983): improved solutions are always accepted, while degrading solutions are accepted with a probability which depends on the degradation and temperature. Temperature is initialized to T init and progressively decreased by a factor f T after each iteration of the LS. A restart strategy resets the temperature to its initial value each time it reaches a lower limit T min . In all experiments, SA parameters were set to T init = 2, T min = 10 −6 , and f T = 0.95. Scale factor α. In the initial problem P 1,1 , temporal data is expressed with a resolution of one-minute time units. The α factor is used to scale down this temporal dimension. The time horizon is scaled down to round(h/α), so that each time step in P α,β has a duration of α minutes. Every temporal input value (travel times d i,j , reveal times Γ r , service times s r , and time windows [e r , l r ]) is scaled from its initial value t to round(t/α). Rounding operations are chosen in such a way that the desired quality of service is never underestimated by scaled data: l r is rounded down while all other values are rounded up. This ensures that a feasible first stage of a simplified problem P α,β always remains feasible once adapted to P 1,1 . Domain reduction factor β The decision variables concerned by domain reductions are waiting time It is both meaningless (for vehicle drivers) and too expensive (for the optimization process) to design first-stage solutions with waiting times that are precise to the minute. Hence, in our experiments the domain of every waiting time decision variable is always reduced by a factor β ≥ 10. When β = 10, waiting times are multiples of 10 minutes. When α = 1 and β = 10, we have Dom 1,10 (τ w ) = {10, 20, 30, . . . , 480}, but temporal data (travel and service times, time windows, etc.) are precise to the minute. variables: V β = {τ w : w ∈ W }. In P 1,1 , we have Dom(τ w ) = [1, h]. Enumerative exact method In order to assess the ability of our algorithms to find (near-) optimal solutions, we devise a simple enumerative optimization method which is able to compute optimal solutions on small instances. To that end, the solution space is restricted to the solutions that (a) use all available vehicles and (b) use all the available waiting time. Indeed, if K ≤ |W |, then on the basis of any optimal solution which uses only a subset of the available vehicles, a solution of the same cost can be obtained by assigning an idle vehicle to either a non-visited waiting vertex (if any) or the last visited vertex of any non-empty route (visiting at least two waiting locations), so that (a) does not remove any optimal solution. Furthermore, if an optimal first-stage solution contains a route for which the vehicle returns to the depot before the end of the horizon, adding the remaining time to the last visited waiting vertex will never increase the expected cost of the solution, so that (b) is also valid. The resulting solution space is then recursively enumerated in order to find the first-stage solution with the optimal expected cost. Wait-and-serve policy In order to assess the contribution of our recourse strategies, we compare them with a policy ignoring anticipative actions. This wait-and-serve (w&s) policy takes place as follows. Vehicles begin the day at the depot. Whenever an online request r appears, it is accepted if at least one of the vehicles is able to satisfy it, otherwise it is rejected. If accepted, it is assigned to the closest such vehicle which then visits it as soon as it becomes idle. If there are several closest candidates, the least loaded vehicle is chosen. After servicing r (which lasts s r time units), the vehicle simply stays idle at r's location until it is assigned another request or until it must return to the depot. Note that a request cannot be assigned to a vehicle if satisfying it prevents the vehicle from returning to the depot before the end of the horizon. Note that, whereas our recourse strategies for the SS-VRPTW-CR generalize to requests such that the time window starts later than the reveal time, in our instances we consider only requests where e r = Γ r . Doing it the other way would in fact require a more complex wait-and-serve policy, since the current version would be far less efficient and unrealistic in the case of requests with e r significantly greater than Γ r . In what follows, average results are always reported for the w&s policy. We randomly generate 10 6 scenarios according to the p r probabilities. For each scenario, we apply the w&s approach to compute a number of rejected requests; finally, the average number of rejected requests is reported. The results of PFS and the exact method are always reported by means of average relative gains, in percentages, with respect to the w&s policy: the gain of a first-stage solution s computed with PFS or the exact method is avg−E avg , where E is the expected cost of s and avg is the average cost under the w&s policy. Experiments on small instances We consider small test instances, having n ∈ {10, 20} customer vertices. Furthermore, PFS is here instantiated such that we perform only a single optimization step (lines 2-7 of Algorithm 2): α 0 = α min and β 0 = β min . The simplified problem P α,β is therefore first optimized for a duration of T seconds, and the returned solution is adapted with respect to the initial problem P 1,1 , ensuring that all results are expressed according Article submitted to ; manuscript no. (Please, provide the manuscript number!) Exact (% gain after 30 minutes) PFS (% gain after 5 minutes) 8.0 -46.9* -26.5 -46.9* -26.5* -55.9* -43.2* -46.9 -29.9 -43.1 -30.6 -43.1 -32.9 10c-5w-4 10.5 -10.9* 0.9 -10.9* 0.9* -10.9* 0.9* -10.9 -8.7 -10.9 -4.9 -10.9 -2.2 10c-5w-5 8.4 -17.9* 2.5 -17.9* 2.5 -20.5* 0.5* -17.9 -6.1 -18.9 -2.8 -19.5 1.1 #eval 10 4 10 4 10 4 10 4 3 * 10 4 3 * 10 3 7 * 10 4 5 * 10 3 2 * 10 5 2 * 10 4 10c+w-1 12. 28.5 #eval 3 * 10 4 3 * 10 3 7 * 10 4 6 * 10 3 2 * 10 5 2 * 10 4 Table 2 Results on small instances (n = 10, K = 2, Q = ∞) when α ∈ {1, 2, 5} and β = 60. For each instance, we give the average cost over 10 6 sampled scenarios using the wait-and-serve policy ( to the original input data. This limited experimental setting, while ignoring the impact of performing several optimization steps in PFS, aims at determining: α = 1 α = 2 α = 5 α = 1 α = 2 α = 5 w&s R q R q+ R q R q+ R q R q+ R q R q+ R q R q+ R q R q+ 10c-5w 1. Whether the loss of precision, introduced by α and β, is counterbalanced by the fact that the approximation P α,β is easier to solve than the initial problem. 2. The impact of avoiding pointless trips in recourse strategy R q+ , compared with simpler (but computationally less demanding) strategy R q . 3. The interest of exploiting stochastic knowledge, by comparing the expected costs of SS-VRPTW-CR solutions with their average costs under the w&s policy. 4. The quality of the solutions computed by the LS algorithm under different scale factors. These are compared with optimal solutions obtained with the exact method. When α > 1 or β > 1, the exact method solves P α,β , and the results are reported according to the final solution, scaled back to P 1,1 . 6.1. Impact of the scale factor α Table 2 shows the average gains, in percentages, of using an SS-VRPTW-CR solution instead of the w&s policy, for small instances composed of n = 10 customer vertices with K = 2 uncapacitated vehicles. We consider three different values for α. When α = 1 (resp. α = 2, α = 5), the time horizon is h = 480 (resp. h α = 240, h α = 96) and each time unit corresponds to one minute (resp. two and five minutes). In all cases, the domain reduction factor β is set to 60: waiting times are restricted to multiples of 60 minutes. Unlike the recourse strategies, which must to deal with a limited set of predefined waiting locations, the w&s policy makes direct use of the customer vertices. Therefore, the relative gain of using an optimized Instance: 10c-5w-1 10c-5w-2 10c-5w-3 10c-5w-4 10c-5w-5 Table 3 Statistics on instances 10c-5w-i: the first (resp. second) line gives the average travel time between customer vertices (resp. between a customer and waiting vertices); the last line gives the average duration of a time window. SS-VRPTW-CR first-stage solution is highly dependent on the locations of the waiting vertices. Gains are always greater for 10c+w-i instances, where any customer vertex can be used as a waiting vertex: for these instances, gains with the best-performing strategy are always greater than 23%, whereas for 10c-5w-i instances, the largest gain is 16%, and is negative in some cases. The results obtained on instance 10c-5w-3 are quite interesting: gains are always negative; i.e., waiting strategies always lead to higher expected numbers of rejected requests than the w&s policy. By looking further into the average travel times in each instance, in Table 3, we find that the average travel time between customer vertices in instance 10c-5w-3 is rather small (12.5), and very close to the average duration of time windows (12.3). In this case, anticipation is of less importance and the w&s policy appears to perform better. Furthermore, average travel time between waiting and customer vertices (19.5) is much larger than the average travel time between customer vertices. We note that the exact enumerative method runs out of time under R q+ for all instances, when α = 1. Increasing α to 2 speeds up the solution process and makes it possible to prove optimality on all 10c-5w-i instances except instance 5. Setting α = 5 allows to find all optimal solutions. However, optimizing with coarser scales may degrade the solution quality. This is particularly true for 10c-5w-i instances which are easier, in terms of solution space, than 10c+w-x instances as they have half the number of waiting locations: for 10c-5w-i instances, gains are often decreased when α is increased because, whatever the scale is, the search finds optimal or near-optimal solutions. For PFS, gains with recourse strategy R q+ are always greater than gains with recourse strategy R q on 10c-5w-i instances. However, we observe the opposite on 10c+w-i instances. This comes from the fact that expected costs are much more expensive to compute under R q+ than under R q . Table 2 displays the average number of times the objective function Q R (x, τ ) is evaluated (#eval), that is the number of solutions considered by either the local search or the exact method, in which case it corresponds to the size of the solution space (when enumeration is complete and under assumptions (a) and (b) discussed in Section 5.2). We note that the number of LS iterations is ±10 times smaller when using R q+ compared to R q . As 10c-5w-i instances are easier than 10c+w-i instances, around 10 4 iterations is enough to allow the LS optimizer of PFS to find near-optimal solutions. In this case, gains obtained with R q+ are much larger than those obtained with R q . However, on 10c+w-i instances, 10 4 iterations are not enough to find near-optimal solutions. For these instances, better results are obtained with R q . Exact (% gain after 30 minutes) PFS (% gain after 5 minutes) 10.5 -10.9 0.9 -10.9 0.9 -10.9 0.9 -10.9 0.9 -10.9 0.9 -10.9 0.9 10c-5w-5 8.4 -17.9 2.5 -17.9 2.5 -20.5 0.5 -17.9 2.5 -18 Table 4 Comparison of R q with the hybrid strategy R q/q+ (that uses strategy R q as evaluation function during the optimization process, and evaluates the final solution with strategy R q+ ) on the small instances used in Table 2, with β = 60. α = 1 α = 2 α = 5 α = 1 α = 2 α = 5 w&s R q R q/q+ R q R q/q+ R q R q/q+ R q R q/q+ R q R q/q+ R q R q/ When optimality has been proven by Exact, we note that PFS often finds solutions with the same gain. With α ∈ {2, 5}, PFS may even find better solutions: this is due to the fact that optimality is only proven for the simplified problem P α,β , whereas the final gain is computed after scaling back to the original horizon at scale 1. When optimality has not been proven, PFS often finds better solutions (with larger gains). Combining recourse strategies: R q/q+ Results obtained from Table 2 show that although it leads to larger gains, the computation of expected costs is much more expensive under recourse strategy R q+ than under R q , which eventually penalizes the optimization process as it performs fewer iterations within the same time limit (for both Exact and PFS). We now introduce a pseudo-strategy that we call R q/q+ , which combines R q and R q+ . For both Enum and PFS, strategy R q/q+ refers to the process that uses R q as the evaluation function during all the optimization process. When stopping at a final solution, we reevaluate it using R q+ . Table 4 reports the gains obtained by applying R q/q+ on instances 10c-5w-i and 10c+w-i. By using R q/q+ , we actually use R q to guide the LS optimization, which permits the algorithm to consider a significantly bigger part of the solution space. For both Enum and PFS, R q/q+ always leads to better results than R q . From now on, we will only consider strategies R q/q+ and R q+ in the next experiments. 6.3. Impact of the domain reduction factor β Table 5 considers instances involving 20 customer vertices and either 10 separated waiting locations (20c-10w-i) or one waiting location at each customer vertex (20c+w-i). It compares results obtained by PFS for two different computation time limits, with β ∈ {10, 30, 60}. When β = 10 (resp. β = 30 and β = 60), domains of waiting time variables contain 48 (resp. 16 and 8) values, corresponding to multiples of 10 (resp. 30 and 60) minutes. In all cases, the scale factor α is set to 2. When considering the recourse strategy R q+ with a five-minute computation time limit, we observe that better results are obtained with β = 60, as domains are much smaller. When the computation time is Table 5 Relative gains 5 and 30 minutes, using three domain reduction factors (β ∈ {10, 30, 60}), with K = 2 uncapacitated vehicles and a scale factor α = 2. Instances involve n = 20 customer locations and either 10 or 20 available waiting locations. increased to 30 minutes, or when considering strategy R q/q+ , which is cheaper to compute, then better results are obtained with β = 10, as domains contain finer-grained values. We observe that R q/q+ always provides better results than pure R q+ , whatever the waiting time multiple β used. Except when switching to significantly greater computational times, R q/q+ seems more adequate as it combines the limited computational cost incurred by R q with the nicer expected performances of the cleverer strategy R q+ . Experiments on large instances We now consider instances with n = 50 customer vertices. Instances 50c-30w-i and 50c • PFS-α*β10: the scale factor α is progressively decreased from 5 to 2 and 1 while the domain reduction factor β remains fixed to 10. More precisely, α 0 = 5, α min = 1, and β 0 = β min = 10. The update policy U successively returns α 1 = 2 and α 2 = 1, while β 1 = β 2 = 10. The computation time policy T always returns 3600 seconds, so that the three LS optimizations have the same CPU time limit of one hour. • PFS-α1β*: α remains fixed to 1 while β is progressively decreased from 60 to 30 and 10. More precisely, α 0 = α min = 1, β 0 = 60, and β min = 10. The update policy U successively returns β 1 = 30 and β 2 = 10, while α 1 = α 2 = 1. The computation time policy T always returns 3600 seconds. • PFS-α*β*: both α and β are progressively decreased. We set α 0 = 5, α min = 1, β 0 = 60, and β min = 10. Comparison of the different PFS instantiations The performances of the seven PFS instantiations and the baseline w&s approach are compared in Figure 3 by using performance profiles. Performance profiles (Dolan and Moré, 2002) provide, for each considered approach, a cumulative distribution of its performance compared to other approaches. For a given method A, a point (x, y) on A's curve means that in (100 · y)% of the instances, A performed at most x times worse than the best method on each instance taken separately. A method A is strictly better than another method B if A's curve always stays above B's curve. In the left part, we clearly recognize the nine different optimization phases of PFS-α*β*. A drop in the expected cost happens whenever the current solution s is converted to a higher scale factor. This happens twice during the run: from α 2 = 1 to α 3 = 5 (point a) and from α 5 = 1 to α 6 = 5 (point b). In both cases, the resulting solution becomes infeasible and the algorithm needs some time to restore feasibility. A sudden leap happens when converting to a lower scale. This happens six times (points c): from α i = 5 to α i+1 = 2 and from α i+1 = 2 to α i+2 = 1, with i ∈ {0, 3, 6}. This is a direct consequence of the fact that rounding operations are always performed in a pessimistic way, as explained in Section 5.2. Whereas the quality of s under R q at scale α appears to be worse than that of PFS-α1β10 (e.g., at point b), the true gain of s (evaluated under R q+ , α = 1) remains always better with PFS-α*β*. Finally, Figure 5 of the deviation (%) between costs computed with α = 1, and α ∈ {2, 5}, under strategy R q . On right, the deviation between costs computed with R q and R q+ , with α = 1 in both cases. Scale α = 2 (left, long dashed) always provides a better approximations, closer to the one as computed under α = 1, than scale α = 5 (left, dashed). We also notice a significant increase in the gaps as the algorithm finds better solutions: under 100 seconds, costs computed at scale α = 2 (resp. α = 5) remain at maximum 10% (resp. 20%) from what would be computed under α = 1, and tend to stabilize at around 20% (resp. 45%) in the long term. Similar observations can be made ( Figure 5, right) regarding the gap between costs computed with R q at α = 1 and those computed with R q+ , α = 1. Similarly, the cost difference subsequent to the recourse strategy tends to increase progressively with the quality of the solutions. This could be explained by the time discrepancies generated by rounding operations when a solution is scaled. Better solutions having complex, tighter schedules are then less robust to such time approximations, and more sensible to the discrepancy effects which propagate and impact on the customer time windows. Results on large instances We now analyze how our SS-VRPTW-CR model behaves compared to the w&s policy, when varying both vehicle fleet size and the urgency of requests. We consider algorithm PFS-α*β* only. Table 6 shows how the performance of the SS-VRPTW-CR model relative to the w&s policy varies with the waiting locations and the number of vehicles. For 5, 10, and 20 vehicles, the average over each of the instance classes (15 instances per class) is reported. Influence of the number of vehicles It shows us that the more vehicles are involved, the more important clever anticipative decisions are, and therefore the more beneficial a SS-VRPTW-CR solution is compared to the w&s policy. It is likely that, as conjectured in Saint-Guillain, Solnon, and Deville (2017) On the other hand, we also observe that due to the lack of anticipative actions, the w&s policy globally fails at tacking the advantage of a larger number of vehicles. Indeed, allowing 20 vehicles does not significantly improve the performances of the baseline policy compared to only 10 vehicles. Influence of the time windows We now consider less urgent requests, by conducting the same experiments as in section 7.3.1 while modifying the time windows only. Table 7 shows the average gain of using an SS-VRPTW-CR model when the service quality is reduced by multiplying all the original time window durations by two. The results show that for K = 5 vehicles, the w&s policy always performs better. With K = 20 vehicles, however, the average relative gain achieved by using the SS-VRPTW-CR model remains significant: there are 33.7% fewer rejected requests on average for the class of instances 50c+w-i. approximations that are used (scaling factor, waiting time multiples), allowing the vehicles to wait directly at customer vertices always leads to better results than using separated waiting vertices. Unless the set of possible waiting locations is restricted, e.g., big vehicles cannot park anywhere in the city, placing waiting vertices in such a way that they coincide with customer vertices appears to be the best choice. Conclusions and research directions In this paper, we consider the SS-VRPTW-CR problem previously introduced by Saint-Guillain, Solnon, and Deville (2017). We extend the model with two additional recourse strategies: R q and R q+ . These take customer demands into account and allow the vehicles to save operational time, traveling directly between customer vertices when possible. We show how, under these recourse strategies, the expected cost of a second-stage solution is computable in pseudo-polynomial time. Proof of concept experiments on small and reasonably large test instances compare these anticipative models with each other and show their interest compared to a basic "wait-and-serve" policy. These preliminary results confirm that, although computationally more demanding, optimal first-stage solutions obtained with R q+ generally show significantly better expected behavior. The LS algorithm presented in Saint-Guillain, Solnon, and Deville (2017) produces near-optimal solutions on small instances. In this paper, we also introduce PFS, a meta-heuristic particularly suitable for our problem when coupled with the LS algorithm. More generally, PFS is applicable to any problem in which: a) the objective function is particularly complex to compute but depends on the accuracy of the data and b) the size of the solution space can be controlled by varying the granularity of the operational decisions. We show that PFS allows to efficiently tackle larger problems for which an exact approach is not possible. We show that SS-VRPTW-CR recourse strategies provide significant benefits compared to a basic, non- anticipative but yet realistic policy. Results for a variety of large instances show that the benefit of using the SS-VRPTW-CR increases with the number of vehicles involved and the urgency of the requests. Finally, all our experiments indicate that allowing the vehicles to wait directly at potential customer vertices, when applicable, leads to better expected results than using separated relocation vertices. Future work and research avenues On solution methods. An adaptive version of PFS, therefore improving the algorithm by making dynamic the decision about changing the scale factor α or the domain reduction factor β, could be designed. Exact Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!) 27 optimal methods should also be investigated. However, the black box nature of the evaluation function Q R makes classical (stochastic) integer programming approaches (e.g. branch-and-cut, L-shaped method, etc.) unsuitable for the SS-VRPTW-CR unless efficient valid inequalities that are active at fractional solutions can be devised (such as those proposed by Hjorring and Holt, 1999, for the SS-VRP-D). Amongst other possible candidates for solving this problem, we could consider set-partitioning methods such as column generation, which are becoming commonly used for stochastic VRPs. Approximate Dynamic Programming (ADP, Powell (2009)) is also widely used to solve routing problems in presence of uncertainty. Combined with scaling techniques, ADP is likely to provide interesting results. On scaling techniques. We have shown through experiments that the computational complexity of the objective function is an issue that can be successfully addressed by scaling down problem instances. However, the scale is only performed in terms of temporal data, decreasing the accuracy of the time horizon. It may also be valuable to consider a reduced, clustered set of potential requests, which would also allow us to significantly reduce computational effort when evaluating a first-stage solution. Further application to online optimization. As already pointed out by Saint-Guillain, Solnon, and Deville Towards better recourse strategies. The expected cost of a first-stage solution obviously depends on how the recourse strategy fits the operational problem. Improving these strategies may tremendously improve the quality of the upper bound they provide to exact reoptimization. The recourse strategies presented in this paper are of limited operational complexity, yet their computational complexity is already very expensive. One potential improvement which would limit the increase in computational requirements would be to rethink the way in which the potential requests are assigned to waiting locations, e.g. by taking their probabilities and demands into account. Another direction would be to think about better, more intelligent, vehicle operations. However, an important question remains: how intelligent could a recourse strategy be such that its expected cost stays efficiently computable? Article submitted to ; manuscript no. (Please, provide the manuscript number!) Appendix A Stochastic Integer Programming formulation of the SS-VRPTW-CR The problem stated by (1)-(2) refers to a nonlinear stochastic integer program with recourse, which can be modeled as the following simple extended three-index vehicle flow formulation: Minimize x,τ Q R (x, τ )(6) subject to j∈W 0 x ijk = j∈W 0 x jik = y ik ∀ i ∈ W 0 , k ∈ [1, K] (7) k∈[1,K] y 0k ≤ K (8) k∈[1,K] y ik ≤ 1 ∀ i ∈ W (9) i∈S j∈W \S x ijk ≥ y vk ∀ S ⊆ W, v ∈ S, k ∈ [1, K] (10) l∈H τ ilk = y ik ∀ i ∈ W, k ∈ [1, K] (11) i∈W 0 j∈W 0 x ijk d i,j + i∈W l∈H τ ilk l + 1 ≤ h ∀ k ∈ [1, K](12)y ik ∈ {0, 1} ∀ i ∈ W 0 , k ∈ [1, K] (13) x ijk ∈ {0, 1} ∀ i, j ∈ W 0 : i = j, k ∈ [1, K](14)τ ilk ∈ {0, 1} ∀ i ∈ W, l ∈ H, k ∈ [1, K](15) Our formulation uses the following binary decision variables: • y ik equals 1 iff vertex i ∈ W 0 is visited by vehicle (or route) k ∈ [1, K]; • x ijk equals 1 iff the arc (i, j) ∈ W 2 0 is part of route k ∈ K; • τ ilk equals 1 iff vehicle k waits for 1 ≤ l ≤ h time units at vertex i. Whereas variables y ik are only of modeling purposes, yet x ijk and τ ilk variables solely define a SS-VRPTW-CR first stage solution. Constraints (7) to (10) together with (14) define the feasible space of the asymmetric Team Orienteering Problem (Chao, Golden, and Wasil, 1996). In particular, constraint (8) limits the number of available vehicles. Constraints (9) ensure that each waiting vertex is visited at most once. Subtour elimination constraints (10) forbid routes that do not include the depot. Constraint (11) ensures that exactly one waiting time 1 ≤ l ≤ h is selected for each visited vertex. Finally, constraint (12) states that the total duration of each route, starting at time unit 1, cannot exceed h. Appendix B Expected cost of second-stage solutions under R q In this section we explain how the expected cost of second stage solutions, provided a first stage solution (x, τ ) to the SS-VRPTW-CR, can be efficiently computed in the case of recourse strategy R q . As a Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!) 29 reminder, R q generalizes strategy R ∞ , introduced in Saint-Guillain, Solnon, and Deville (2017), by considering vehicle capacity constraints. Recall also that, once the request ordering and assignment phase finished, we end up with a partition {π ⊥ , π 1 , ..., π K } of R, where π k is the ordered sequence of potential requests assigned to the waiting vertices visited by vehicle k, and π ⊥ is the set of unassigned requests (such that w(r) = ⊥). We note π w , the set of requests assigned to a waiting vertex w ∈ W x . We note fst(π w ) and fst(π k ), the first requests of π w and π k , respectively, according to the order < R . For each request r ∈ π k such that r = fst(π k ), we note prv(r), the request of π k that immediately precedes r according to the order < R . Table 9 summarizes the main notations introduced in this section. Remember that they are all specific to a first-stage solution (x, τ ). In the case of strategy R q , t min r,w and t max r,w are computed according to the definition provided in Saint- Table 9 Notations summary: material for recourse strategies. ⊥ The null vertex: ∀r ∈ R, w(r) = ⊥ ⇔ r is unassigned w(r) Waiting vertex of W x to which r ∈ R is assigned π k Potential request assigned to vehicle k: π k = {r ∈ R : w(r) ∈ x k } π w Potential request assigned to waiting location w ∈ W x : π w = {r ∈ R : w(r) = w} fst(π w ) Smallest request of π w according to < R . fst(π k ) Smallest request of π k according to < R . prv(r) Request of π k which immediately precedes r according to < R , if any t min r,w Min. time from which a vehicle can handle request r ∈ R from w ∈ W x t max r,w Max. time from which a vehicle can handle request r ∈ R from w ∈ W x Guillain, Solnon, and Deville (2017) for strategy R ∞ . Hence, t min r,w = max{on(w), Γ r , e r − d w,r } and t max r,w = min{l r − d w,r , on(w) − d w,r − s r − d r,w }. We assume that request probabilities to be independent of each other; i.e., for any couple of requests r, r ∈ R, the probability p r∧r that both requests will appear is given by p r∧r = p r · p r . Q R q (x, τ ) is equal to the expected number of rejected requests, which in turn is equal to the expected number of requests that are found to appear minus the expected number of accepted requests. Under the independence hypothesis, the expected number of revealed requests is given by the sum of all request probabilities, whereas the expected number of accepted requests is equal to the cumulative sum, for every request r, of the probability that it belongs to A h , i.e., Q R q (x, τ ) = r∈R p r − r∈R Pr{r ∈ A h } = r∈R p r − Pr{r ∈ A h }(16) The probability Pr{r ∈ A h } is computed by considering every feasible time t ∈ [t min r,w , t max r,w ] and every possible load configuration q ∈ [0, Q − q r ] that satisfies r: Pr{r ∈ A h } = t max r,w t=t min r,w Q−qr q=0 g 1 (r, t, q).(17) Article submitted to ; manuscript no. (Please, provide the manuscript number!) g 1 (r, t, q) is the probability that r has appeared and that vehicle k leaves w(r) at time t with load q to serve r, i.e., g 1 (r, t, q) ≡ Pr{r appeared, departureTime(r) = t and load(k, t) = q} where load(k, t) is the load of vehicle k ∈ [1, K] at time t ∈ H, and departureTime(r) = max{available(r), t min r,w(r) } is the time at which it actually leaves the waiting vertex w(r) in order to serve r (the vehicle may have to wait if available(r) is smaller than the earliest time for leaving w(r) to serve r). Computation of probability g 1 (r, t, q) Recall that π k is the set of potential requests on route k ∈ [1, K], ordered by < R . The base case for computing g 1 is concerned with the very first potential request on the entire route, r = fst(π k ), which must be considered as soon as vehicle k arrives at w = w(r), that is, at time on(w), except if on(w) < t min r,w : if r = fst(π k ) then g 1 (r, t, q) = p r if t = max{on(w), t min r,w } ∧ q = 0 0 otherwise.(18) For any q ≥ 1, g 1 (r, t, q) is equal to zero as vehicle k necessarily carries an empty load when considering the first request r. The more general case of a request r which is not the first request of a waiting vertex w ∈ W x , (i.e., w = fst(π w )), depends on the time and load configuration at which vehicle k is available for r, Although available(r) and load(k, t) are both deterministic when we know the set A Γ prv(r) of previously accepted requests, this is not true anymore when computing probability g 1 (r, t, q). As a consequence, g 1 (r, t, q) depends on the probability f (r, t, q) that vehicle k is available for r at time t with load q: f (r, t, q) ≡ Pr{finishToServe(prv(r)) = t and load(k, t) = q}. Note that for any such request r ∈ R : r = fst(π w(r) ), the time finishToServe(prv(r)) is equivalent to available(r). On the contrary, this is not the case for a request that is the first of its waiting vertex. The computation of f is detailed below. Given this probability f , the general case for computing g 1 is: if r = fst(π w(r) ) then g 1 (r, t, q) =      p r · f (r, t, q) if t > t min r,w(r) p r · t min r,w(r) t =on(w(r)) f (r, t , q) if t = t min r,w(r) 0 otherwise(19) Indeed, if t > t min r,w(r) , then vehicle k leaves w(r) to serve r as soon as it becomes available. If t < t min r,w(r) , the probability that vehicle k leaves w(r) at time t is null since t min r,w(r) is the earliest time for serving r from w(r). Finally, at time t = t min r,w(r) , we must consider the possibility that vehicle k has been waiting to serve r since an earlier time on(w(r)) ≤ t < t min r,w(r) . In this case, the probability that vehicle k leaves w(r) to serve r at time t is p r times the probability that vehicle k has actually been available from a time on(w(r)) ≤ t ≤ t min r,w(r) . We complete the computation of g 1 with the particular case of a request r which is not the first of the route (i.e., r = fst(π k )) but is the first assigned to the waiting vertex associated with r (i.e., r = fst(π w(r) )). As the arrival time on w(r) is fixed by the first-stage solution, departureTime(r) is necessarily max(on(w(r)), t min r,w(r) ). In particular, time finishToServe(prv(r)) is no longer equivalent to available(r). Unlike departureTime(r), load(k, t) is not deterministic but rather depends on what happened previously. More precisely, load(k, t) depends on the load carried by vehicle k when it has finished serving prv(r) at the previous waiting location w(prv(r)). For every first request of a waiting vertex, but not the first of the route, we then have: if r = fst(π w(r) ) and r = fst(π k ) then g 1 (r, t, q) = p r · on(w(prv(r))) t =on(w(prv(r)) f (r, t , q), if t = max(on(w(r)), t min r,w(r) ) 0 otherwise , where we see that all possible time units for vehicle k to serve prv(r) belong to on(w(prv(r))), on(w(prv(r))) . Computation of probability f (r, t, q) Let us now define how to compute f (r, t, q), the probability that vehicle k becomes available for r at time t with load q. This depends on what happened to the previous request r − = prv(r). We have to consider three cases: (a) r − appeared and was satisfied, (b) r − appeared but was rejected, and (c) r − did not appear. Let us introduce our last probability g 2 (r, t, q), which is the probability that a request r did not appear and is discarded at time t while the associated vehicle carries load q. We note discardedTime(r) = max{available(r), Γ r }, the time at which the vehicle becomes available for r whereas r does not appear: g 2 (r, t, q) ≡ Pr{r did not appear, discardedTime(r) = t and load(k, t) = q}. The computation of g 2 is detailed below. Given g 2 , we compute f as follows: f (r, t, q) = g 1 (r − , t − S r − , q − q r − ) · δ(r − , t − S r − , q − q r − ) + g 1 (r − , t, q) · 1 − δ(r − , t, q) + g 2 (r − , t, q).(21) where the indicator function δ(r, t, q) returns 1 if and only if request r is satisfiable from vertex w(r) at time t with load q; i.e., δ(r, t, q) = 1 if t ≤ t max r,w(r) and q + q r ≤ Q, whereas δ(r, t, q) = 0 otherwise. The first term in the summation of the right hand side of equation (21) gives the probability that request r − actually appeared and was satisfied (case a). In such a case, departureTime(r − ) must be the current time t minus Article submitted to ; manuscript no. (Please, provide the manuscript number!) the delay S r − needed to serve r − . The second and third terms of equation (21) add the probability that the vehicle was available at time t but that request r − did not consume any operational time. There are only two possible reasons for that: either r − actually appeared but was not satisfiable (case b, corresponding to the second term) or r − did not appear at all (case c, corresponding to the third term). Note that f (r, t, q) must be defined only when r is not the first potential request of a waiting location. Computation of probability g 2 (r, t, q) This probability is computed recursively, as for g 1 . For the very first request of the route of vehicle k, we have: if r = fst(π k ) then g 2 (r, t, q) = 1 − p r , if t = max(on(w(r)), Γ r ) ∧ q = 0 0 otherwise.(22) The general case of a request which is not the first of its waiting vertex is quite similar to the one of function g 1 . We just consider the probability 1 − p r that r is found not to appear and replace t min r,w(r) by the reveal time Γ r : if r = fst(π w(r) ) then g 2 (r, t, q) =      (1 − p r ) · f (prv(r), t, q) if t > max(on(w(r)), Γ r ) (1 − p r ) · max(on(w(r)),Γr) t =on(w(r)) f (prv(r), t , q) if t = max(on(w(r)), Γ r ) 0 otherwise. Finally, for the first request of a waiting location that is not the first of its route, we have: if r = fst(π w(r) ) and r = fst(π k ) then g 2 (r, t, q) = (1 − p r ) · on(w(prv(r))) t =on(w(prv(r))) f (prv(r), t , q), if t = max(on(w(r)), Γ r ) 0 otherwise. Computational complexity. The complexity of computing Q R q (x, τ ) is equivalent to that of filling up K matrices of size |π k |×h×Q containing all the g 1 (r, t, q) probabilities. In particular, once the probabilities in cells (prv(r), 1 · · · t, 1 · · · q) are known, the cell (r, t, q) such that r = fst(π w ) can be computed in O(h) according to equation (19). Given n customer vertices and a time horizon of length h, there are at most |R| = nh ≥ K k=1 |π k | potential requests in total, leading to an overall worst case complexity of O(nh 2 Q). Incremental computation. Since we are interested in computing Pr{r ∈ A h } for each request r separately, by following the definition of g 1 and f , the probability of satisfying r only depends on the g 1 and g 2 probabilities associated with prv(r). As a consequence, two similar first-stage solutions are likely to share equivalent subsets of probabilities. This is of particular interest when considering LS-based methods generating sequences of (first-stage) solutions, where each new solution is usually quite similar to the previous one. In fact, for every two similar solutions, subsets of equivalent probabilities can easily be deduced, hence allowing an incremental update of the expected cost. This does not change the time complexity, as in the worst case (i.e., when the first waiting vertex of each sequence in x has been changed), all probabilities must be recomputed. However, this greatly improves the efficiency in practice. B.1 Relation with SS-VRP-C As presented in section , the SS-VRP-C differs by having stochastic binary demands (which represent the random customer presence) and no time window. In this case, the goal is to minimize the expected distance traveled, provided that when a vehicle reaches its maximal capacity, it unloads by making a round trip to the depot. In order to compute the expected length of a first-stage solution that visits all customers, a key point is to compute the probability distribution of the vehicle's current load when reaching a customer. In fact, this is directly related to the probability that the vehicle makes a round trip to the depot to unload, which is denoted by the function "f (m, r)" in Bertsimas (1992). Here we highlight the relation between SS-VRPTW-CR and SS-VRP-C by showing how Bertsimas's "f (m, r)" equation can be derived from equation (21) when time windows are not taken into account. Since there is no time window consideration, we can state that Γ r = t min r,w = 1 and t max r,w = +∞ for any request r. Also, each demand q r is equal to 1. Consequently, the δ-function used in the computation of the f probabilities depends only on q and is equal to 1 if q ≤ Q. Therefore, the f probabilities are defined by: f (r, t, q) = g 1 (r − , t − S r − , q − 1) + g 2 (r − , t, q) with r − = prv(r). Now let f (r, q) = t∈H f (r, t, q). As f (r, t, q) is the probability that the vehicle is available for r at time t with load q, f (r, q) is the probability that the vehicle is available for r with load q during the day. It is also true that f (r, q) gives the probability that exactly q requests among the r 1 , ..., r − potential ones actually appear (with a unit demand). We have: f (r, q) = t∈H f (r, t, q) = t∈H g 1 (r − , t − S r − , q − 1) + t∈H g 2 (r − , t, q) As we are interested in f (r, q), not the travel distance, we can assume that all potential requests are assigned to the same waiting vertex. Then either r = fst(π k ) or r = fst(π w(r) ). If r = fst(π k ) we naturally obtain: f (r, q) = t∈H p r · f (r, t, q − 1) + t∈H (1 − p r ) · f (r, t, q) = p r + (1 − p r ) = 1, if q = 0 0, otherwise. If r = fst(π w(r) ), since we always have t ≥ t min r,w , we have: f (r, q) = t∈H p r · f (r, t, q − 1) + t∈H (1 − p r ) · f (r, t, q) = p r · f (r, q − 1) + (1 − p r ) · f (r, q). We directly see that the definition of f (r, q) is exactly the same as the corresponding function "f (m, r)" described in Bertsimas (1992) for the SS-VRP-CD with unit demands, that is, the SS-VRP-C. E-Companion Appendix EC.1 Instance generation Data used to generate instances We derive our test instances from the benchmark described in Melgarejo, Laborie, and Solnon (2015) for the Time-Dependent TSP with Time Windows (TD-TSPTW). This benchmark has been created using real accurate delivery and travel time data coming from the city of Lyon, France. Travel times have been computed from vehicle speeds that have been measured by 630 sensors over the city (each sensor measures the speed on a road segment every 6 minutes). For road segments without sensors, travel speed has been estimated with respect to speed on the closest road segments of similar type. Figure EC.1 displays the set of 255 delivery addresses extracted from real delivery data, covering two full months of time-stamped and geo-localized deliveries from three freight carriers operating in Lyon. For each couple of delivery addresses, travel duration has been computed by searching for a quickest path between the two addresses. In the original benchmark, travel durations are computed for different starting times (by steps of 6 minutes), to take into account the fact that travel durations depend on starting times. In our case, we remove the time-dependent dimension by simply computing average travel times (for all possible starting times). We note V the set of 255 delivery addresses, and d i,j the duration for traveling from i to j with i, j ∈ V . This allows us to have realistic travel times between real delivery addresses. Note that in this real-world context, the resulting travel time matrix is not symmetric. Instance generation We have generated two different kinds of instances: instances with separated waiting locations, and instances without separated waiting locations. Each instance with separated waiting locations is denoted nc-mw-i, where n ∈ {10, 20, 50} is the number of customer vertices, m ∈ {5, 10, 30, 50} is the number of waiting vertices, and x ∈ [1, 15] is the random seed. It is constructed as follows: Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times 2 Article submitted to ; manuscript no. (Please, provide the manuscript number!) 1. We first partition the 255 delivery addresses of V in m clusters, using the k-means algorithm with k = m. During this clustering phase, we have considered symmetric distances, by defining the distance between two points i and j as the minimum duration among d i,j and d j,i . 2. For each cluster, we select the median delivery address, i.e., the address in the cluster such that its average distance to all other addresses in the cluster is minimal. The set W of waiting vertices is defined by the set of m median addresses. 3. We randomly and uniformly select the depot and the set C of n customer vertices in the remaining set V \ W . Each instance without separated waiting locations is denoted nc+w-i. It is constructed by randomly and uniformly selecting the depot and the set C in the entire set V and by simply setting W = C. In other words, in these instances vehicles do not wait at separated waiting vertices, but at customer vertices, and every customer vertex is also a waiting location. Furthermore, instances sharing the same number of customers n and the same random seed x (e.g. 50c-30w-1, 50c-50w-1 and 50c+w-1) always share the exact same set of customer vertices C. Operational day, horizon and time slots We fix the duration of an operational day to 8 hours in all instances. We fix the horizon resolution to h = 480, which corresponds to one minute time steps. As it is not realistic to detail request probabilities for each time unit of the horizon (i.e., every minute), we introduce time slots of 5 minutes each. We thus have n TS = 96 time slots over the horizon. To each time slot corresponds a potential request at each customer vertex. Customer potential requests and attributes. For each customer vertex c, we generate the request probabilities associated with c as follows. First, we randomly and uniformly select two integer values µ 1 and µ 2 in [1, n TS ]. Then, we randomly generate 200 integer values: 100 with respect to a normal distribution the mean of which is µ 1 and 100 with respect to a normal distribution the mean of which is µ 2 . Let us note nb[i] the number of times value i ∈ [1, n TS ] has been generated among the 200 trials. Finally, for each reveal time Γ ∈ H, if Γ mod 5 = 0, then we set p (c,Γ) = 0 (as we assume that requests are revealed every 5 minute time slots). Otherwise, we set p (c,Γ) = min(1, nb[Γ/5] 100 ). Hence, the expected number of requests at each customer vertex is smaller than or equal to 2 (in particular, it is smaller than 2 when some of the 200 randomly generated numbers do not belong to the interval [1, n TS ], which may occur when µ 1 or µ 2 are close to the boundary values). Figure request. Note that the beginning of the time window of a request r is equal to its reveal time Γ r . This aims at simulating operational contexts similar to the practical application example described in section (the on-demand health care service at home), requiring immediate responses within small time windows. Input data. We consider a complete directed graph G = (V, A) and a discrete time horizon H = [1, h], where the interval [a, b] denotes the set of all integer values i such that a ≤ i ≤ b. A travel time (or distance) d i,j ∈ N is associated with every arc (i, j) ∈ A. The set of vertices V = {0} ∪ W ∪ C is composed of a depot 0, a set of m waiting locations W = [1, m], and a set of n customer vertices C = [m + 1, m + n]. Pr s(w) then travel from v to s(← set of requests of π w not yet satisfied, either accepted or not yet revealed; 4 if P = ∅ then travel to the next waiting vertex (or the depot); next ← smallest element of P according to the predefined order on R; 7 Figure 2 2Comparative examples of strategies R ∞ (left) and R q+ (right). The depot, waiting vertices and customer vertices are represented by a square, circles and crosses, respectively. Arrows represent vehicle routes. A filled cross represents a revealed request. Under R ∞ , some requests (r3, r6, r15, . . .) can be missed. By avoiding pointless journeys when possible, R q+ is likely to end up with a lower number of missed requests. For example, if request r3 is revealed by the time request r1 is satisfied, then traveling directly to r3 could help satisfy it. Similarly, on a different route, by traveling directly to the waiting vertex associated with request r20, the vehicle could save enough time to satisfy r20.is not affordable in practice. This section gives an overview of how we efficiently compute the expected number of rejected requests under the recourse strategy R q+ . Further developments of the closed-form expressions are then provided in Appendix C. Domains are reduced by selecting a subset of |Dom(τ w )|/β values, evenly distributed in [1, h]. As the temporal dimension is also scaled with respect to α, selected values are scaled down: Dom α,β (τ w ) = {round(i/α) : i ∈ [1, h], i mod β = 0}. w&s) and the gain of the best solution found by the exact approach within a time limit of 30 minutes and PFS within a time limit of 5 minutes (average on 10 runs). Results marked with a star ( * ) have been proved optimal. #eval gives the average number of expectation computations for each run: solutions enumerated (Exact) or LS iterations (PFS). -50w-i have m = 30and m = 50 separated waiting locations, respectively. Instances 50c+w-i have m = 50 waiting vertices which correspond to the customer vertices. Each class is composed of 15 instances such that, for each seed i ∈ [1, 15], the three instances classes 50c-30w-i, 50c-50w-i, and 50c+w-i contain the same set of 50 customer vertices and thus only differ in terms of the number and/or positions of waiting vertices. For each instance, the vehicle's capacity is set to Q = 20, and we consider three different numbers of vehicles K ∈ {5, 10, 20}. In total, we thus have 45 × 3 = 135 different configurations.We first compare and discuss the behaviors of different instantiations of PFS. Then, based on the PFS variant that appears to perform best, further experiments (Section 7.3) measure the contribution of a twostage stochastic model, through the use of a SS-VRPTW-CR formulation and our recourse strategies.7.1. Instantiations of PFSAll runs of PFS are limited to T = 10800 seconds (three hours). We compare seven instantiations of PFS, which have different update and computation time policies U and T , while all other parameters are set as described in Section 6.2.1. Strategy R q/q+ is used for all experiments. The different instantiations are: 22 Article submitted to ; manuscript no. (Please, provide the manuscript number!) The update policy U returns the following couples of values for (α i , β i ): 10). The computation time policy T always returns 1200 seconds. The PFS optimization process is hence composed of nine LS optimizations of 20 minutes each.• PFS-αaβb which performs only a single LS optimization step with T = 10800 and α 0 = α min = a and β 0 = β min = b, as experimented in Section 7. We consider two different values for α, i.e., a ∈ {1, 2}, and two different values for β, i.e., b ∈ {10, 60}, thus obtaining four different instantiations. According to Figure 3 ( 3left), algorithms PFS-α*β10 and PFS-α*β* show the best performances when tested on the 15 instances of class 50c+w-i with K = 20 vehicles. More experiments are conducted and reported inFigure 3(right) in order to distinguish between the algorithms PFS-α*β10, PFS-α1β* and PFSα*β* on all 135 instances. In comparison to the other approaches, algorithms PFS-α*β10 and PFS-α*β* clearly obtain the best performances on average over the 135 configurations. Figure 4 4illustrates, on a single instance (50c-50w-1 with K = 10 vehicles), the evolution through time of the gain of the expected cost of the current solution s, with respect to the average cost of the w&s policy, during a single run of PFS-α1β10, PFS-α*β10, PFS-α1β*, and PFS-α*β*. For each incumbent solution s, the left part ofFigure 4plots the gain of s under R q at its current scale α. It corresponds to the quality of s as evaluated by the LS algorithm. The right part plots the corresponding gain under R q+ at scale α = 1. Figure 3 3Performance profiles. Left: comparison of the seven PFS instantiations and the w&s policy on the 15 instances of class 50c+w-i, using K = 20 vehicles. Right: comparison of PFS instantiations PFSα*β10, PFS-α1β* and PFS-α*β* on the 3 classes (50c-30w-i, 50c-50w-i, 50c+w-i), with K ∈ {5, 10, 20} vehicles (135 instances). Figure 4 4Evolution through time of the gain of the expected cost of the current solution with respect to the average cost of the w&s policy, during a single execution of four PFS instantiations for instance 50c-50w-1 (with K = 10 vehicles). Left: gain evaluated under R q at current scale α. Right: gain evaluated under R q+ at scale α = 1. Figure 5 5compares the expected costs when varying either the scale α (left) or the recourse strategy (right), using the same sequences of solutions than those used for Figure 4. On left, the evolution Scale approximation quality and impact of recourse strategies. For each solution s encountered while running PFS-α*β*, on instance 50c-50w-1 as displayed in Figure 4, left curves show the evolution of the gap (in %) between costs computed with α ∈ {2, 5} and those computed with α = 1. On right, the gap between R q and R q+ , both with α = 1. , a higher number of vehicles leads to a less uniform objective function, most probably with the steepest local optima. Because it requires much more anticipation than when there are only five vehicles, using the SS-VRPTW-CR model instead of the w&s policy is found to be particularly beneficial provided that there are at least 10 or 20 vehicles. With 20 vehicles, our model decreases the average number of rejected requests by 52.2% when vehicles are allowed to wait at customer vertices (i.e. for the class of instances 50c+w-i). ( 2017 ) 2017, another potential application of the SS-VRPTW-CR is to online optimization problems such as the Dynamic and Stochastic VRPTW (DS-VRPTW). Most of the approaches that have been proposed in order to solve the DS-VRPTW rely on reoptimization. However, because perfect online reoptimization isintractable, heuristic methods are often preferred. Approaches based on sampling, such as Sample AverageApproximation(Ahmed and Shapiro, 2002), are very common and consist in restricting the set of scenario to a random subset. Because the computed costs depend on the quality and size of the subset of scenarios, they do not provide any guarantee. Thanks to recourse strategies, the expected cost of a first-stage SS-VRPTW-CR solution provides an upper bound on the expected cost under perfect reoptimization, as it also enforce the nonanticipativity constraints (see Saint-Guillain, Deville, and Solnon, 2015, for a description of these constraints). The SS-VRPTW-CR can therefore be exploited when solving the DS-VRTPW. ec1 Figure ec1Figure EC.1 Lyon's road network. In purple, the 255 customer vertices. Figure EC.2 shows a representation of the distributions in an instance involving 10 customer vertices.For a same customer vertex, there may be several requests on the same day at different time slots, and their probabilities are assumed independent. To each potential request r = (c r , Γ r ) is assigned a deterministic demand q r taken uniformly in [0, 2], a deterministic service duration s r = 5 and a time window [Γ r , Γ r + ∆ − 1], where ∆ is taken uniformly in {5, 10, 15, 20} that is, either 5, 10, 15 or 20 minutes to meet the EC.2 Probability distributions in instance 10-c5w-1. Each cell represents one of the 96 time slots, for each customer vertex. The darker a cell, the more likely a request to appear at the corresponding time slot. A white cell represents a zero probability request that is, no potential request. Table 1 1Notation summary: graph and potential requests.G = (V, A) Complete directed graph R = C × H Set of potential requests V = {0} ∪ W ∪ C Set of vertices (depot is 0) Γ r Reveal time of request r ∈ R W = [1, m] Waiting vertices c r Customer vertex hosting request r ∈ R C = [m + 1, m + n] Customer vertices s r Service time of request r ∈ R d i,j Travel time of arc (i, j) ∈ A [e r , l r ] Time window of request r ∈ R K Number of vehicles q r Demand of request r ∈ R Q Vehicle capacity p r Probability associated with request r H = [1, h] Discrete time horizon Table 8 8illustrates how the average gain is impacted when time windows are multiplied by three. Given20 vehicles, the SS-VRPTW-CR model still improves the w&s policy by 14% when vehicles are allowed to wait directly at customer vertices. Together with Table 6, Tables 7 and 8 show that the SS-VRPTW-CR model is more beneficial when the number of vehicles is high and the time windows are small, that is, in instances that are particularly hard in terms of quality of service and thus require much more anticipation. 7.3.3. Positions of the waiting locations From all the experiments conducted on our benchmark, it immediately appears that, no matter the operational context (number of customer vertices, vehicles) or the Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!)33 Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times Article submitted to ; manuscript no. (Please, provide the manuscript number!) Article submitted to ; manuscript no. (Please, provide the manuscript number!) AcknowledgmentsComputational resources have been provided by the Consortium des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11. Christine Solnon is supported by the LABEX IMU (ANR-10-LABX-0088) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Finally, we thank Anthony Papavasiliou for its sound suggestions and advices during the early stages of the study.Appendix C Computing g v 1 (r, t, q) under strategy R q+ This section develops equation (5) introduced in Section 3.2 for computing the expected cost of second stage solutions under recourse strategy R q+ , provided a first stage solution (x, τ ). Since the method is (more complex but) similar to the one developed in the case of strategy R q (see Section 5.2.1), for a better understanding we recommend to read through Appendix B before considering this section.We are interested in the computation of g v 1 (r, t, q), the probability that request r appeared at a time t ≤ t and that, if it is accepted, the vehicle serves it by leaving vertex v ∈ W ∪ C at time t whilst carrying a load of q. Let the following additional random functions: h w (r, t, q) ≡ Pr{the vehicle gets rid of request r at time t with a load of q and at waiting location w} h r (r, t, q) ≡ Pr{the vehicle gets rid of request r at time t with a load of q and at location c r } and:g w 2 (r, t, q) ≡ Pr{request r did not appear and the vehicle discards it at time t with a load of q while being at waiting location w} g r 2 (r, t, q) ≡ Pr{request r did not appear and the vehicle discards it at time t with a load of q while being at location c r }.For the very first request r k 1 = fst(π k ) of the route, trivially the current load q of the vehicle must be zero, and it seems normal for the waiting location w = w(r k 1 ) to be the only possible location from which the vehicle can be available to handling r k 1 if the request appears, or to discard it if it doesn't:The vehicle thus cannot be available for r k 1 at any other location r < R r: Concerning r 1 = fst(π w ) the first request of any other waiting location w = w(r k 1 ), we use the same trick as for strategy R q in order to obtain the probabilities for each possible vehicle load q:with w = w(prv(r 1 )). From any other request r < R r we still have:For a request r > R fst(π w ), w ∈ W x :when replacing v by either w = w(r) or r ∈ π w , r < R r, w = w(r).At a waiting location w ∈ W x :The aforementioned terms of the sum are: and t w = t − d w,r − s r − d r,w , t r = t − d r ,r − s r − d r,w and Γ next r = Γ r if nxt(r) exists, zero otherwise. The second term h w 2 is:h w 2 (r, t, q) = g w 1 (r, t, q) · 1 − δ(r, t, w) + g w 2 (r, t, q).Finally:where bool(a) returns 1 if the Boolean expression a is true, 0 otherwise.The probability that the vehicle gets rid of request r at r's location is:h r (r, t, q) = g w 1 (r, t − d w,r − s r , q − q r ) · δ w (r, t − d w,r − s r , q − q r ) + r ∈πw r < R r g r 1 (r, t − d r ,r − s r , q − q r ) · δ r (r, t − d r ,r − s r , q − q r )if t ≥ Γ next r , otherwise h r (r, t, q) = 0. 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N Secomandi, F Margot, http:/pubsonline.informs.org/doi/abs/10.1287/opre.1080.0520Operations Research. 57Secomandi N, Margot F, 2009 Reoptimization approaches for the vehicle-routing problem with stochastic demands. Operations Research 57:214-230, URL http://pubsonline.informs.org/doi/abs/10.1287/ opre.1080.0520. A model and algorithm for the courier delivery problem with uncertainty. I Sungur, Y Ren, http:/pubsonline.informs.org/doi/abs/10.1287/trsc.1090.0303Transportation science. 442Sungur I, Ren Y, 2010 A model and algorithm for the courier delivery problem with uncertainty. Transportation sci- ence 44(2):193-205, URL http://pubsonline.informs.org/doi/abs/10.1287/trsc.1090. A tabu search heuristic for the vehicle routing problem with soft time windows. É Taillard, P Badeau, M Gendreau, F Guertin, J Y Potvin, http:/pubsonline.informs.org/doi/abs/10.1287/trsc.31.2.170Transportation Science. 312Taillard É, Badeau P, Gendreau M, Guertin F, Potvin JY, 1997 A tabu search heuristic for the vehicle routing problem with soft time windows. Transportation Science 31(2):170-186, URL http://pubsonline.informs. org/doi/abs/10.1287/trsc.31.2.170. The sample average approximation method applied to stochastic routing problems: a computational study. B Verweij, S Ahmed, A J Kleywegt, G Nemhauser, A Shapiro, Computational Optimization and Applications. 242-3Verweij B, Ahmed S, Kleywegt AJ, Nemhauser G, Shapiro A, 2003 The sample average approximation method applied to stochastic routing problems: a computational study. Computational Optimization and Applications 24(2- 3):289-333. Stochastic Vehicle Routing Problem with Restocking. W H Yang, K Mathur, R H Ballou, 10.1287/trsc.34.1.99.12278Transportation Science. 341Yang WH, Mathur K, Ballou RH, 2000 Stochastic Vehicle Routing Problem with Restocking. Transportation Science 34(1):99-112, URL http://dx.doi.org/10.1287/trsc.34.1.99.12278. Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times. e-companion to Saint-Guillain, Solnon, and Deville: PFS for the SS-VRPTW with both Random Customers and Reveal Times

The full compressible Navier-Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid is studied in a threedimensional simply connected bounded domain with smooth boundary having a finite number of two-dimensional connected components. For the initial-boundaryvalue problem with slip boundary conditions on the velocity and Neumann boundary one on the temperature, the global existence of classical and weak solutions which are of small energy but possibly large oscillations is established. In particular, both the density and temperature are allowed to vanish initially. Finally, the exponential stability of the density, velocity, and temperature is also obtained. Moreover, it is shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially. This is the first result concerning the global existence of classical solutions to the full compressible Navier-Stokes equations with vacuum in general three-dimensional bounded smooth domains.

2207.00441

e9c1edc9b9e8302f232b1db6abe4aff388f5e13a

Global Existence of Classical Solutions to Full Compressible Navier-Stokes System with Large Oscillations and Vacuum in 3D Bounded Domains Jul 2022 Jing Li Department of Mathematics & Institute of Mathematics and Interdisciplinary Sciences Nanchang University 330031NanchangP. R. China School of Mathematical Sciences University of Chinese Academy of Sciences 100049BeijingP. R. China Institute of Applied Mathematics AMSS Boqiang Lü Department of Mathematics & Institute of Mathematics and Interdisciplinary Sciences Nanchang University 330031NanchangP. R. China Xue Wang School of Mathematical Sciences University of Chinese Academy of Sciences 100049BeijingP. R. China Hua Loo-Keng Key Laboratory of Mathematics Chinese Academy of Sciences 100190BeijingP. R. China Global Existence of Classical Solutions to Full Compressible Navier-Stokes System with Large Oscillations and Vacuum in 3D Bounded Domains Jul 2022full compressible Navier-Stokes systemglobal existenceslip boundary conditionvacuumlarge oscillations The full compressible Navier-Stokes system describing the motion of a viscous, compressible, heat-conductive, and Newtonian polytropic fluid is studied in a threedimensional simply connected bounded domain with smooth boundary having a finite number of two-dimensional connected components. For the initial-boundaryvalue problem with slip boundary conditions on the velocity and Neumann boundary one on the temperature, the global existence of classical and weak solutions which are of small energy but possibly large oscillations is established. In particular, both the density and temperature are allowed to vanish initially. Finally, the exponential stability of the density, velocity, and temperature is also obtained. Moreover, it is shown that for the classical solutions, the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially. This is the first result concerning the global existence of classical solutions to the full compressible Navier-Stokes equations with vacuum in general three-dimensional bounded smooth domains. Introduction The motion of a compressible viscous, heat-conductive, and Newtonian polytropic fluid occupying a spatial domain Ω ⊂ R 3 is governed by the following full compressible Navier-Stokes system:      ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) + ∇P = divS, (ρE) t + div(ρEu + P u) = div(κ∇θ) + div(Su), (1.1) where S and E are respectively the viscous stress tensor and the total energy given by S = 2µD(u) + λdivuI 3 , E = e + 1 2 |u| 2 , with D(u) = (∇u + (∇u) tr )/2 and I 3 denoting the deformation tensor and the 3 × 3 identity matrix respectively. Here, t ≥ 0 is time, x ∈ Ω is the spatial coordinate, and ρ, u = u 1 , u 2 , u 3 tr , e, P, and θ represent respectively the fluid density, velocity, specific internal energy, pressure, and absolute temperature. The viscosity coefficients µ and λ are constants satisfying the physical restrictions: µ > 0, 2µ + 3λ ≥ 0. (1.2) The heat-conductivity coefficient κ is a positive constant. We consider the ideal polytropic fluids so that P and e are given by the state equations: P (ρ, e) = (γ − 1)ρe = Rρθ, e = Rθ γ − 1 ,(1.3) where γ > 1 is the adiabatic constant and R is a positive constant. Let Ω ⊂ R 3 be a simply connected bounded domain. Note that for the classical solutions, the system (1.1) can be rewritten as      ρ t + div(ρu) = 0, ρ(u t + u · ∇u) = µ∆u + (µ + λ)∇(divu) − ∇P, R γ−1 ρ(θ t + u · ∇θ) = κ∆θ − P divu + λ(divu) 2 + 2µ|D(u)| 2 . (1.4) We consider the system (1.4) subjected to the given initial data (ρ, ρu, ρθ)(x, t = 0) = (ρ 0 , ρ 0 u 0 , ρ 0 θ 0 )(x), x ∈ Ω, (1.5) and boundary conditions u · n = 0, curlu × n = 0, ∇θ · n = 0 on ∂Ω × (0, T ), (1.6) where n = (n 1 , n 2 , n 3 ) tr is the unit outward normal vector on ∂Ω. There is a lot of literature on the global existence and large time behavior of solutions to (1.1). The one-dimensional problem with strictly positive initial density and temperature has been studied extensively by many people (see [2,18,19] and the references therein). For the multi-dimensional case, the local existence and uniqueness of classical solutions are known in [24,29] in the absence of vacuum. The global classical solutions were first obtained by Matsumura-Nishida [23] for initial data close to a nonvacuum equilibrium in some Sobolev space H s . Later, Hoff [12] studied the global weak solutions with strictly positive initial density and temperature for discontinuous initial data. On the other hand, in the presence of vacuum, this issue becomes much more complicated. Concerning viscous compressible fluids in a barotropic regime, where the state of these fluids at each instant t > 0 is completely determined by the density ρ = ρ(x, t) and the velocity u = u(x, t), the pressure P being an explicit function of the density, the major breakthrough is due to Lions [22] (see also Feireisl [9,10]), where he obtained global existence of weak solutions, defined as solutions with finite energy, when the pressure P satisfies P (ρ) = aρ γ (a > 0, γ > 1) with suitably large γ. The main restriction on initial data is that the initial energy is finite, so that the density vanishes at far fields, or even has compact support. Recently, Huang-Li-Xin [16] and Li-Xin [21] established the global well-posedness of classical solutions to the Cauchy problem for the 3D and 2D barotropic compressible Navier-Stokes equations in whole space with smooth initial data that are of small energy but possibly large oscillations, in particular, the initial density is allowed to vanish. More recently, for slip boundary condition in bounded domains, Cai-Li [6] obtained the global classical solutions with initial vacuum, provided that the initial energy is suitably small. Compared with the barotropic flows, it seems much more difficult and complicated to study the global well-posedness of solutions to full compressible Navier-Stokes system (1.1) with vacuum, where some additional difficulties arise, such as the degeneracy of both momentum and energy equations, the strong coupling between the velocity and temperature, et al. For specific pressure laws excluding the perfect gas equation of state, the question of existence of so-called "variational" solutions in dimension d ≥ 2 has been recently addressed in [8,9], where the temperature equation is satisfied only as an inequality which justifies the notion of variational solutions. Moreover, for a very particular form of the viscosity coefficients depending on the density, Bresch-Desjardins [5] obtained global stability of weak solutions. For the global well-posedness of classical solutions to the full compressible Navier-Stokes system (1.1), it is shown in Xin [35] that there is no solution in C 1 [0, ∞), H s R d for large s to the Cauchy problem for the full compressible Navier-Stokes system without heat conduction provided that the initial density has compact support. See also the recent generalizations to the case for non-compact but rapidly decreasing at far field initial densities ( [27]). Recently, Huang-Li [14] established the global existence and uniqueness for the classical solutions to the 3D Cauchy problem with interior vacuum provided the initial energy is small enough. Later, Wen-Zhu [32] obtained the global existence and uniqueness of the classical solutions for vanishing far-field density under the assumption that the initial mass is sufficiently small or both viscosity and heat-conductivity coefficients are large enough. It should be mentioned here that the results of [14,32] hold only for the Cauchy problem. However, the global existence of classical solutions or even weak ones with vacuum to multi-dimensional full compressible Navier-Stokes system (1.1) in general bounded domains remains completely open except for spherically or cylindrically symmetric initial data (see [33,34]). In fact, one of the aims of this paper is to study the global well-posedness of classical solutions to full compressible Navier-Stokes system (1.1) in general bounded domains. Before stating the main results, we explain the notations and conventions used throughout this paper. We denote f dx Ω f dx, and f 1 |Ω| Ω f dx, which is the average of a function f over Ω. For 1 ≤ p ≤ ∞ and integer k ≥ 0, we adopt the simplified notations for Sobolev spaces as follows: L p = L p (Ω), W k,p = W k,p (Ω), H k = W k,2 (Ω), H i ω = f ∈ H i f · n = 0, curlf × n = 0 on ∂Ω (i = 1, 2). Without loss of generality, we assume that ρ 0 = 1 |Ω| ρ 0 dx = 1. (1.7) We then define the initial energy C 0 as follows: C 0 1 2 ρ 0 |u 0 | 2 dx + R (1 + ρ 0 log ρ 0 − ρ 0 ) dx + R γ − 1 ρ 0 (θ 0 − log θ 0 − 1) dx. (1.8) The first main result in this paper can be stated as follows: Theorem 1.1. Let Ω ⊂ R 3 be a simply connected bounded smooth domain, whose boundary ∂Ω has a finite number of 2-dimensional connected components. For given numbers M > 0 (not necessarily small), q ∈ (3, 6),ρ > 2, andθ > 1, suppose that the initial data (ρ 0 , u 0 , θ 0 ) satisfies ρ 0 ∈ W 2,q , u 0 ∈ H 2 ω , θ 0 ∈ f ∈ H 1 ∇f · n = 0 on ∂Ω , (1.9) 0 ≤ inf ρ 0 ≤ sup ρ 0 <ρ, 0 ≤ inf θ 0 ≤ sup θ 0 ≤θ, ∇u 0 L 2 ≤ M, (1.10) and the compatibility condition − µ∆u 0 − (µ + λ)∇divu 0 + R∇(ρ 0 θ 0 ) = √ ρ 0 g, (1.11) with g ∈ L 2 . Then there exists a positive constant ε depending only on µ, λ, κ, R, γ,ρ, θ, Ω, and M such that if C 0 ≤ ε, (1.12) the problem (1.4)-(1.6) admits a unique global classical solution (ρ, u, θ) in Ω × (0, ∞) satisfying 0 ≤ ρ(x, t) ≤ 2ρ, θ(x, t) ≥ 0, x ∈ Ω, t ≥ 0, (1.13) and for any 0 < τ < T < ∞ andp ∈ [1,6). Moreover, for any p ∈ [1, ∞) and r ∈ [1,6], there exist positive constants C, α 0 , and θ ∞ depending only on µ, λ, κ, R, γ,ρ,θ, Ω, p, r, and M such that for any t ≥ 1,               ρ − 1 L p + u 2 W 1,r + θ − θ ∞ 2 H 2 ≤ Ce −α 0 t . (1.15) The next result of this paper concerns weak solutions whose definition is as follows. Definition 1.1. We say that (ρ, u, E = 1 2 |u| 2 + R γ−1 θ) is a weak solution to Cauchy problem (1.1) (1.5) (1.6) provided that ρ ∈ L ∞ loc ([0, ∞); L ∞ (Ω)), u, θ ∈ L 2 loc ([0, ∞); H 1 (Ω)), and that for all test functions ψ ∈ D(Ω × (−∞, ∞)), Ω ρ 0 ψ(·, 0)dx + ∞ 0 Ω (ρψ t + ρu · ∇ψ) dxdt = 0, (1.16) Ω ρ 0 u j 0 ψ(·, 0)dx + ∞ 0 Ω ρu j ψ t + ρu j u · ∇ψ + P (ρ, θ)ψ x j dxdt − ∞ 0 Ω µ∇u j · ∇ψ + (µ + λ)(divu)ψ x j dxdt = 0, j = 1, 2, 3, (1.17) Ω 1 2 ρ 0 |u 0 | 2 + R γ − 1 ρ 0 θ 0 ψ(·, 0)dx + ∞ 0 Ω (ρEψ t + (ρE + P )u · ∇ψ) dxdt − ∞ 0 Ω κ∇θ + 1 2 µ∇(|u| 2 ) + µu · ∇u + λdivuu · ∇ψdxdt = 0. ( ρ ∈ C([0, ∞); L p ), (ρu, ρ|u| 2 , ρθ) ∈ C([0, ∞); H −1 ), (1.20) u ∈ L ∞ (0, ∞; H 1 ) ∩ C((0, ∞); L 2 ), θ ∈ C((0, ∞); W 1,p ), (1.21) u(·, t), curlu(·, t), ((2µ + λ)divu − (P − P ))(·, t), ∇θ(·, t) ∈ H 1 , t > 0, (1.22) ρ ∈ [0, 2ρ] a.e., θ ≥ 0 a.e., (1.23) and the exponential decay property (1.15) with p ∈ [1, ∞),p ∈ [1,6), and r ∈ [1,6]. In addition, there exists some positive constant C depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that, for σ(t) min{1, t}, the following estimates hold sup t∈(0,∞) u 2 H 1 + ∞ 0 |(ρu) t + div(ρu ⊗ u)| 2 dxdt ≤ C, (1.24) sup t∈(0,∞) (ρ − 1) 2 + ρ|u| 2 + ρ(Rθ − P ) 2 dx + ∞ 0 ∇u 2 L 2 + ∇θ 2 L 2 dt ≤ CC 1/4 0 , (1.25) sup t∈(0,∞) σ ∇u 2 L 6 + σ 2 θ 2 H 2 + ∞ 0 σ u t 2 L 2 + σ 2 ∇u 2 L 2 + σ 2 θ t 2 H 1 dt ≤ C. (1.26) Moreover, (ρ, u, θ) satisfies (1.4) 3 in the weak form, that is, for any test function ψ ∈ D(Ω × (−∞, ∞)), R γ − 1 ρ 0 θ 0 ψ(·, 0)dx + R γ − 1 ∞ 0 ρθ (ψ t + u · ∇ψ) dxdt = κ ∞ 0 ∇θ · ∇ψdxdt + R ∞ 0 ρθdivuψdxdt − ∞ 0 λ(divu) 2 + 2µ|D(u)| 2 ψdxdt. (1.27) Next, as a direct application of (1.15), the following Corollary 1.3, whose proof is similar to that of [6,Theorem 1.2], shows that the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum appears (even at a point) initially. Corollary 1.3. In addition to the conditions of Theorem 1.1, assume further that there exists some point x 0 ∈ Ω such that ρ 0 (x 0 ) = 0. Then for anyp > 3, there exists some positive constant C depending on µ, λ, κ, R, γ,ρ,θ, Ω,p and M such that the unique global classical solution (ρ, u, θ) to the problem (1.4)-(1.6) obtained in Theorem 1.1 satisfies that for any t ≥ 1, ∇ρ(·, t) Lp ≥ Ce Ct . A few remarks are in order: Remark 1.1. It is easy to deduce from (1.14) and the Sobolev imbedding theorem that for any 0 < τ < T < ∞, (ρ, ∇ρ, u) ∈ C(Ω × [0, T ]), (θ, ∇θ, ∇ 2 θ) ∈ C(Ω × (0, T ]),(1.28) and [20], which is much weaker than those in [7,14,32] where not only (1.11) but also the following compatibility condition on the temperature (∇u, ∇ 2 u) ∈ C([τ, T ]; L 2 ) ∩ L ∞ (τ, T ; W 1,q ) ֒→ C(Ω × [τ, T ]),(1.κ∆θ 0 + µ 2 |∇u 0 + (∇u 0 ) tr | 2 + λ(divu 0 ) 2 = √ ρ 0 g 1 , g 1 ∈ L 2 (1.31) is needed. This reveals that the compatibility condition on the temperature (1.31) is not necessary for establishing the classical solutions with vacuum to the full Navier-Stokes equations, which is just the same as the barotropic case [16,21]. Remark 1.4. It should be mentioned here that the boundary condition for velocity u: u · n = 0, curlu × n = 0 on ∂Ω, (1.32) is a special case of the following general Navier-type slip condition (see Navier [25]) u · n = 0, (2D(u) n + ϑu) tan = 0 on ∂Ω, which as indicated by [6,Remark 1.1], is in fact a particular case of the following slip boundary one: u · n = 0, curlu × n = −Au on ∂Ω, (1.33) where ϑ is a scalar friction function, the symbol v tan represents the projection of tangent plane of the vector v on ∂Ω, and A = A(x) is a given 3 × 3 symmetric matrix defined on ∂Ω. Indeed, our result still holds for more general slip boundary condition (1.33) with A being semi-positive and regular enough. The proof is similar to [6] and omitted here. We now comment on the analysis of this paper. We mainly take the strategy that we first extend the standard local classical solutions with strictly positive initial density (see Lemma 2.1) globally in time just under the condition that the initial energy is suitably small (see Proposition 5.1), then let the lower bound of the initial density go to zero. To do so, one needs to establish global a priori estimates, which are independent of the lower bound of the density, on smooth solutions to (1.4)-(1.6) in suitable higher norms. As indicated in [13,14], the key issue is to obtain the time-independent upper bound of the density. The first main difficulty arises in deriving the basic energy estimate, which indeed is obtained directly for the Cauchy problem [14]. However, in our case, the basic energy equality reads: E ′ (t) + λ(divu) 2 + 2µ|D(u)| 2 θ + κ |∇θ| 2 θ 2 dx = −µ |curlu| 2 + 2(divu) 2 − 2|D(u)| 2 dx,(1.34) where the basic energy E(t) is defined by E(t) 1 2 ρ|u| 2 + R(1 + ρ log ρ − ρ) + R γ − 1 ρ(θ − log θ − 1) dx. (1.35) Note that the right-hand term in (1.34) is sign-undetermined due to the slip boundary condition (1.32), thus it seems difficult to obtain directly the usual standard energy estimate E(t) ≤ CC 0 , (1.36) where the smallness (with the same order of initial energy) of basic energy plays a key role in the whole analysis of the global existence of classical solutions with vacuum not only for Cauchy problem/IBVP of barotropic flows [6,16,21] but also for Cauchy problem of full compressible Navier-Stokes equations [14]. To overcome this difficulty, we first assume that A 2 (T ) (see (3.3)) a priori satisfies A 2 (T ) ≤ 2C 1/4 0 (see (3.9)) and obtain the following "weaker" basic energy estimate (see also (3.15)): E(t) ≤ CC 1/4 0 , (1.37) which compared with (1.36), however, is not enough and indeed will bring us some essential difficulties to obtain all the a priori estimates (see Proposition 3.1). Then, the first observation is that the average of the pressure P is uniformly bounded with positive lower and upper bounds (see (3.11)) by both "weaker" basic energy estimate (1.37) and Jensen's inequality (see Lemma 2.2). Combining this with the fact that the quantity P plays a similar role as θ (see (3.22)) implies that we can replaceθ byP whose positive lower and upper bounds play an important role in further analysis. Next, the second difficulty lies in the estimation on the energy-like term A 2 (T ) (see (3.2)) which includes the key bounds on the L 2 (Ω × (0, T ))-norm of the spatial derivatives of both the velocity and the temperature. To proceed, first, we adopt the ideas due to [6,12] to estimateu andθ (see Lemma 3.3), whereḟ f t + u · ∇f denotes the material derivative of f. Indeed, in this process, one needs to deal with the boundary integrals in (3.43), for example ∂Ω G(u · ∇)u · ∇n · udS, which by the classical trace theorem seems to be bounded by some good terms and the L p -norm of ∇ 2 u, which is unavailable in this step. To overcome this difficulty, we adopt some idea due to [6], that is, u = u ⊥ × n with u ⊥ −u × n, which combined with the following fact: div(∇u i × u ⊥ ) = −∇u i · ∇ × u ⊥ , (1.38) yields that the above boundary integral can be indeed bounded by some suitable norms on both ∇u and ∇G (see (3.45) for details). Next, after observing that the evolution of P can be derived from the temperature equation (see (3.26)), combining a careful analysis on the system (1.4) with the L 1 (0, min{1, T }; L ∞ )-norm of the temperature gives the desired basic energy estimate for small time (see Lemma 3.5). Moreover, we observe that Rθ − P can be bounded by the combination of the initial energy with the spatial L 2 -norm of the spatial derivatives of the temperature (see (3.74)), which together with a suitable combination of kinetic energy and thermal energy (see (3.91)) yields A 2 (T ) ≤ CC 7/24 0 (see (3.99)) which implies A 2 (T ) ≤ C 1/4 0 (see (3.88)), provided the initial energy is suitably small. Next, the third difficulty is to obtain the key time-independent upper bound of the density. It should be noted that the methods used in Cauchy problem [14], which heavily relies on the nontrivial far field states, can not be applied to the IBVP directly. Here, by inserting the key quantity P , we rewrite the continuity equation in the following way (2µ + λ)D t ρ = −P ρ(ρ − 1) − ρ 2 (Rθ − P ) − ρG, where D t is the material derivative and G is the effective viscous flux, defined by D t f ḟ f t + u · ∇f, G (2µ + λ)divu − (P − P ), (1.39) respectively. With the aid of the uniform bound of P (see (3.11)), the upper bound of the density follows directly by applying the Grönwall-type inequality (see Lemma 2.8) and using the estimates on Rθ − P and G. Finally, with the lower-order estimates including the time-independent upper bound of the density at hand, we can obtain the higher-order estimates just under the compatibility condition on the velocity (1.11). Note that all the a priori estimates are independent of the lower bound of the density, thus after a standard approximate procedure, we can obtain the global existence of classical solutions with vacuum. Moreover, we can as well establish the global weak solutions almost the same way as we established the classical one with a new modified approximate initial data. The rest of the paper is organized as follows: In Section 2, we collect some basic facts and inequalities which will be used later. Section 3 is devoted to deriving the lower-order a priori estimates on classical solutions which are needed to extend the local solutions to all time. The higher-order estimates are established in Section 4. Finally, with all a priori estimates at hand, the main results, Theorems 1.1 and 1.2, are proved in Section 5. Preliminaries First, the following local existence theory with strictly positive initial density can be shown by the standard contraction mapping arguments as in [7,23,30]. Lemma 2.1. Let Ω be as in Theorem 1.1. Assume that (ρ 0 , u 0 , θ 0 ) satisfies    (ρ 0 , u 0 , θ 0 ) ∈ H 3 , inf x∈Ω ρ 0 (x) > 0, inf x∈Ω θ 0 (x) > 0, u 0 · n = 0, curlu 0 × n = 0, ∇θ 0 · n = 0 on ∂Ω. (2.1) Then there exist a small time 0 < T 0 < 1 and a unique classical solution (ρ, u, θ) to the problem (1.4)-(1.6) on Ω × (0, T 0 ] satisfying inf (x,t)∈Ω×(0,T 0 ] ρ(x, t) ≥ 1 2 inf x∈Ω ρ 0 (x),(2. 2) and (ρ, u, θ) ∈ C([0, T 0 ]; H 3 ), ρ t ∈ C([0, T 0 ]; H 2 ), (u t , θ t ) ∈ C([0, T 0 ]; H 1 ), (u, θ) ∈ L 2 (0, T 0 ; H 4(tu t , tθ t ) ∈ L 2 (0, T 0 ; H 3 ), (tu tt , tθ tt ) ∈ L 2 (0, T 0 ; H 1 ), (t 2 u tt , t 2 θ tt ) ∈ L 2 (0, T 0 ; H 2 ), (t 2 u ttt , t 2 θ ttt ) ∈ L 2 (0, T 0 ; L 2 ). (2.4) Moreover, for any (x, t) ∈ Ω × [0, T 0 ], the following estimate holds: θ(x, t) ≥ inf x∈Ω θ 0 (x) exp −(γ − 1) T 0 0 divu L ∞ dt . (2.5) Next, we state the classical Jensen's inequality (see [28,Theorem 3.3]), which guarantees the key uniform upper and lower bounds of P . Lemma 2.2. Let µ be a positive measure on a σ-algebra M in a set Ω, so that µ(Ω) = 1. If f is a real function in L 1 (µ), a < f (x) < b for all x ∈ Ω, and Φ is convex on (a, b), then Φ Ω f dµ ≤ Ω (Φ • f )dµ. (2.6) Next, the following well-known Gagliardo-Nirenberg-Sobolev-type inequality (see [26]) will be used later frequently. Lemma 2.3. Assume that Ω ⊂ R 3 is a bounded Lipschitz domain. For r ∈ [2, 6], p ∈ (1, ∞), and q ∈ (3, ∞), there exist positive constants C, C ′ , and C ′′ which may depend on r, p, q, and Ω such that for f ∈ H 1 , g ∈ L p ∩ W 1,q , and ϕ, ψ ∈ H 2 , f L r ≤ C f (6−r)/2r L 2 ∇f (3r−6)/2r L 2 + C ′ f L 2 , (2.7) g C(Ω) ≤ C g p(q−3)/(3q+p(q−3)) L p ∇g 3q/(3q+p(q−3)) L q + C ′′ g L 2 , (2.8) ϕψ H 2 ≤ C ϕ H 2 ψ H 2 . (2.9) Moreover, if f · n| ∂Ω = 0 or f = 0, one has C ′ = 0. Similarly, if g · n| ∂Ω = 0 or g = 0, it holds C ′′ = 0. Then, the following div-curl type inequality (see [31,Theorem 3.2]) is used to get the estimates on the spatial derivatives of velocity. Lemma 2.4. Assume that Ω ⊂ R 3 is a simply connected bounded domain with C 1,1 boundary ∂Ω. Then, for v ∈ W 1,q with q ∈ (1, ∞) and v · n| ∂Ω = 0, there exists a positive constant C = C(q, Ω) such that v W 1,q ≤ C ( divv L q + curlv L q ) . (2.10) Now, we deduce from (1.4) 2 that G defined in (1.39) satisfies the following elliptic equation: ∆G = div(ρu), x ∈ Ω, ∇G · n = ρu · n, x ∈ ∂Ω. (2.11) The standard L p -estimate for (2.11) together with the div-curl type inequalities (see [3,31]) yields the following essential estimates (see also [6,Lemma 2.9]). Lemma 2.5. Let Ω ⊂ R 3 be the same as in Theorem 1.1 and (ρ, u, θ) a smooth solution of (1.4)-(1.6). Then there exists a generic positive constant C depending only on p, µ, λ, and Ω such that, for any p ∈ [2,6], ∇u L p ≤ C ( divu L p + curlu L p ) , (2.12) ∇G L p ≤ C ρu L p , (2.13) ∇curlu L p ≤ C( ρu L p + ∇u L 2 ), (2.14) G L p ≤ C ρu (3p−6)/(2p) L 2 ∇u L 2 + P − P L 2 (6−p)/(2p) , (2.15) curlu L p ≤ C ρu (3p−6)/(2p) L 2 ∇u (6−p)/(2p) L 2 + C ∇u L 2 . (2.16) Moreover, it holds that G L p + curlu L p ≤ C( ρu L 2 + ∇u L 2 ), (2.17) ∇u L p ≤C ρu (3p−6)/(2p) L 2 ∇u L 2 + P − P L 2 (6−p)/(2p) + C( ∇u L 2 + P − P L p ). (2.18) Next, we state the following estimates onu with u · n| ∂Ω = 0, whose proof can be found in [6, Lemma 2.10]. Lemma 2.6. Let Ω ⊂ R 3 with C 1,1 boundary. Assume that u is smooth enough and u · n| ∂Ω = 0, then there exists a generic positive constant C depending only on Ω such that u L 6 ≤ C( ∇u L 2 + ∇u 2 L 2 ), (2.19) ∇u L 2 ≤ C( divu L 2 + curlu L 2 + ∇u 2 L 4 ). (2.20) Furthermore, by the classical elliptic theory owing to Agmon-Douglis-Nirenberg [1], one has the following estimates for smooth solution to the Lamé's system: −µ∆u − (λ + µ)∇divu = −ρu − ∇P, x ∈ Ω, u · n = 0, curlu × n = 0, x ∈ ∂Ω,(2.21) where Ω ⊂ R 3 is a bounded smooth domain, and µ, λ satisfy the condition (1.2). Lemma 2.7. Let u be a smooth solution of the Lamé's system (2.21). Then for p ∈ [2,6] and k ≥ 2, there exists a positive constant C depending only on λ, µ, p, k, and Ω such that u W k,p ≤ C( ρu W k−2,p + ∇P W k−2,p + ∇u L 2 ). (2.22) Next, the following Grönwall-type inequality will be used to get the uniform (in time) upper bound of the density ρ, whose proof can be found in [14,Lemma 2.5]. Lemma 2.8. Let the function y ∈ W 1,1 (0, T ) satisfy y ′ (t) + α(t)y(t) ≤ g(t) on [0, T ], y(0) = y 0 , where 0 < α 0 ≤ α(t) for any t ∈ [0, T ] and g ∈ L p (0, T 1 ) ∩ L q (T 1 , T ) for some p, q ≥ 1, T 1 ∈ [0, T ]. Then it has sup 0≤t≤T y(t) ≤ |y 0 | + (1 + α −1 0 ) g L p (0,T 1 ) + g L q (T 1 ,T ) . (2.23) Next, to derive the exponential decay property of the solutions, we consider the following auxiliary problem divv = f, x ∈ Ω, v = 0, x ∈ ∂Ω. 1)The operator B : {f ∈ L p | f = 0} → W 1,p 0 is a bounded linear one, that is, B[f ] W 1,p 0 ≤ C(p) f L p , for any p ∈ (1, ∞). 2) The function v = B[f ] solves the problem (2.24). 3) If, moreover, for f = divg with a certain g ∈ L r , g · n| ∂Ω = 0, then for any r ∈ (1, ∞), B[f ] L r ≤ C(r) g L r . Finally, in order to estimate ∇u L ∞ for the further higher order estimates, we need the following Beale-Kato-Majda-type inequality, which was first proved in [4,17] when divu ≡ 0, whose detailed proof in the case of slip boundary condition can be found in [6, Lemma 2.7] (see also [13,15]). Lemma 2.10. Let Ω ⊂ R 3 be a bounded smooth domain. For 3 < q < ∞, assume that u ∈ {f ∈ W 2,q |f · n = 0, curlf × n = 0 on ∂Ω}, then there is a positive constant C = C(q, Ω) such that ∇u L ∞ ≤ C ( divu L ∞ + curlu L ∞ ) ln(e + ∇ 2 u L q ) + C ∇u L 2 + C. A priori estimates (I): lower-order estimates In this section, we will establish a priori bounds for the local-in-time smooth solution to problem (1.4)-(1.6) obtained in Lemma 2.1. Let (ρ, u, θ) be a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] for some fixed time T > 0, with initial data (ρ 0 , u 0 , θ 0 ) satisfying (2.1). For σ(t) min{1, t}, we define A i (T )(i = 1, 2, 3) as follows: A 1 (T ) sup t∈[0,T ] ∇u 2 L 2 + T 0 ρ|u| 2 dxdt, (3.1) A 2 (T ) 1 2(γ − 1) sup t∈[0,T ] ρ(Rθ − P ) 2 dx + T 0 ∇u 2 L 2 + ∇θ 2 L 2 dt, (3.2) A 3 (T ) sup t∈(0,T ] σ ∇u 2 L 2 + σ 2 ρ|u| 2 dx + σ 2 ∇θ 2 L 2 + T 0 σρ|u| 2 + σ 2 |∇u| 2 + σ 2 ρ|θ| 2 dxdt. (3.3) We have the following key a priori estimates on (ρ, u, θ). Proposition 3.1. For given numbers M > 0,ρ > 2, andθ > 1, assume further that (ρ 0 , u 0 , θ 0 ) satisfies 0 < inf ρ 0 ≤ sup ρ 0 <ρ, 0 < inf θ 0 ≤ sup θ 0 ≤θ, ∇u 0 L 2 ≤ M. (3.4) Then there exist positive constants K, C * , α, θ ∞ , and ε 0 all depending on µ, λ, κ, R, γ, ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying 0 < ρ ≤ 2ρ, A 1 (T ) ≤ 3K, A 2 (T ) ≤ 2C 1/4 0 , A 3 (T ) ≤ 2C 1/6 0 , (3.5) the following estimates hold: 0 < ρ ≤ 3ρ/2, A 1 (T ) ≤ 2K, A 2 (T ) ≤ C 1/4 0 , A 3 (T ) ≤ C 1/6 0 , (3.6) and for any t ≥ 1, ρ − 1 L 2 + u 2 W 1,6 + θ − θ ∞ 2 H 2 ≤ C * e −αt ,(3.7) provided C 0 ≤ ε 0 . (3.8) Proof. Proposition 3.1 is a straight consequence of the following Lemmas 3.2, 3.4, 3.6, 3.7, and 3.9 with ε 0 as in (3.124). In this section, we always assume that C 0 ≤ 1 and let C denote some generic positive constant depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M, and we write C(α) to emphasize that C may depend on α. To begin with, we have the following uniform estimate on P , which plays an important role in the whole analysis. 0 < ρ ≤ 2ρ, A 2 (T ) ≤ 2C 1/4 0 , (3.9) the following estimates hold: sup 0≤t≤T ρ|u| 2 + (ρ − 1) 2 dx ≤ CC 1/4 0 ,(3. 10) and 0 < π 1 ≤ P (t) ≤ π 2 , for any t ∈ [0, T ], (3.11) where π 1 and π 2 are positive constants depending only on µ, γ, R, and Ω. Proof. First, it follows from (3.4) and (2.5) that, for all ( x, t) ∈ Ω × (0, T ), θ(x, t) > 0. (3.12) Note that ∆u = ∇divu − ∇ × curlu, one can rewrite (1.4) 2 as ρ(u t + u · ∇u) = (2µ + λ)∇divu − µ∇ × curlu − ∇P. (3.13) Adding (3.13) multiplied by u to (1.4) 3 multiplied by 1 − θ −1 and integrating the resulting equality over Ω by parts, we obtain after using (1.4) 1 , (1.2), (3.12), and the boundary conditions (1.6) that E ′ (t) = − λ(divu) 2 + 2µ|D(u)| 2 θ + κ |∇θ| 2 θ 2 dx − µ |curlu| 2 + 2(divu) 2 − 2|D(u)| 2 dx ≤ 2µ |∇u| 2 dx, (3.14) where E(t) is the basic energy defined by (1.35). Then, integrating (3.14) with respect to t over (0, T ) and using (3.9), one has sup 0≤t≤T E(t) ≤ C 0 + 2µ T 0 |∇u| 2 dxdt ≤ CC 1/4 0 , (3.15) which together with (ρ − 1) 2 ≥ 1 + ρ log ρ − ρ = (ρ − 1) 2 1 0 1 − α α(ρ − 1) + 1 dα ≥ (ρ − 1) 2 2(2ρ + 1) (3.16) gives (3.10). Next, it is easy to deduce from (1.4) 1 and (1.7) that for any t ∈ [0, T ], ρ(t) = ρ 0 = 1. (3.17) Denote dµ |Ω| −1 ρdx. Then dµ is a positive measure satisfying µ(Ω) = 1 due to (3.9) and (3.17). Moreover, observe that y − log y − 1 is a convex function in (0, ∞), it thus follows directly from Jensen's inequality (2.6) that for any t ∈ [0, T ], (3.15). This in particular gives (3.11) and finishes the proof of Lemma 3.1. ρθ(t) − log ρθ(t) − 1 ≤ (θ − log θ − 1) ρdx |Ω| ≤ C due to Remark 3.1. It should be pointed out that the following term in (3.14) − µ |curlu| 2 + 2(divu) 2 − 2|D(u)| 2 dx (3.18) is a sign-undetermined term due to the slip boundary condition (1.32), which is in sharp contrast to the Cauchy problem [14] where the term (3.18) vanishes after integration by parts. Thus, in this case, we can not bound the basic energy only by the initial energy. However, this term obviously can be bounded by C |∇u| 2 dx, which implies a "weaker" basic energy estimate (3.15). The next lemma provides an estimate on the term A 1 (T ). 0 < ρ ≤ 2ρ, A 2 (T ) ≤ 2C 1/4 0 , A 1 (T ) ≤ 3K, (3.19) the following estimate holds: A 1 (T ) ≤ 2K, (3.20) provided C 0 ≤ ε 1 . Proof. First, integrating (3.13) multiplied by 2u t over Ω by parts gives d dt µ|curlu| 2 + (2µ + λ)(divu) 2 dx + ρ|u| 2 dx ≤ 2 P divu t dx + ρ|u · ∇u| 2 dx = 2 d dt (P − P )divudx − 2 (P − P ) t divudx + ρ|u · ∇u| 2 dx = 2 d dt (P − P )divudx − 1 2µ + λ d dt (P − P ) 2 dx − 2 2µ + λ (P − P ) t Gdx + ρ|u · ∇u| 2 dx, (3.21) where in the last equality we have used (1.39). Next, straight calculations show that for any p ∈ [2,6], Rθ − P L p ≤ R θ − θ L p + C|Rθ − P | ≤ C(ρ) ∇θ L 2 ,(3.22) where one has used (2.7) and the following fact: |Rθ − P | = R |Ω| (1 − ρ)θdx = R |Ω| (1 − ρ)(θ − θ)dx ≤C ρ − 1 L 2 θ − θ L 2 ≤C(ρ)C 1/8 0 ∇θ L 2 dueP − P L p = ρ(Rθ − P ) + (ρ − 1)P L p ≤ ρ(Rθ − P ) (6−p)/(2p) L 2 ρ(Rθ − P ) 3(p−2)/(2p) L 6 + π 2 ρ − 1 L p ≤ C(ρ)C (6−p)/(16p) 0 ∇θ 3(p−2)/(2p) L 2 + C(ρ)C 1/(4p) 0 , (3.23) which together with (2.18) and (3.19) yields ∇u L 6 ≤ C(ρ) ρ 1/2u L 2 + ∇u L 2 + ∇θ L 2 + C 1/24 0 . (3.24) Note that (1.4) 3 implies P t = − div(P u) − (γ − 1)P divu + (γ − 1)κ∆θ + (γ − 1) λ(divu) 2 + 2µ|D(u)| 2 ,(3.25) which along with (1.6) gives P t = − (γ − 1)P divu + (γ − 1) λ(divu) 2 + 2µ|D(u)| 2 . (3.26) We thus obtain after using integration by parts, (3.11), (2.7), (2.13), and (3.23)-(3.25) that P t Gdx ≤ C P (|G||∇u| + |u||∇G|)dx + C |∇θ||∇G| + |∇u| 2 |G| dx ≤ C |P − P |(|G||∇u| + |u||∇G|)dx + CP (|G||∇u| + |u||∇G|)dx + C ∇G L 2 ∇θ L 2 + C G L 6 ∇u 3/2 L 2 ∇u 1/2 L 6 ≤ C(ρ)( ∇θ 1/2 L 2 + 1) ∇G L 2 ∇u L 2 + C ∇G L 2 ∇θ L 2 + C(ρ) ∇G L 2 ∇u 3/2 L 2 ρ 1/2u L 2 + ∇u L 2 + ∇θ L 2 + 1 1/2 ≤ δ ∇G 2 L 2 + δ ρ 1/2u 2 L 2 + C(δ,ρ) ∇u 2 L 2 + ∇θ 2 L 2 + ∇u 6 L 2 ≤ C(ρ)δ ρ 1/2u 2 L 2 + C(δ,ρ) ∇u 2 L 2 + ∇θ 2 L 2 + ∇u 6 L 2 ,(3.27) and P t Gdx ≤C(ρ)( ∇u L 2 + ∇u 2 L 2 ) G L 2 ≤δ ∇G 2 L 2 + C(δ,ρ)( ∇u 2 L 2 + ∇u 4 L 2 ) ≤C(ρ)δ ρ 1/2u 2 L 2 + C(δ,ρ)( ∇u 2 L 2 + ∇u 4 L 2 ), (3.28) where one has used |P t | ≤ C P − P L 2 ∇u L 2 + C ∇u 2 L 2 ≤ C(ρ)(C 1/8 0 ∇u L 2 + ∇u 2 L 2 ) (3.29) owing to (3.26) and (3.23). Then, it follows from (2.7), (3.19), and (3.24) that ρ|u · ∇u| 2 dx ≤ C(ρ) u 2 L 6 ∇u L 2 ∇u L 6 ≤ δ ρ 1/2u 2 L 2 + C(δ,ρ) ∇u 2 L 2 + ∇θ 2 L 2 + ∇u 6 L 2 .d dt µ|curlu| 2 + (2µ + λ)(divu) 2 dx + 1 2µ + λ d dt P − P 2 L 2 + 1 2 ρ|u| 2 dx ≤ 2 d dt (P − P )divudx + C(ρ) ∇u 2 L 2 + ∇θ 2 L 2 + ∇u 6 L 2 . (3.31) Note that it holds P − P 2 L 2 (0) = R 2 (ρ 0 θ 0 − ρ 0 θ 0 ) 2 dx ≤ C(ρ) ρ 0 (θ 0 − ρ 0 θ 0 ) 2 dx + C (ρ 0 − 1) 2 dx ≤ C(ρ,θ)C 0 ,(3.32) where we have used (3.11), (3.16), and the following fact: ρ 0 (θ 0 − ρ 0 θ 0 ) 2 dx ≤ C ρ 0 (θ 0 − 1) 2 dx + C|1 − ρ 0 θ 0 | 2 ≤ C ρ 0 (θ 0 − 1) 2 dx + C ρ 0 (1 − θ 0 )dx 2 ≤ C(ρ,θ) ρ 0 (θ 0 − log θ 0 − 1)dx ≤ C(ρ,θ)C 0 (3.33) due to (1.7) and θ − log θ − 1 = (θ − 1) 2 1 0 α α(θ − 1) + 1 dα ≥ 1 2( θ(·, t) L ∞ + 1) (θ − 1) 2 .(3.∇u 2 L 2 + T 0 ρ|u| 2 dxdt ≤ CM 2 + C(ρ,θ)C 1/4 0 + C(ρ)C 1/4 0 sup 0≤t≤T ∇u 4 L 2 + C(ρ)C 1/8 0 sup 0≤t≤T ∇u L 2 ≤ CM 2 + C(ρ,θ)C 1/12 0 + C(ρ)C 1/4 0 sup 0≤t≤T ∇u 4 L 2 ≤ K + 9C(ρ)C 1/4 0 K 2 ≤ 2K, with K CM 2 + C(ρ,θ) + 1, provided C 0 ≤ ε 1 min 1, (9C(ρ)K) −4 . The proof of Lemma 3.2 is completed. Next, to estimate A 3 (T ), we adopt the approach due to Hoff [12] (see also Huang-Li [14]) to establish the following elementary estimates onu andθ, where the boundary terms are handled by the ideas due to [6]. The estimate of A 3 (T ) will be postponed to Lemma 3.4. Then there exist positive constants C, C 1 , and C 2 depending only on µ, λ, k, R, γ,ρ,θ, Ω, and M such that, for any η ∈ (0, 1] and m ≥ 0, the following estimates hold: (σB 1 ) ′ (t) + 1 2 σ ρ|u| 2 dx ≤ CC 1/4 0 σ ′ + C ∇u 2 L 2 + ∇θ 2 L 2 , (3.35) σ m ρ 1/2u 2 L 2 t + C 1 σ m ∇u 2 L 2 ≤ −2 ∂Ω σ m (u · ∇n · u)GdS t + C(σ m−1 σ ′ + σ m ) ρ 1/2u 2 L 2 + C 2 σ m ρ 1/2θ 2 L 2 + C ∇u 2 L 2 + Cσ m ∇u 4 L 4 + Cσ m θ∇u 2 L 2 ,(3. 36) and (σ m B 2 ) ′ (t) + σ m ρ|θ| 2 dx ≤ Cησ m ∇u 2 L 2 + C ∇θ 2 L 2 + Cσ m ∇u 4 L 4 + C(η)σ m θ∇u 2 L 2 , (3.37) where B 1 (t) µ curlu 2 L 2 + (2µ + λ) divu 2 L 2 + 1 2µ + λ P − P 2 L 2 − 2 divu(P − P )dx,(3. 38) and B 2 (t) γ − 1 R κ ∇θ 2 L 2 − 2 (λ(divu) 2 + 2µ|D(u)| 2 )θdx .(3.σ m 2 ρ|u| 2 dx t − m 2 σ m−1 σ ′ ρ|u| 2 dx = ∂Ω σ mu · nG t dS − σ m [divuG t + u · ∇u · ∇G]dx − µ σ muj (∇ × curlu) j t + div(u(∇ × curlu) j ) dx 3 i=1 N i . (3.41) Noticing that u · ∇u · n = −u · ∇n · u on ∂Ω (3.42) due to u · n| ∂Ω = 0, one can deduce from (1.6) and (3.42) that N 1 = − ∂Ω σ m (u · ∇n · u)G t dS = − ∂Ω σ m (u · ∇n · u)GdS t + mσ m−1 σ ′ ∂Ω (u · ∇n · u)GdS + ∂Ω σ m (u · ∇n · u)GdS + ∂Ω σ m (u · ∇n ·u)GdS − ∂Ω σ m G(u · ∇)u · ∇n · udS − ∂Ω σ m Gu · ∇n · (u · ∇)udS ≤ − ∂Ω σ m (u · ∇n · u)GdS t + Cσ m−1 σ ′ ∇u 2 L 2 ∇G L 2 + δσ m u 2 H 1 + C(δ)σ m ∇u 2 L 2 ∇G 2 L 2 − ∂Ω σ m G(u · ∇)u · ∇n · udS − ∂Ω σ m Gu · ∇n · (u · ∇)udS,(3.43) where one has used ∂Ω (u · ∇n · u + u · ∇n ·u)GdS ≤ C u H 1 u H 1 G H 1 ≤ C u H 1 ∇u L 2 ∇G L 2 , and ∂Ω (u · ∇n · u)GdS ≤ C ∇u 2 L 2 ∇G L 2 . (3.44) Now, we will adopt the idea in [6] to deal with the last two boundary terms in (3.43). In fact, denote u ⊥ −u × n, it follows from u · n| ∂Ω = 0 that u = u ⊥ × n on ∂Ω, which along with (2.7), (1.38), and integration by parts yields − ∂Ω G(u · ∇)u · ∇n · udS = − ∂Ω Gu ⊥ × n · ∇u i ∇ i n · udS = − ∂Ω Gn · (∇u i × u ⊥ )∇ i n · udS = − div(G(∇u i × u ⊥ )∇ i n · u)dx = − ∇(∇ i n · uG) · (∇u i × u ⊥ )dx − div(∇u i × u ⊥ )∇ i n · uGdx = − ∇(∇ i n · uG) · (∇u i × u ⊥ )dx + G∇u i · ∇ × u ⊥ ∇ i n · udx ≤ C |∇G||∇u||u| 2 dx + C |G|(|∇u| 2 |u| + |∇u||u| 2 )dx ≤ C ∇G L 6 ∇u L 2 u 2 L 6 + C G L 3 ∇u 2 L 4 u L 6 + C G L 6 ∇u L 2 u 2 L 6 ≤ δ ∇G 2 L 6 + C(δ) ∇u 6 L 2 + C ∇u 4 L 4 + C ∇G 2 L 2 ( ∇u 2 L 2 + 1). (3.45) Similarly, it holds that − ∂Ω Gu · ∇n · (u · ∇)udS ≤ δ ∇G 2 L 6 + C(δ) ∇u 6 L 2 + C ∇u 4 L 4 + C ∇G 2 L 2 ( ∇u 2 L 2 + 1). (3.46) Next, it follows from (1.39) that G t =(2µ + λ)divu t − (P t − P t ) =(2µ + λ)divu − (2µ + λ)div(u · ∇u) − Rρθ + div(P u) + Rρθ =(2µ + λ)divu − (2µ + λ)∇u : (∇u) tr − u · ∇G + P divu − Rρθ + Rρθ, (3.47) where one has used P t = (Rρθ) t = Rρθ − div(P u), P t = Rρθ. (3.48) Then, integration by parts combined with (3.47) gives N 2 = − σ m [divuG t + u · ∇u · ∇G]dx = − (2µ + λ) σ m (divu) 2 dx + (2µ + λ) σ m divu∇u : (∇u) tr dx + σ m divuu · ∇Gdx − σ m divuP divudx + R σ m divuρθdx − Rρθ σ m divudx − σ m u · ∇u · ∇Gdx ≤ − (2µ + λ) σ m (divu) 2 dx + Cσ m ∇u L 2 ∇u 2 L 4 + Cσ m ∇u L 2 ∇G 1/2 L 2 ∇G 1/2 L 6 u L 6 + C(ρ)σ m ∇u L 2 θ∇u L 2 + C(ρ)σ m ∇u L 2 ρ 1/2θ L 2 . (3.49) Note that curlu t = curlu − u · ∇curlu − ∇u i × ∇ i u, which together with some straight calculations yields N 3 = −µ σ m |curlu| 2 dx + µ σ m curlu · (∇u i × ∇ i u)dx + µ σ m u · ∇curlu · curludx + µ σ m u · ∇u · (∇ × curlu)dx ≤ −µ σ m |curlu| 2 dx + δσ m ( ∇u 2 L 2 + ∇curlu 2 L 6 ) + C(δ)σ m ∇u 4 L 4 + C(δ)σ m ∇u 4 L 2 ∇curlu 2 L 2 . (3.50) Finally, it is easy to deduce from Lemmas 2.5 and 2.6 that G H 1 + curlu H 1 ≤ C( ρu L 2 + ∇u L 2 ), (3.51) and that Finally, we will prove (3.37). ∇G L 6 + ∇curlu L 6 + u H 1 ≤ C( ρu L 6 + ∇u L 2 ) + u H 1 ≤ C(ρ)( ∇u L 2 + ∇u L 2 + ∇u 2 L 2 ).σ m 2 ρ 1/2u 2 L 2 t + (2µ + λ)σ m divu 2 L 2 + µσ m curlu 2 L 2 ≤ − ∂Ω σ m (u · ∇n · u)GdS t + C(ρ)δσ m ∇u 2 L 2 + C(δ,ρ)σ m ρ 1/2θ 2 L 2 + C(δ,ρ, M )(σ m−1 σ ′ + σ m ) ρ 1/2u 2 L 2 + C(δ,ρ, M ) ∇u 2 L 2 + C(δ)σ m ∇u 4 L 4 + C(δ,ρ)σ m θ∇u 2 L 2 . For m ≥ 0, multiplying (1.4) 3 by σ mθ and integrating the resulting equality over Ω yield that κσ m 2 ∇θ 2 L 2 t + Rσ m γ − 1 ρ|θ| 2 dx = −κσ m ∇θ · ∇(u · ∇θ)dx + λσ m (divu) 2θ dx + 2µσ m |D(u)| 2θ dx − Rσ m ρθdivuθdx 4 i=1 I i . (3.54) First, combining (2.7) and (3.5) gives I 1 ≤ Cσ m ∇u L 2 ∇θ 1/2 L 2 ∇ 2 θ 3/2 L 2 ≤ δσ m ρ 1/2θ 2 L 2 + σ m ∇u 4 L 4 + θ∇u 2 L 2 + C(δ,ρ, M )σ m ∇θ 2 L 2 ,(3.55) where in the last inequality we have used the following estimate: ∇ 2 θ L 2 ≤ C(ρ) ρ 1/2θ L 2 + ∇u 2 L 4 + θ∇u L 2 , (3.56) which is derived from the standard L 2 -estimate to the following elliptic problem: κ∆θ = R γ−1 ρθ + Rρθdivu − λ(divu) 2 − 2µ|D(u)| 2 , ∇θ · n| ∂Ω×(0,T ) = 0. (3.57) Next, it holds that for any η ∈ (0, 1], I 2 =λσ m (divu) 2 θ t dx + λσ m (divu) 2 u · ∇θdx =λσ m (divu) 2 θdx t − 2λσ m θdivudiv(u − u · ∇u)dx + λσ m (divu) 2 u · ∇θdx =λσ m (divu) 2 θdx t − 2λσ m θdivudivudx + 2λσ m θdivu∂ i u j ∂ j u i dx + λσ m u · ∇ θ(divu) 2 dx ≤λ σ m (divu) 2 θdx t − λmσ m−1 σ ′ (divu) 2 θdx + ησ m ∇u 2 L 2 + C(η)σ m θ∇u 2 L 2 + Cσ m ∇u 4 L 4 ,(3.58) and I 3 ≤ 2µ σ m |D(u)| 2 θdx t − 2µmσ m−1 σ ′ |D(u)| 2 θdx + ησ m ∇u 2 L 2 + C(η)σ m θ∇u 2 L 2 + Cσ m ∇u 4 L 4 . (3.59) Finally, Cauchy's inequality gives .6) on Ω × (0, T ] satisfying (3.5) with K as in Lemma 3.2, the following estimate holds: |I 4 | ≤ δσ m ρ|θ| 2 dx + C(δ,ρ)σ m θ∇u 2 L 2 .A 3 (T ) ≤ C 1/6 0 , (3.61) provided C 0 ≤ ε 2 . Proof. First, it follows from (3.38), (2.12), and (3.23) that B 1 (t) ≥ C ∇u 2 L 2 − C P − P 2 L 2 ≥ C ∇u 2 L 2 − C(ρ)C 1/4 0 , which together with (3.35) and (3.5) implies that sup t∈(0,T ] σ ∇u 2 L 2 + T 0 σρ|u| 2 dxdt ≤ C(ρ, M )C 1/4 0 . (3.62) For C 2 as in (3.36), adding (3.37) multiplied by C 2 + 1 to (3.36) and choosing η suitably small give (σ m ϕ) ′ (t) + σ m C 1 2 |∇u| 2 + ρ|θ| 2 dx ≤ −2 ∂Ω σ m (u · ∇n · u)GdS t + C(ρ, M )(σ m−1 σ ′ + σ m ) ρ 1/2u 2 L 2 + C(ρ, M )( ∇u 2 L 2 + ∇θ 2 L 2 ) + Cσ m ∇u 4 L 4 + C(ρ)σ m θ∇u 2 L 2 ,(3.63) where ϕ(t) is defined by ϕ(t) ρ 1/2u 2 L 2 + (C 2 + 1)B 2 (t). (3.64) Then it follows from (3.39) that ϕ(t) ≥ 1 2 ρ 1/2u 2 L 2 + κ(γ − 1) 2R ∇θ 2 L 2 − C(ρ, M ) ∇u 2 L 2 ,(3.65) where one has used that for any δ ∈ (0, 1], θ|∇u| 2 dx ≤ C |Rθ − P ||∇u| 2 dx + CP |∇u| 2 dx ≤ C Rθ − P L 6 ∇u 3/2 L 2 ∇u 1/2 L 6 + C ∇u 2 L 2 ≤ C(ρ) ∇θ L 2 ∇u 3/2 L 2 ρ 1/2u L 2 + ∇u L 2 + ∇θ L 2 + 1 1/2 + C ∇u 2 L 2 ≤ δ ∇θ 2 L 2 + ρ 1/2u 2 L 2 + C(δ,ρ, M ) ∇u 2θ∇u 2 L 2 ≤ C Rθ − P 2 L 6 ∇u L 2 ∇u L 6 + CP 2 ∇u 2 L 2 ≤ C(ρ, M ) ∇u 2 L 2 + ∇θ 2 L 2 ρ 1/2u 2 L 2 + ∇θ 2 L 2 + 1 .∇u 4 L 4 ≤ C ρu 3 L 2 ( ∇u L 2 + P − P L 2 ) + C( ∇u 4 L 2 + P − P 4 L 4 ) ≤ C(ρ) ρ 1/2u 3 L 2 ( ∇u L 2 + 1) + C(ρ) ∇θ 3 L 2 + C ρ − 1 4 L 4 + C ∇u 4 L 2 ≤ C(ρ, M ) ρ 1/2u 3 L 2 + ∇θ 3 L 2 + C(ρ) ρ − 1 2 L 2 + C(ρ, M ) ∇u 2 L 2 ,(3.68) which together with (3.5) yields σ ∇u 4 L 4 ≤ C(ρ, M ) ρ 1/2u 2 L 2 + ∇θ 2 L 2 + ∇u 2 L 2 + C(ρ)σ ρ − 1 2 L 2 . (3.69) Thus, taking m = 2 in (3.63), one obtains after using (3.5), (3.67), and (3.69) that σ 2 ϕ ′ (t) + σ 2 C 1 2 |∇u| 2 + ρ|θ| 2 dx ≤ −2 ∂Ω σ 2 (u · ∇n · u)GdS t + C(ρ, M )σ ρ 1/2u 2 L 2 + C(ρ, M ) ∇u 2 L 2 + ∇θ 2 L 2 + C(ρ)σ ρ − 1 2 L 2 .σ 2 u · ∇n · uGdS ≤C(ρ) sup 0≤t≤T (σ ∇u 2 L 2 ) sup 0≤t≤T (σ ρ 1/2u L 2 ) ≤C(ρ)C 1/4 0 . (3.71) Furthermore, note that (1.39) is equivalent to P (ρ − 1) = −G + (2µ + λ)divu − ρ(Rθ − P ),σ ρ − 1 2 L 2 dt ≤ C T 0 σ( G 2 L 2 + ∇u 2 L 2 )dt + C(ρ) T 0 Rθ − P 2 L 2 dt ≤ C(ρ) T 0 σ ρ 1/2u 2 L 2 + ∇u 2 L 2 + ∇θ 2 L 2 dt ≤ C(ρ, M )C 1/4 0 . ) that A 3 (T ) ≤ C(ρ, M )C 1/4 0 ≤ C 1/6 0 , provided C 0 ≤ ε 2 min{1, (C(ρ, M )) −12 }. The proof of Lemma 3.4 is completed. Next, in order to control A 2 (T ), we first re-establish the basic energy estimate for short time [0, σ(T )], and then show that the spatial L 2 -norm of Rθ−P could be bounded by the combination of the initial energy and the spatial L 2 -norm of ∇θ, which is indeed the key ingredient to estimate A 2 (T ). ρ|u| 2 + (ρ − 1) 2 + ρ(θ − log θ − 1) dx ≤ CC 0 ,(3. 73) and (Rθ − P )(·, t) L 2 ≤ C C 1/2 0 + C 1/3 0 ∇θ(·, t) L 2 , (3.74) for all t ∈ (0, σ(T )], provided C 0 ≤ ε 3,1 . Proof. The proof is divided into the following two steps. Step 1: The proof of (3.73). First, multiplying (3.13) by u, one deduces from integration by parts, (1.4) 1 , and (3.34) that d dt 1 2 ρ|u| 2 + R(1 + ρ log ρ − ρ) dx + (µ|curlu| 2 + (2µ + λ)(divu) 2 )dx = R ρ(θ − 1)divudx ≤ δ ∇u 2 L 2 + C(δ,ρ) ρ(θ − 1) 2 dx ≤ δ ∇u 2 L 2 + C(δ,ρ)( θ(·, t) L ∞ + 1) ρ(θ − log θ − 1)dx. (3.75) Using (2.12) and choosing δ small enough in (3.75), it holds that d dt 1 2 ρ|u| 2 + R(1 + ρ log ρ − ρ) dx + C 3 |∇u| 2 dx ≤ C(ρ)( θ(·, t) L ∞ + 1) ρ(θ − log θ − 1)dx. (3.76) Then, adding (3.76) multiplied by (2µ + 1)C 3 −1 to (3.14), one has (2µ + 1)C 3 −1 + 1 d dt 1 2 ρ|u| 2 + R(1 + ρ log ρ − ρ) dx + R γ − 1 d dt ρ(θ − log θ − 1)dx + |∇u| 2 dx ≤ C(ρ)( θ(·, t) L ∞ + 1) ρ(θ − log θ − 1)dx.σϕ + t 0 σ C 1 2 |∇u| 2 + ρ|θ| 2 dxdτ ≤ 2σ ∂Ω (u · ∇n · u)GdS (t) + C(ρ, M ) t 0 ( ρ 1/2u 2 L 2 + ∇u 2 L 2 + ∇θ 2 L 2 )dτ + C(ρ) t 0 σ ρ − 1 2 L 2 dτ + C(ρ) t 0 ∇u 2 L 2 + ∇θ 2 L 2 σϕdτ ≤ C(ρ)(σ ∇u 2 L 2 ρ 1/2u L 2 )(t) + C(ρ, M ) + C(ρ, M ) t 0 ∇u 2 L 2 + ∇θ 2 L 2 σϕdτ ≤ C(ρ, M ) + C(ρ, M ) t 0 ∇u 2 L 2 + ∇θ 2 L 2 σϕdτ. Then Grönwall inequality together with (3.5) and (3.65) yields sup 0≤t≤T σ ρ|u| 2 dx + ∇θ 2 L 2 + T 0 σ |∇u| 2 + ρ|θ| 2 dxdt ≤ C(ρ,L 2 + ρ 1/2u 2 L 2 dt + C(ρ, M ) T 0 ∇u 2 L 2 + ∇θ 2 L 2 + σ ρ − 1 2 L 2 dt ≤ C(ρ, M ).Rθ − P L ∞ ≤C Rθ − P 1/2 L 6 ∇θ 1/2 L 6 + Rθ − P L 2 ≤C(ρ) ∇θ 1/2 L 2 ∇ 2 θ 1/2 L 2 + C(ρ) ∇θ L 2 ,σ(T ) 0 Rθ − P L ∞ dt ≤ C(ρ) σ(T ) 0 ∇θ 1/2 L 2 σ ∇ 2 θ 2 L 2 1/4 σ −1/4 dt + C(ρ) σ(T ) 0 ∇θ 2 L 2 dt 1/2 ≤ C(ρ) σ(T ) 0 ∇θ 2 L 2 dt σ(T ) 0 σ ∇ 2 θ 2 L 2 dt 1/4 + C(ρ)C 1/8 0 ≤ C(ρ, M )C 1/16 0 . (3.82) Combining this with (3.11) yields (3.78) directly. Step 2: The proof of (3.74). Direct calculations together with (3.34) lead to θ − log θ − 1 ≥ 1 8 (θ − 1)1 (θ(·,t)>2) + 1 12 (θ − 1) 2 1 (θ(·,t)<3) , with (θ(·, t) > 2) {x ∈ Ω| θ(x, t) > 2} and (θ(·, t) < 3) {x ∈ Ω| θ(x, t) < 3} . Com- bining this with (3.73) gives sup 0≤t≤σ(T ) ρ(θ − 1)1 (θ(·,t)>2) + ρ(θ − 1) 2 1 (θ(·,t)<3) dx ≤ C(ρ, M )C 0 . (3.83) Next, it follows from (3.83), (3.73), and the Sobolev inequality that for t ∈ (0, σ(T )], θ − 1 2 L 2 (θ(·,t)<3) ≤ ρ(θ − 1) 2 1 (θ(·,t)<3) dx + (ρ − 1)(θ − 1) 2 dx ≤ C(ρ, M )C 0 + C ρ − 1 L 2 θ − 1 1/2 L 2 θ − 1 3/2 L 6 ≤ C(ρ, M )C 0 + C(ρ, M )C 1/2 0 θ − 1 1/2 L 2 ( θ − 1 L 2 + ∇θ L 2 ) 3/2 ≤ C(ρ, M ) C 0 + C(δ)C 2/3 0 ∇θ 2 L 2 + (δ + C 1/2 0 ) θ − 1 2 L 2 ,(3.84) and θ − 1 2 L 2 (θ(·,t)>2) ≤ θ − 1 4/5 L 1 (θ(·,t)>2) θ − 1 6/5 L 6 ≤ C(ρ, M ) C 0 + C 1/2 0 θ − 1 L 2 4/5 ( θ − 1 L 2 + ∇θ L 2 ) 6/5 ≤ C(ρ, M ) C 0 + C(δ)C 2/3 0 ∇θ 2 L 2 + (δ + C 2/5 0 ) θ − 1 2 L 2 ,(3.85) where in the second inequality one has used θ − 1 L 1 (θ(·,t)>2) ≤ ρ(θ − 1)1 (θ(·,t)>2) dx + |(ρ − 1)(θ − 1)| dx ≤C(ρ, M )(C 0 + C 1/2 0 θ − 1 L 2 ). Hence, adding (3.84) with (3.85) together and choosing δ small enough in the resulting inequality, one has for any t ∈ (0, σ(T )], θ − 1 2 L 2 ≤ C(ρ, M ) C 0 + C 2/3 0 ∇θ 2 L 2 + C 2/5 0 θ − 1 2 L 2 , which implies that Finally, note that θ − 1 2 L 2 ≤ C(ρ, M ) C 0 + C 2/3 0 ∇θ 2 L 2 ,(3.Rθ − P L 2 ≤ R θ − 1 L 2 + C|1 − ρθ| ≤ R θ − 1 L 2 + C ρ(1 − θ)dx ≤ C(ρ) θ − 1 L 2 , this together with (3.86) yields (3.74). The proof of Lemma 3.5 is completed. Next, with the help of (3.74), the estimate on A 2 (T ) will be handled smoothly. A 2 (T ) ≤ C 1/4 0 , (3.88) provided C 0 ≤ ε 3 . Proof. To begin with, multiplying (3.13) by u and integrating by parts give that d dt 1 2 ρ|u| 2 + P (1 + ρ log ρ − ρ) dx + µ|curlu| 2 + (2µ + λ)(divu) 2 dx = P t (1 + ρ log ρ − ρ)dx + ρ(Rθ − P )divudx. (3.89) Next, multiplying (1.4) 3 by P −1 (Rθ − P ), one obtains after integrating the resulting equality over Ω by parts that 1 2(γ − 1) d dt P −1 ρ(Rθ − P ) 2 dx + κRP −1 ∇θ 2 L 2 = − 1 γ − 1 P −1 P t ρ(Rθ − P )dx − 1 2(γ − 1) P −2 P t ρ(Rθ − P ) 2 dx − P −1 ρ(Rθ − P ) 2 divudx − ρ(Rθ − P )divudx + P −1 (Rθ − P )(λ(divu) 2 + 2µ|D(u)| 2 )dx. (3.90) Adding (3.89) and (3.90) together yields that d dt 1 2 ρ|u| 2 + P (1 + ρ log ρ − ρ) + 1 2(γ − 1) ρP −1 (Rθ − P ) 2 dx + µ curlu 2 L 2 + (2µ + λ) divu 2 L 2 + κRP −1 ∇θ 2 L 2 = − 1 γ − 1 P −1 P t ρ(Rθ − P )dx − 1 2(γ − 1) P −2 P t ρ(Rθ − P ) 2 dx + P t (1 + ρ log ρ − ρ)dx − P −1 ρ(Rθ − P ) 2 divudx + P −1 (Rθ − P )(λ(divu) 2 + 2µ|D(u)| 2 )dx 5 i=1 J i . (3.91) The terms J i (i = 1, · · · , 5) can be estimated as follows. It follows from (3.11), (3.29), (3.5), and (3.22) that J 1 + J 2 ≤C|P t | ρ(Rθ − P ) L 2 + ρ 1/2 (Rθ − P ) 2 L 2 ≤C(ρ) C 1/8 0 ∇u L 2 + ∇u 2 L 2 ρ 1/2 (Rθ − P ) L 2 ≤C(ρ)C 1/8 0 ∇u 2 L 2 + ∇θ 2 L 2 ,(3.92) and J 4 ≤C ρ 1/2 (Rθ − P ) 1/2 L 2 ρ 1/2 (Rθ − P ) 3/2 L 6 ∇u L 2 ≤C(ρ)A 1/4 2 (T ) ∇θ 3/2 L 2 ∇u L 2 ≤C(ρ, M )C 1/16 0 ( ∇u 2 L 2 + ∇θ 2 L 2 ). (3.93) Furthermore, by virtue of (3.16), (3.29), and (3.10), we have J 3 ≤|P t | (1 + ρ log ρ − ρ)dx ≤C(ρ) C 1/8 0 ∇u L 2 + ∇u 2 L 2 ρ − 1 2 L 2 ≤C(ρ)C 1/4 0 ∇u 2 L 2 + C(ρ)C 1/4 0 ρ − 1 2 L 2 . (3.94) Now, we will estimate the term J 5 for the short time t ∈ [0, σ(T )) and the large time t ∈ [σ(T ), T ], respectively. For t ∈ [0, σ(T )), it follows from (3.11), (3.24), (3.22), (3.74), and (3.5) that J 5 ≤C |Rθ − P ||∇u| 2 dx ≤C Rθ − P 1/2 L 2 Rθ − P 1/2 L 6 ∇u L 2 ∇u L 6 ≤C(ρ) Rθ − P 1/2 L 2 ∇θ 1/2 L 2 ∇u L 2 ρ 1/2u L 2 + ∇u L 2 + ∇θ L 2 + C 1/24 0 ≤C(ρ) Rθ − P 1/2 L 2 ∇θ 1/2 L 2 ∇u L 2 ρ 1/2u L 2 + C(ρ, M )C 1/24 0 ( ∇θ 2 L 2 + ∇u 2 L 2 ) ≤C(ρ, M )C 7/24 0 ρ 1/2u 2 L 2 + C(ρ, M )C 1/24 0 ( ∇u 2 L 2 + ∇θ 2 L 2 ),(3.95) where we have used following calculations: Rθ − P 1/2 L 2 ∇θ 1/2 L 2 ∇u L 2 ρ 1/2u L 2 ≤ C(ρ, M )(C 1/4 0 ∇θ 1/2 L 2 + C 1/6 0 ∇θ L 2 ) ∇u L 2 ρ 1/2u L 2 ≤ C(ρ, M )C 7/24 0 ρ 1/2u 2 L 2 + C(ρ, M )C 1/12 0 ∇u 2 L 2 + C(ρ, M )C 1/24 0 ∇θ 2 L 2 owing to (3.74). For t ∈ [σ(T ), T ], it holds that J 5 ≤C Rθ − P L 3 ∇u L 2 ∇u L 6 ≤ C(ρ)C 1/24 0 ( ∇u 2 L 2 + ∇θ 2 L 2 ),(3.ρ|u| 2 + P (1 + ρ log ρ − ρ) + 1 2(γ − 1) ρP −1 (Rθ − P ) 2 dx + T 0 (µ curlu 2 L 2 + (2µ + λ) divu 2 L 2 + RκP −1 ∇θ 2 L 2 )dt ≤ C(ρ, M )C 1/24 0 T 0 ( ∇u 2 L 2 + ∇θ 2 L 2 )dt + C(ρ)C 1/4 0 T 0 ρ − 1 2 L 2 dt + C(ρ, M )C 7/24 0 σ(T ) 0 ρ 1/2u 2 L 2 dt + C(ρ,θ)C 0 ≤ C(ρ,θ, M )C 7/24 0 ,(3.97) where one has used with ε 3,1 as in (3.87). The proof of Lemma 3.6 is completed. T 0 ρ − 1 2 L 2 dt ≤ sup 0≤t≤σ(T ) ρ − 1 2 L 2 + T σ(T ) ρ − 1 2 L 2 dt ≤ C(ρ, M )C Remark 3.2. It's worth noticing that the energy-like estimate A 2 (T ) is a little subtle, since A 2 (T ) is not a conserved quantity for the full Navier-Stokes system owing to the nonlinear coupling of θ and u. Thus, further consideration is needed to handle this issue. More precisely, • on the one hand, while deriving the kinetic energy (see (3.89)), we need to deal with the following term ρ(Rθ − P )divudx. Unfortunately, this term is troublesome for large time t ∈ [σ(T ), T ]. In fact, this term could be bounded by ∇θ L 2 ∇u L 2 , which will only be of the same order as C 1/4 0 with the help of all a priori estimates (3.5). Therefore, we can not handle this term directly. Here, based on careful analysis on system (1.4), we find that this term can be cancelled by a suitable combination of kinetic energy and thermal energy, see (3.89)-(3.91); • on the other hand, while deriving the thermal energy (see (3.90)), we need to handle the following term (Rθ − P )(divu) 2 dx. Note that for short time t ∈ [0, σ(T )), the "weaker" basic energy estimate (3.10) is not enough, hence it's necessary to re-establish the basic energy estimate (3.73), which is obtained by the a priori L 1 (0, σ(T ); L ∞ )-norm of θ (see (3.78)). Consequently, we can obtain (3.74) as a consequence of (3.73) and then handle this term for short time t ∈ [0, σ(T )) (see (3.95)). Moreover, it should be mentioned that the uniform positive lower and upper bounds of P also play a critical role in estimating A 2 (T ). We now proceed to derive a uniform (in time) upper bound for the density, which turns out to be the key to obtaining all the higher order estimates and thus extending the classical solution globally. sup 0≤t≤T ρ(·, t) L ∞ ≤ 3ρ 2 , (3.100) provided C 0 ≤ ε 4 . Proof. First, it follows from (3.80), (3.81), and (3.5) that T σ(T ) Rθ − P 2 L ∞ dt ≤C(ρ) T σ(T ) ∇θ 2 L 2 dt 1/2 T σ(T ) ∇ 2 θ 2 L 2 dt 1/2 + C(ρ) T σ(T ) ∇θ 2 L 2 dt ≤C(ρ, M )C 1/8 0 .(3.σ(T ) 0 G L ∞ dt ≤ C σ(T ) 0 ∇G 1/2 L 2 ∇G 1/2 L 6 dt ≤ C(ρ) σ(T ) 0 ρu 1/2 L 2 ( ∇u L 2 + ∇u 2 L 2 ) 1/2 dt ≤ C(ρ) σ(T ) 0 (σ ρu L 2 ) 1/4 σ ρu 2 L 2 1/8 σ ∇u 2 L 2 1/4 σ −5/8 dt + C(ρ) σ(T ) 0 (σ ρu L 2 ) 1/2 ∇u L 2 σ −1/2 dt ≤ C(ρ, M )C 1/48 0 σ(T ) 0 σ ∇u 2 L 2 dt 1/4 σ(T ) 0 σ −5/6 dt 3/4 + C(ρ, M )C 1/24 0 σ(T ) 0 σ −1/2 dt ≤ C(ρ, M )C 1/48 0 , (3.102) and T σ(T ) G 2 L ∞ dt ≤ C T σ(T ) ∇G L 2 ∇G L 6 dt ≤ C(ρ, M ) T σ(T ) ρ 1/2u 2 L 2 + ∇u 2 L 2 + ∇u 2 L 2 dt ≤ C(ρ, M )C 1/6 0 . (3.103) Denoting D t ρ = ρ t + u · ∇ρ and using (1.39), one can rewrite (1.4) 1 as follows (2µ + λ)D t ρ = −P ρ(ρ − 1) − ρ 2 (Rθ − P ) − ρG ≤ −P (ρ − 1) + C(ρ) Rθ − P L ∞ + C(ρ) G L ∞ , which gives D t (ρ − 1) + P 2µ + λ (ρ − 1) ≤ C(ρ) Rθ − P L ∞ + C(ρ) G L ∞ . (3.104) Finally, applying Lemma 2.8 with The proof of Lemma 3.7 is completed. y = ρ − 1, α = P 2µ + λ , g = C(ρ) Rθ − P L ∞ + C(ρ) G L ∞ , T 1 = σ(T ), Next, we summarize some uniform estimates on (ρ, u, θ) which will be useful for higher-order ones in the next section. Moreover, it holds that sup 0<t≤T σ ∇u 2 L 6 + σ 2 θ 2 H 2 + T 0 (σ ∇u 4 L 4 + σ ∇θ 2 H 1 + σ u t 2 L 2 + σ 2 θ t 2 H 1 + ρ − 1 2 L 2 )dt ≤ C. (3.106) Proof. First, applying the operator ∂ t + div(u·) to (1.4) 3 and using (1.4) 1 , one gets R γ − 1 ρ ∂ tθ + u · ∇θ = κ∆θ t + κdiv(∆θu) + λ(divu) 2 + 2µ|D(u)| 2 divu + Rρθ∂ k u l ∂ l u k − Rρθdivu − Rρθdivu + 2λ divu − ∂ k u l ∂ l u k divu + µ(∂ i u j + ∂ j u i ) ∂ iu j + ∂ ju i − ∂ i u k ∂ k u j − ∂ j u k ∂ k u i . (3.107) Direct calculations show that (∆θ t + div(∆θu))θdx = − (∇θ t · ∇θ + ∆θu · ∇θ)dx = − |∇θ| 2 dx + (∇(u · ∇θ) · ∇θ − ∆θu · ∇θ)dx. (3.108) Multiplying (3.107) byθ and integrating the resulting equality over Ω, it holds that R 2(γ − 1) ρ|θ| 2 dx t + κ ∇θ 2 L 2 ≤ C |∇θ| |∇ 2 θ||u| + |∇θ||∇u| dx + C ρ|Rθ − P ||∇u||θ|dx + C(ρ) |∇u| 2 |θ| |∇u| + |Rθ − P | dx + C |∇u|ρ|θ|dx + C(ρ) |∇u| 2 |θ| + ρ|θ| 2 |∇u| + |∇u||∇u||θ| dx ≤ C ∇u 1/2 L 2 ∇u 1/2 L 6 ∇ 2 θ L 2 ∇θ L 2 + C(ρ) ∇θ L 2 ∇u L 2 θ L 6 + C(ρ) ∇u L 2 ∇u L 6 ( ∇u L 6 + ∇θ L 2 ) θ L 6 + C ∇u L 2 ρθ L 2 + C(ρ) ∇u 1/2 L 6 ∇u 1/2 L 2 θ L 6 ∇u L 2 + ρθ L 2 + ∇u L 2 ≤ κ 2 ∇θ 2 L 2 + C(ρ) ∇u 2 L 2 ∇u 4 L 6 + ∇θ 4 L 2 + C(ρ, M ) ∇u L 6 ∇u 2 L 2 + C(ρ, M ) 1 + ∇u L 6 + ∇θ 2 L 2 ∇ 2 θ 2 L 2 + ∇u 2 L 2 + ρ 1/2θ 2 L 2 , (3.109) where we have used (3.108), (2.7), (2.8), (3.5), (3.22), and the following Poincaré-type inequality ( [9, Lemma 3.2]): f L p ≤ C(ρ)( ρ 1/2 f L 2 + ∇f L 2 ), p ∈ [2, 6], (3.110) for any f ∈ {h ∈ H 1 ρ 1/2 h ∈ L 2 } . Multiplying (3.109) by σ 2 and integrating the resulting inequality over (0, T ), we obtain after integrating by parts that sup 0≤t≤T σ 2 ρ|θ| 2 dx + T 0 σ 2 ∇θ 2 L 2 dt ≤ C(ρ) sup 0≤t≤T σ 2 ( ∇u 4 L 6 + ∇θ 4 L 2 ) T 0 ∇u 2 L 2 dt + C(ρ, M ) sup 0≤t≤T σ 1 + ∇u L 6 + ∇θ 2 L 2 · T 0 σ ∇ 2 θ 2 L 2 + ∇u 2 L 2 + ρ 1/2θ 2 L 2 dt + C(ρ, M ) sup 0≤t≤T (σ ∇u L 6 ) T 0 ∇u 2 L 2 dt + C T 0 σ ρ 1/2θ 2 L 2 dt ≤ C(ρ, M ),L 2 dt ≤ C T 0 σ( u 2 L 2 + u · ∇u 2 L 2 )dt ≤ C(ρ) T 0 σ( ρ 1/2u 2 L 2 + ∇u 2 L 2 + u 2 L ∞ ∇u 2 L 2 )dt ≤ C(ρ, M ), (3.113) T 0 σ 2 θ t 2 L 2 dt ≤ C T 0 σ 2 ( θ 2 L 2 + u · ∇θ 2 L 2 )dt ≤ C(ρ) T 0 σ 2 ( ρ 1/2θ 2 L 2 + ∇θ 2 L 2 + u 2 L 6 ∇θ 2 L 3 )dt ≤ C(ρ, M ), (3.114) and T 0 σ 2 ∇θ t 2 L 2 dt ≤ C T 0 σ 2 ∇θ 2 L 2 dt + C T 0 σ 2 ∇(u · ∇θ) 2 L 2 dt ≤ C(ρ, M ) + C T 0 σ 2 ∇u 2 L 3 + u 2 L ∞ ∇ 2 θ 2 L 2 dt ≤ C(ρ, M ).1 2 W ′ (t) + µ curlu 2 L 2 + (2µ + λ) divu 2 L 2 + κRP −1 ∇θ 2 L 2 ≤ C|P t |( Rθ − P L 2 + Rθ − P 2 L 2 ) + |P t | ρ − 1 2 L 2 + C Rθ − P L ∞ ( Rθ − P 2 L 2 + ∇u 2 L 2 ) ≤ C(C 1/8 0 ∇u L 2 + ∇u 2 L 2 )( ∇θ L 2 + ∇θ 2 L 2 + ρ − 1 2 L 2 ) + C( ∇θ 1/2 L 2 ∇ 2 θ 1/2 L 2 + ∇θ L 2 )( ∇θ 2 L 2 + ∇u 2 L 2 ) ≤ CC 1/24 0 ( ∇u 2 L 2 + ∇θ 2 L 2 + ρ − 1 2 L 2 ), (3.116) where W (t) ρ|u| 2 + 2P (1 + ρ log ρ − ρ) + 1 γ − 1 ρP −1 (Rθ − P ) 2 dx ≤Ĉ 3 ( ∇u 2 L 2 + ∇θ 2 L 2 + ρ − 1 2 L 2 )(W ′ (t) +Ĉ 1 ( ∇u 2 L 2 + ∇θ 2 L 2 ) ≤Ĉ 2 C 1/24 0 ( ∇u 2 L 2 + ∇θ 2 L 2 + ρ − 1 2 L 2 ).P (ρ − 1) 2 dx = ρu · B[ρ − 1]dx t − ρu · B[ρ t ]dx − ρu · ∇B[ρ − 1] · udx + µ ∂ j u · ∂ j B[ρ − 1]dx + (µ + λ) (ρ − 1)divudx − ρ(Rθ − P )(ρ − 1)dx ≤ ρu · B[ρ − 1]dx t + C ρu 2 L 2 + C u 2 L 4 ρ − 1 L 2 + C ρ − 1 L 2 ∇u L 2 + C ρ − 1 L 2 Rθ − P L 2 ≤ ρu · B[ρ − 1]dx t + π 1 2 ρ − 1 2 L 2 + C( ∇u 2 L 2 + ∇θ 2 L 2 ), which as well as (3.11) leads to ρ − 1 2 L 2 ≤ 2 π 1 ρu · B[ρ − 1]dx t +Ĉ 4 ( ∇u 2 L 2 + ∇θ 2 L 2 ). (3.119) By virtue of (3.11), (3.16), and Lemma 2.9, it holds ρu · B[ρ − 1]dx ≤ C ρu 2 L 2 + ρ − 1 2 L 2 ≤Ĉ 5 ρ 1/2 u 2 L 2 + 2P (1 + ρ log ρ − ρ) . (3.120) Adding (3.118) to (3.119) multiplied byĈ 6 withĈ 6 = min{ π 1 4Ĉ 5 ,Ĉ 1 4Ĉ 4 } yields W ′ 1 (t) + 3Ĉ 1 4 ( ∇u 2 L 2 + ∇θ 2 L 2 ) +Ĉ 6 ρ − 1 2 L 2 ≤Ĉ 2 C 1/24 0 ( ∇u 2 L 2 + ∇θ 2 L 2 + ρ − 1 2 L 2 ), (3.121) where W 1 (t) W (t) − 2Ĉ 6 π 1 ρu · B[ρ − 1]dx, satisfies 1 2 W (t) ≤ W 1 (t) ≤ 2W (t) (3.122) due to (3.120). Thus we infer from (3.121) that W ′ 1 (t) +Ĉ 1 2 ( ∇u 2 L 2 + ∇θ 2 L 2 ) +Ĉ 6 2 ρ − 1 2 L 2 ≤ 0, (3.123) provided C 0 ≤ ε 0 min    ε 1 , · · · , ε 4 , Ĉ 6 2Ĉ 2 24 , Ĉ 1 4Ĉ 2 24    . (3.124) Then by (3.117), one derives that for α = 1 3 min{Ĉ 1 2Ĉ 3 ,Ĉ 6 2Ĉ 3 }, W ′ 1 (t) + 3αW 1 (t) ≤ 0, which along with (3.122), (3.16), (3.11), (3.117), and (3.5) shows that for any t ≥ 1, ρ 1/2 u 2 L 2 + ρ − 1 2 L 2 + ρ 1/2 (Rθ − P ) 2 L 2 ≤ CW 1 (t) ≤ Ce −3αt . (3.125) Moreover, we deduce from (3.123) and (3.125) that for any 1 ≤ t ≤ T < ∞, T 1 e αt ( ∇u 2 L 2 + ∇θ 2 L 2 )dt ≤ C. (3.126) Next, multiplying (3.31) by e αt and using (3.5) imply that for B 1 defined in (3.38), (e αt B 1 (t)) ′ + 1 2 e αt ρ|u| 2 dx ≤ Ce αt ∇u 2 L 2 + ∇θ 2 L 2 + P − P 2 L 2 . (3.127) Note that by (3.5), (3.11), and (3.125), P − P L 2 ≤ ρ(Rθ − P ) L 2 + P ρ − 1 L 2 ≤ Ce −αt ,(3.ϕ ′ (t) + 2 ∂Ω (u · ∇n · u)GdS t + C 1 2 |∇u| 2 + ρ|θ| 2 dx ≤ C( ρ 1/2u 2 L 2 + ∇u 2 L 2 + ∇θ 2 L 2 + ρ − 1 2 L 2 ), where ϕ(t) is defined in ( e αt ( ρ 1/2u 2 L 2 + ∇θ 2 L 2 ) + T 1 e αt ( ∇u 2 L 2 + ρ 1/2θ 2 L 2 )dt ≤ C.e αt ( ρ 1/2θ 2 L 2 + ∇ 2 θ 2 L 2 ) + T 1 e αt ∇θ 2 L 2 dt ≤ C. (3.131) Finally, it remains to determine the limit of θ as t tends to infinity. Combining (3.26), (3.128), and (3.129) shows that for any t ≥ 1, |P t | ≤ C( ∇u 2 L 2 + P − P 2 L 2 ) ≤ Ce −αt , which implies there exists a constant P ∞ such that lim t→∞ P = P ∞ and |P − P ∞ | ≤ Ce −αt . (3.132) Denoting θ ∞ P ∞ /R, we have θ − θ ∞ 2 L 2 ≤ C Rθ − P 2 L 2 + C|P − Rθ ∞ | 2 ≤ Ce −αt ,(3. A priori estimates (II): higher-order estimates In this section, we will derive the higher-order estimates of smooth solution (ρ, u, θ) to problem (1.4)-(1.6) on Ω × (0, T ] with initial data (ρ 0 , u 0 , θ 0 ) satisfying (1.9) and (3.4). We shall assume that (3.5) and (3.8) both hold as well. To proceed, we defineg as From now on, the generic constant C will depend only on g ρ −1/2 0 (−µ∆u 0 − (µ + λ)∇divu 0 + R∇(ρ 0 θ 0 )) .T, g L 2 , ρ 0 W 2,q , ∇u 0 H 1 , ∇θ 0 L 2 , besides µ, λ, κ, R, γ,ρ,θ, Ω, and M. We begin with the following estimates on the spatial gradient of the smooth solution (ρ, u, θ). Lemma 4.1. The following estimates hold: sup 0≤t≤T ρ 1/2u 2 L 2 + σ ρ 1/2θ 2 L 2 + θ 2 H 1 + σ ∇ 2 θ 2 L 2 + T 0 ∇u 2 L 2 + ρ 1/2θ 2 L 2 + ∇ 2 θ 2 L 2 + σ ∇θ 2 L 2 dt ≤ C, (4.3) and sup 0≤t≤T ( u H 2 + ρ H 2 ) + T 0 ∇u 3/2 L ∞ + σ ∇ 3 θ 2 L 2 + u 2 H 3 dt ≤ C. (4.4) Proof. The proof is divided into the following two steps. Step 1: The proof of (4.3). First, for ϕ(t) as in (3.64), taking m = 0 in (3.63), one gets ϕ ′ (t) + C 1 2 |∇u| 2 + ρ|θ| 2 dx ≤ −2 ∂Ω G (u · ∇n · u) dS t + C ρ 1/2u 2 L 2 + ∇u 2 L 2 + ∇θ 2 L 2 + C ρ 1/2u 3 L 2 + ∇θ 3 L 2 + ∇u 2 L 2 + ρ − 1 2 L 2 + C ∇u 2 L 2 + ∇θ 2 L 2 ρ 1/2u 2 L 2 + ∇θ 2 L 2 + 1 ≤ −2 ∂Ω G (u · ∇n · u) dS t + C ρ 1/2u 2 L 2 + ∇θ 2 L 2 (ϕ + 1) + C|ϕ(0)| ≤ C g 2 L 2 + C ≤ C. (4.6) Then, integrating (4.5) over (0, t), one obtains after using (3.5), (3.44), (2.13), (3.65), and (4.6) that ϕ(t) + t 0 C 1 2 |∇u| 2 + ρ|θ| 2 dxds ≤ 2 ∂Ω G (u · ∇n · u) dS (t) + C t 0 ρ 1/2u 2 L 2 + ∇θ 2 L 2 ϕds + C ≤ C( ∇u 2 L 2 ρ 1/2u L 2 )(t) + C t 0 ρ 1/2u 2 L 2 + ∇θ 2 L 2 ϕds + C ≤ 1 2 ϕ(t) + C t 0 ρ 1/2u 2 L 2 + ∇θ 2 L 2 ϕds + C. (4.7) Applying Grönwall's inequality to (4.7) and using (3.5) and (3.65), it holds sup 0≤t≤T ρ 1/2u 2 L 2 + ∇θ 2 L 2 + T 0 |∇u| 2 + ρ|θ| 2 dxdt ≤ C,(4.8) which together with (3.11) and (3.22) implies θ L 2 ≤ C( Rθ − P L 2 + P ) ≤ C( ∇θ L 2 + 1) ≤ C. (4.9) Next, multiplying (3.109) by σ and integrating over (0, T ) lead to sup 0≤t≤T σ ρ|θ| 2 dx + T 0 σ ∇θ 2 L 2 dt ≤ C T 0 ∇ 2 θ 2 L 2 + ∇u 2 L 2 + ρ 1/2θ 2 L 2 dt + C ≤ C,sup 0≤t≤T σ ∇ 2 θ 2 L 2 + T 0 ∇ 2 θ 2 L 2 dt ≤ C, which along with (4.8)-(4.10) gives (4.3). Step 2: The proof of (4.4). First, standard calculations show that for 2 ≤ p ≤ 6, ∂ t ∇ρ L p ≤ C ∇u L ∞ ∇ρ L p + C ∇ 2 u L p ≤ C 1 + ∇u L ∞ + ∇ 2 θ L 2 ∇ρ L p + C 1 + ∇u L 2 + ∇ 2 θ L 2 ,(4.12) where we have used ∇ 2 u L p ≤ C( ρu L p + ∇P L p + ∇u L 2 ) ≤ C (1 + ∇u L 2 + ∇θ L p + θ L ∞ ∇ρ L p ) ≤ C 1 + ∇u L 2 + ∇ 2 θ L 2 + ( ∇ 2 θ L 2 + 1) ∇ρ L p∇u L ∞ ≤ C ( divu L ∞ + curlu L ∞ ) log(e + ∇ 2 u L 6 ) + C ∇u L 2 + C ≤ C ( divu L ∞ + curlu L ∞ ) log(e + ∇u L 2 + ∇ 2 θ L 2 ) + C ( divu L ∞ + curlu L ∞ ) log (e + ∇ρ L 6 ) + C. (4.14) Denote f (t) e + ∇ρ L 6 , h(t) 1 + divu 2 L ∞ + curlu 2 L ∞ + ∇u 2 L 2 + ∇ 2 θ 2 L 2 . One obtains after submitting (4.14) into (4.12) with p = 6 that f ′ (t) ≤ Ch(t)f (t) ln f (t), which implies (ln(ln f (t))) ′ ≤ Ch(t). divu 2 L ∞ + curlu 2 L ∞ dt ≤ C T 0 G 2 L ∞ + curlu 2 L ∞ + P − P 2 L ∞ dt ≤ C T 0 G 2 W 1,6 + curlu 2 W 1,6 + θ 2 L ∞ dt + C ≤ C T 0 ∇G 2 L 6 + ∇curlu 2 L 6 + ∇ 2 θ 2 L 2 dt + C ≤ C T 0 ( ∇u 2 L 2 + ∇ 2 θ 2 L 2 )dt + C ≤ C,u H 2 ≤C sup 0≤t≤T ( ρu L 2 + ∇P L 2 + ∇u L 2 ) ≤ C. (4.19) Next, applying operator ∂ ij (1 ≤ i, j ≤ 3) to (1.4) 1 gives (∂ ij ρ) t + div(∂ ij ρu) + div(ρ∂ ij u) + div(∂ i ρ∂ j u + ∂ j ρ∂ i u) = 0. (4.20) Multiplying (4.20) by 2∂ ij ρ and integrating the resulting equality over Ω, it holds d dt ∇ 2 ρ 2 L 2 ≤ C(1 + ∇u L ∞ ) ∇ 2 ρ 2 L 2 + C ∇u 2 H 2 ≤ C(1 + ∇u L ∞ + ∇ 2 θ 2 L 2 )(1 + ∇ 2 ρ 2 L 2 ) + C ∇u 2 L 2 ,(4.21) where one has used (3.5), (4.17), and the following estimate: u H 3 ≤ C ( ∇(ρu) L 2 + ρu L 2 + ∇P H 1 + ∇u L 2 ) ≤ C ∇ρ L 3 u L 6 + C ∇u L 2 + C ∇θ H 1 + C |∇ρ||∇θ| L 2 + C θ L ∞ ∇ρ H 1 + C ≤ C ∇u L 2 + C(1 + ∇ 2 θ L 2 )(1 + ∇ 2 ρ L 2 ) + C∇ 2 θ H 1 ≤ C ρθ H 1 + ρθdivu H 1 + |∇u| 2 H 1 ≤ C 1 + ∇θ L 2 + ρ 1/2θ L 2 + ∇(ρθdivu) L 2 + |∇u||∇ 2 u| L 2 ≤ C 1 + ∇θ L 2 + ρ 1/2θ L 2 + ∇ 2 θ L 2 + ∇ 3 u L 2 ,sup 0≤t≤T ρ t H 1 + T 0 u t 2 H 1 + σ θ t 2 H 1 + ρu t 2 H 1 + σ ρθ t 2 H 1 dt ≤ C, (4.26) and T 0 σ (ρu t ) t 2 H −1 + (ρθ t ) t 2 H −1 dt ≤ C. (4.27) Proof. First, it follows from (4.3) and (4.4) that sup 0≤t≤T ρ|u t | 2 + σρθ 2 t dx + T 0 ∇u t 2 L 2 + σ ∇θ t 2 L 2 dt ≤ sup 0≤t≤T ρ|u| 2 + σρ|θ| 2 dx + T 0 ∇u 2 L 2 + σ ∇θ 2 L 2 dt + sup 0≤t≤T ρ |u · ∇u| 2 + σ|u · ∇θ| 2 dx + T 0 ∇(u · ∇u) 2 L 2 + σ ∇(u · ∇θ) 2 L 2 dt ≤ C,(4.28) which together with (4.3) and (4.4) gives T 0 ∇(ρu t ) 2 L 2 + σ ∇(ρθ t ) 2 L 2 dt ≤ C T 0 ∇u t 2 L 2 + ∇ρ 2 L 3 u t 2 L 6 + σ ∇θ t 2 L 2 + σ ∇ρ 2 L 3 θ t 2 L 6 dt ≤ C,(4.29) where we have used θ t L 6 ≤ C ρ 1/2 θ t L 2 + C ∇θ t L 2 (4.30) due to (3.110). Next, one deduces from (1.4) 1 , (4.4), and (2.9) that ρ t H 1 ≤ div(ρu) H 1 ≤ C u H 2 ρ H 2 ≤ C, which as well as (4.28)-(4.30) shows (4.26). Finally, differentiating (1.4) 2 with respect to t yields that (ρu t ) t = −(ρu · ∇u) t + (µ∆u + (µ + λ)∇divu) t − ∇P t . (ρu · ∇u) t L 2 = ρ t u · ∇u + ρu t · ∇u + ρu · ∇u t L 2 ≤ C ρ t L 6 ∇u L 3 + C u t L 6 ∇u L 3 + C u L ∞ ∇u t L 2 ≤ C + C ∇u t L 2 ,(4.32) and ∇P t L 2 = R ρ t ∇θ + ρ∇θ t + ∇ρ t θ + ∇ρθ t L 2 ≤ C ( ρ t L 6 ∇θ L 3 + ∇θ t L 2 + θ L ∞ ∇ρ t L 2 + ∇ρ L 6 θ t L 3 ) ≤ C + C ∇θ t L 2 + C ρ 1/2 θ t L 2 .sup 0≤t≤T σ ∇u t 2 L 2 + ρ tt 2 L 2 + u 2 H 3 + T 0 σ ρ 1/2 u tt 2 L 2 + ∇u t 2 H 1 dt ≤ C. (4.35) Proof. Differentiating (3.13) with respect to t leads to Multiplying (4.36) 1 by u tt and integrating the resulting equality by parts, one gets and      (2µ + λ)∇divu t − µ∇ × curlu t = ρu tt + ρ t u t + ρ t u · ∇u + ρu t · ∇u + ρu · ∇u t + ∇P t f , in Ω × [0, T ],1 2 d dt µ|curlu t | 2 + (2µ + λ)(divu t ) 2 dx + ρ|u tt | 2 dx = d dt − 1 2 ρ t |u t | 2 dx − ρ t u · ∇u · u t dx + P t divu t dx + 1 2 ρ tt |u t | 2 dx + (ρ t u · ∇u) t · u t dx − ρu t · ∇u · u tt dx − ρu · ∇u t · u tt dx − (P tt − κ(γ − 1)∆θ t ) divu t dx + κ(γ − 1) ∇θ t · ∇divu t dx d dtĨ 0 + 6 i=1Ĩ i .|Ĩ 0 | = − 1 2 ρ t |u t | 2 dx − ρ t u · ∇u · u t dx + P t divu t dx ≤ C ρ|u||u t ||∇u t |dx + C ρ t L 3 ∇u L 2 u t L 6 + C (ρθ) t L 2 ∇u t L 2 ≤ C ρ 1/2 u t L 2 ∇u t L 2 + C(1 + ρ 1/2 θ t L 2 + ρ t L 3 θ L 6 ) ∇u t L 2 ≤ C(1 + ρ 1/2 θ t L 2 ) ∇u t L 2 ,(4.38)2|Ĩ 1 | = ρ tt |u t | 2 dx ≤ C ρ tt L 2 u t 2 L 4 ≤ C ρ tt 2 L 2 + C ∇u t 4 L 2 , (4.39) |Ĩ 2 | = (ρ t u · ∇u) t · u t dx = (ρ tt u · ∇u · u t + ρ t u t · ∇u · u t + ρ t u · ∇u t · u t ) dx ≤ C ρ tt L 2 ∇u L 6 u L 6 u t L 6 + C ρ t L 2 u t 2 L 6 ∇u L 6 + C ρ t L 3 u L ∞ ∇u t L 2 u t L 6 ≤ C ρ tt 2 L 2 + C ∇u t 2 L 2 ,|Ĩ 3 | + |Ĩ 4 | = ρu t · ∇u · u tt dx + ρu · ∇u t · u tt dx ≤ C ρ 1/2 u tt L 2 ( u t L 6 ∇u L 3 + u L ∞ ∇u t L 2 ) ≤ 1 4 ρ 1/2 u tt 2 L 2 + C ∇u t 2 L 2 .P tt − κ(γ − 1)∆θ t L 2 ≤ C (u · ∇P ) t L 2 + C (P divu) t L 2 + C |∇u||∇u t | L 2 ≤ C u t L 6 ∇P L 3 + C u L ∞ ∇P t L 2 + C P t L 6 ∇u L 3 + C P L ∞ ∇u t L 2 + C ∇u L ∞ ∇u t L 2 ≤ C 1 + ∇u L ∞ + ∇ 2 θ L 2 ∇u t L 2 + C 1 + ∇θ t L 2 + ρ 1/2 θ t L 2 , which yields |Ĩ 5 | = (P tt − κ(γ − 1)∆θ t ) divu t dx ≤ P tt − κ(γ − 1)∆θ t L 2 ∇u t L 2 ≤ C 1 + ∇u L ∞ + ∇ 2 θ L 2 ∇u t 2 L 2 + C 1 + ∇θ t 2 L 2 + ρ 1/2 θ t 2 L 2 . (4.42) Next, combining Lamé's system (4.36) with Lemma 2.7, (4.4), (4.26), and (4.33) gives ∇ 2 u t L 2 ≤ C f L 2 + C ∇u t L 2 ≤ C ρu tt L 2 + C ρ t L 3 u t L 6 + C ρ t L 3 ∇u L 6 u L ∞ + C u t L 6 ∇u L 3 + C ∇u t L 2 u L ∞ + C ∇P t L 2 ≤ C ρu tt L 2 + ∇u t L 2 + ρ 1/2 θ t L 2 + ∇θ t L 2 + 1 ,(4.43) which immediately leads to |Ĩ 6 | = κ(γ − 1) ∇θ t · ∇divu t dx ≤ C ∇ 2 u t L 2 ∇θ t L 2 ≤ 1 4 ρ 1/2 u tt 2 L 2 + C 1 + ∇u t 2 L 2 + ρ 1/2 θ t 2 L 2 + ∇θ t 2 L 2 . (4.44) Putting (4.39)-(4.42) and (4.44) into (4.37) yields d dt µ|curlu t | 2 + (2µ + λ)(divu t ) 2 − 2Ĩ 0 dx + ρ|u tt | 2 dx ≤ C 1 + ∇u L ∞ + ∇u t 2 L 2 + ∇ 2 θ 2 L 2 ∇u t 2 L 2 + C 1 + ρ tt 2 L 2 + ρ 1/2 θ t 2 L 2 + ∇θ t 2 L 2 . (4.45) Furthermore, we infer from (1.4) 1 , (4.4), and (4.26) that ρ tt L 2 = div(ρu) t L 2 ≤ C ( ρ t L 6 ∇u L 3 + ∇u t L 2 + u t L 6 ∇ρ L 3 + ∇ρ t L 2 ) ≤ C + C ∇u t L 2 .L 2 + u 2 H 3 + T 0 σ ∇u t 2 H 1 dt ≤ C, which along with (4.47) gives (4.35). We complete the proof of Lemma 4.3. ρ W 2,q + T 0 ∇ 2 u p 0 W 1,q dt ≤ C,(4.∇ 2 u W 1,q ≤C ( ρu W 1,q + ∇P W 1,q + ∇u L 2 ) ≤C( ∇u L 2 + ∇(ρu) L q + ∇ 2 θ L 2 + θ∇ 2 ρ L q + |∇ρ||∇θ| L q + ∇ 2 θ L q + 1) ≤C ∇u L 2 + ∇(ρu) L q + ∇ 2 θ L q + 1 + C(1 + ∇ 2 θ L 2 ) ∇ 2 ρ L q . (4.50) Next, multiplying (4.20) by q|∂ ij ρ| q−2 ∂ ij ρ and integrating the resulting equality over Ω, we obtain after using (4.4) and (4.50) that d dt ∇ 2 ρ q L q ≤ C ∇u L ∞ ∇ 2 ρ q L q + C ∇ 2 u W 1,q ∇ 2 ρ q−1 L q ( ∇ρ L q + 1) ≤ C u H 3 ∇ 2 ρ q L q + C ∇ 2 u W 1,q ∇ 2 ρ q−1 L q ≤ C u H 3 + ∇u L 2 + ∇(ρu) L q + ∇ 2 θ L q + 1 ∇ 2 ρ q L q + 1 .∇(ρu) L q ≤ C ∇ρ L 6 u L 6q/(6−q) + C ∇u L q ≤ C u W 1,6q/(6+q) + C ∇u L q ≤ C ∇u t L q + C ∇(u · ∇u) L q + C ≤ C ∇u t (6−q)/2q L 2 ∇u t 3(q−2)/2q L 6 + C ∇u L q ∇u L ∞ + C u L ∞ ∇ 2 u L q + C ≤ Cσ −1/2 σ ∇u t 2 H 1 3(q−2)/4q + C u H 3 + C,(4.52) and ∇ 2 θ L q ≤C ∇ 2 θ (6−q)/2q L 2 ∇ 3 θ 3(q−2)/2q L 2 + C ∇ 2 θ L 2 ≤Cσ −1/2 σ ∇ 3 θ 2 L 2 3(q−2)/4q + C ∇ 2 θ L 2 , which combined with Lemma 4.1 and (4.35) shows that, for p 0 as in (4.49), T 0 ∇(ρu) p 0 L q + ∇ 2 θ p 0 L q dt ≤ C.sup 0≤t≤T σ θ t H 1 + ∇ 2 θ H 1 + u t H 2 + u W 3,q + T 0 σ 2 ∇u tt 2 L 2 dt ≤ C. (4.54) Proof. First, differentiating (4.36) with respect to t gives            ρu ttt + ρu · ∇u tt − (2µ + λ)∇divu tt + µ∇ × curlu tt = 2div(ρu)u tt + div(ρu) t u t − 2(ρu) t · ∇u t −(ρ tt u + 2ρ t u t ) · ∇u − ρu tt · ∇u − ∇P tt , in Ω × [0, T ], u tt · n = 0, curlu tt × n = 0, on ∂Ω × [0, T ]. (4.55) Multiplying (4.55) 1 by u tt and integrating the resulting equality over Ω by parts imply that 1 2 d dt ρ|u tt | 2 dx + (2µ + λ)(divu tt ) 2 + µ|curlu tt | 2 dx = −4 u i tt ρu · ∇u i tt dx − (ρu) t · (∇(u t · u tt ) + 2∇u t · u tt ) dx − (ρ tt u + 2ρ t u t ) · ∇u · u tt dx − ρu tt · ∇u · u tt dx + P tt divu tt dx 5 i=1J i . (4.56) It follows from Lemmas 4.1-4.3, (4.28), (4.30), and (4.46) that, for η ∈ (0, 1], and |J 1 | ≤ C ρ 1/2 u tt L 2 ∇u tt L 2 u L ∞ ≤ η ∇u tt 2 L 2 + C(η) ρ 1/2 u tt 2 L 2 , (4.57) |J 2 | ≤ C ( ρu t L 3 + ρ t u L 3 ) ( ∇u tt L 2 u t L 6 + u tt L 6 ∇u t L 2 ) ≤ C ρ 1/2 u t 1/2 L 2 u t 1/2 L 6 + ρ t L 6 u L 6 ∇u tt L 2 ∇u t L 2 ≤ η ∇u tt 2 L 2 + C(η) ∇u t 3 L 2 + C(η) ≤ η ∇u tt 2 L 2 + C(η)σ −3/2 , (4.58) |J 3 | ≤ C ( ρ tt L 2 u L 6 + ρ t L 2 u t L 6 ) ∇u L 6 u tt L 6 ≤ η ∇u tt 2 L 2 + C(η)σ −1 ,|J 4 | + |J 5 | ≤ C ρu tt L 2 ∇u L 3 u tt L 6 + C (ρ t θ + ρθ t ) t L 2 ∇u tt L 2 ≤ η ∇u tt 2 L 2 + C(η) ρ 1/2 u tt 2 L 2 + ρ tt θ 2 L 2 + ρ t θ t 2 L 2 + ρ 1/2 θ tt 2 L 2 ≤ η ∇u tt 2 L 2 + C(η) ρ 1/2 u tt 2 L 2 + ∇θ t 2 L 2 + ρ 1/2 θ tt 2 L 2 + σ −2 . (4.60) Substituting (4.57)-(4.60) into (4.56), we obtain after using (2.10) and choosing η suitably small that d dt ρ|u tt | 2 dx + C 4 |∇u tt | 2 dx ≤ Cσ −2 + C ρ 1/2 u tt 2 L 2 + C ∇θ t 2 L 2 + C 5 ρ 1/2 θ tt 2 L 2 . (4.61) Next, differentiating (3.57) with respect to t infers            − κ(γ−1) R ∆θ t + ρθ tt = −ρ t θ t − ρ t (u · ∇θ + (γ − 1)θdivu) − ρ (u · ∇θ + (γ − 1)θdivu) t + γ−1 R λ(divu) 2 + 2µ|D(u)| 2 t , ∇θ t · n| ∂Ω×(0,T ) = 0. (4.62) Multiplying (4.62) 1 by θ tt and integrating the resulting equality over Ω lead to κ(γ − 1) 2R ∇θ t 2 L 2 + H 0 t + ρθ 2 tt dx = 1 2 ρ tt θ 2 t + 2 (u · ∇θ + (γ − 1)θdivu) θ t dx + ρ t (u · ∇θ + (γ − 1)θdivu) t θ t dx − ρ (u · ∇θ + (γ − 1)θdivu) t θ tt dx − γ − 1 R λ(divu) 2 + 2µ|D(u)| 2 tt θ t dx 4 i=1 H i ,(4.63) where H 0 1 2 ρ t θ 2 t dx + ρ t (u · ∇θ + (γ − 1)θdivu) θ t dx − γ − 1 R λ(divu) 2 + 2µ|D(u)| 2 t θ t dx. It follows from (1.4) 1 , (4.28), (4.30), (4.46), and Lemmas 4.1-4.3 that |H 0 | ≤C ρ|u||θ t ||∇θ t |dx + C ρ t L 3 θ t L 6 ( ∇θ L 2 u L ∞ + θ L 6 ∇u L 3 ) + C ∇u L 3 ∇u t L 2 θ t L 6 ≤C( ρ 1/2 θ t L 2 + ∇θ t L 2 ) ρ 1/2 θ t L 2 + ∇u t L 2 + 1 ≤ κ(γ − 1) 4R ∇θ t 2 L 2 + Cσ −1 ,(4.64) and |H 1 | ≤ C ρ tt L 2 θ t 2 L 4 + θ t L 6 ( u · ∇θ L 3 + θdivu L 3 ) ≤ C ρ tt L 2 ρ 1/2 θ t 2 L 2 + ∇θ t 2 L 2 + σ −1 ≤ C(1 + ∇u t L 2 ) ∇θ t 2 L 2 + Cσ −3/2 .(u · ∇θ + (γ − 1)θdivu) t L 2 ≤ C ( u t L 6 ∇θ L 3 + ∇θ t L 2 + θ t L 6 ∇u L 3 + θ L ∞ ∇u t L 2 ) ≤ C ∇u t L 2 ( ∇ 2 θ L 2 + 1) + C ∇θ t L 2 + C ρ 1/2 θ t L 2 ,|H 2 | + |H 3 | ≤ C σ −1/2 ( ∇u t L 2 + 1) + ∇θ t L 2 ( ρ t L 3 θ t L 6 + ρθ tt L 2 ) ≤ 1 2 ρθ 2 tt dx + C ∇θ t 2 L 2 + Cσ −1 ∇u t 2 L 2 + Cσ −1 . (4.67) One deduces from (4.4), (4.28), (4.30), and (4.35) that |H 4 | ≤ C |∇u t | 2 + |∇u||∇u tt | |θ t |dx ≤ C ∇u t 3/2 L 2 ∇u t 1/2 L 6 + ∇u L 3 ∇u tt L 2 θ t L 6 ≤ δ ∇u tt 2 L 2 + C ∇ 2 u t 2 L 2 + C(δ)( ∇θ t 2 L 2 + σ −1 ) + Cσ −2 ∇u t 2 L 2 .κ(γ − 1) 2R ∇θ t 2 L 2 + H 0 t + 1 2 ρθ 2 tt dx ≤ δ ∇u tt 2 L 2 + C(δ)((1 + ∇u t L 2 ) ∇θ t 2 L 2 + σ −3/2 ) + C( ∇ 2 u t 2 L 2 + σ −2 ∇u t 2 L 2 ). (4.69) Finally, for C 5 as in (4.61), adding (4.69) multiplied by 2(C 5 + 1) to (4.61) and choosing δ suitably small yield that 2(C 5 + 1) κ(γ − 1) 2R ∇θ t 2 L 2 + H 0 + ρ|u tt | 2 dx t + ρθ 2 tt dx + C 4 2 |∇u tt | 2 dx ≤ C(1 + ∇u t 2 L 2 )(σ −2 + ∇θ t 2 L 2 ) + C ρ 1/2 u tt 2 L 2 + C ∇ 2 u t 2 L 2 . Multiplying this by σ 2 and integrating the resulting inequality over (0, T ), we obtain after using (4.64), (4.35), (4.26), and Grönwall's inequality that Lemma 4.6. The following estimate holds: sup 0≤t≤T σ 2 |∇θ t | 2 + ρ|u tt | 2 dx + T 0 σ 2 ρθ 2 tt + |∇u tt | 2 dxdt ≤ C,(4.σ ∇u t H 1 + ∇ 2 θ H 1 + ∇ 2 u W 1,q ≤ C.sup 0≤t≤T σ 2 ∇ 2 θ H 2 + θ t H 2 + ρ 1/2 θ tt L 2 + T 0 σ 4 ∇θ tt 2 L 2 dt ≤ C. (4.72) Proof. First, differentiating (4.62) with respect to t yields Multiplying (4.73) 1 by θ tt and integrating the resulting equality over Ω yield that                      ρθ ttt − κ(γ−1) R ∆θ tt = −ρu · ∇θ tt + 2div(ρu)θ tt − ρ tt (θ t + u · ∇θ + (γ − 1)θdivu) −2ρ t (u · ∇θ + (γ − 1)θdivu) t −ρ (u tt · ∇θ + 2u t · ∇θ t + (γ − 1)(θdivu) tt ) + γ−1 R λ(divu) 2 + 2µ|D(u)| 2 tt , ∇θ tt · n| ∂Ω×(0,T ) = 0.1 2 d dt ρ|θ tt | 2 dx + κ(γ − 1) R |∇θ tt | 2 dx = −4 θ tt ρu · ∇θ tt dx − ρ tt (θ t + u · ∇θ + (γ − 1)θdivu) θ tt dx − 2 ρ t (u · ∇θ + (γ − 1)θdivu) t θ tt dx − ρ (u tt · ∇θ + 2u t · ∇θ t + (γ − 1)(θdivu) tt ) θ tt dx + γ − 1 R λ(divu) 2 + 2µ|D(u)| 2 tt θ tt dx 5 i=1 K i .σ 4 |K 1 | ≤ Cσ 4 ρ 1/2 θ tt L 2 ∇θ tt L 2 u L ∞ ≤ δσ 4 ∇θ tt 2 L 2 + C(δ)σ 4 ρ 1/2 θ tt 2 L 2 , (4.75) σ 4 |K 2 | ≤ Cσ 4 ρ tt L 2 θ tt L 6 ( θ t H 1 + ∇θ L 3 + ∇u L 6 θ L 6 ) ≤ Cσ 2 ( ∇θ tt L 2 + ρ 1/2 θ tt L 2 ) ≤ δσ 4 ∇θ tt 2 L 2 + C(δ)(σ 4 ρ 1/2 θ tt 2 L 2 + 1), (4.76) σ 4 |K 4 | ≤ Cσ 4 θ tt L 6 ( ∇θ L 3 ρu tt L 2 + ∇θ t L 2 u t L 3 ) + Cσ 4 θ tt L 6 ( ∇u L 3 ρθ tt L 2 + ∇u t L 2 θ t L 3 ) + Cσ 4 θ L ∞ ρθ tt L 2 ∇u tt L 2 ≤ δσ 4 ∇θ tt 2 L 2 + C(δ) σ 4 ρ 1/2 θ tt 2 L 2 + σ 3 ∇u tt 2 L 2 + C(δ), (4.77) σ 4 |K 5 | ≤ Cσ 4 θ tt L 6 ∇u t 3/2 L 2 ∇u t 1/2 L 6 + ∇u L 3 ∇u tt L 2 ≤ δσ 4 ∇θ tt 2 L 2 + C(δ)σ 4 ρ 1/2 θ tt 2 L 2 + ∇u tt 2 L 2 + C(δ),(4.78) and σ 4 |K 3 | ≤ Cσ 4 ρ t L 3 θ tt L 6 σ −1/2 ∇u t L 2 + ρ 1/2 θ t L 2 + ∇θ t L 2 ≤ δσ 4 ∇θ tt 2 L 2 + Cσ 4 ρ 1/2 θ tt 2 L 2 + C(δ),(4.79) where in the last inequality we have used (4.66). Then, multiplying (4.74) by σ 4 , substituting (4.75)-(4.79) into the resulting equality and choosing δ suitably small, one obtains d dt σ 4 ρ|θ tt | 2 dx + κ(γ − 1) R σ 4 |∇θ tt | 2 dx ≤ Cσ 2 ρ 1/2 θ tt 2 L 2 + ∇u tt 2 L 2 + C, which together with (4.70) gives sup 0≤t≤T σ 4 ρ|θ tt | 2 dx + T 0 σ 4 |∇θ tt | 2 dxdt ≤ C. (4.80) Finally, applying the standard L 2 -estimate to (4.62), one obtains after using Lemmas 4.1-4.3, 4.5, (4.28), and (4.80) that sup 0≤t≤T σ 2 ∇ 2 θ t L 2 ≤ C sup 0≤t≤T σ 2 ( ρθ tt L 2 + ρ t L 3 θ t L 6 + ρ t L 6 ( ∇θ L 3 + θ L 6 ∇u L 6 )) + C sup 0≤t≤T σ 2 ρ 1/2 θ t L 2 + ∇θ t L 2 + (1 + ∇ 2 θ L 2 ) ∇u t L 2 + ∇u t L 6 ≤ C. ∇ 2 θ H 2 ≤ C ρθ t H 2 + ρu · ∇θ H 2 + ρθdivu H 2 + |∇u| 2 H 2 ≤ C ( ρ H 2 θ t H 2 + ρ H 2 u H 2 ∇θ H 2 ) + C ρ H 2 θ H 2 divu H 2 + C ∇u 2 H 2 + C ≤ Cσ −1 + C ∇ 3 θ L 2 + C θ t H 2 . Combining this with (4.54), (4.81), and (4.80) shows (4.72). The proof of Lemma 4.6 is completed. Proof of Theorems 1.1 and 1.2 With all the a priori estimates in Sections 3 and 4 at hand, we are ready to prove the main results of this paper in this section. Thus, Lemma 2.1 implies that there exists some T * * > T * , such that (3.5) holds for T = T * * , which contradicts ( θ t (·, 0) −u 0 · ∇θ 0 + γ − 1 R ρ −1 0 κ∆θ 0 − Rρ 0 θ 0 divu 0 + λ(divu 0 ) 2 + 2µ|D(u 0 )| 2 , which along with (2.1) gives θ t (·, 0) L 2 ≤C. ∇ 2 u H 2 ≤C ( ρu H 2 + ∇P H 2 + ∇u L 2 ) ≤C ( ρ H 2 u t H 2 + ρ H 2 u H 2 ∇u H 2 ) +C ( ∇ρ H 2 θ H 2 + ρ H 2 ∇θ H 2 + 1) ≤C(1 + ∇ 2 u t L 2 + ∇ 3 ρ L 2 + ∇ 3 θ L 2 ), which along with some standard calculations leads to ∇ 3 ρ L 2 t ≤C |∇ 3 u||∇ρ| L 2 + |∇ 2 u||∇ 2 ρ| L 2 + |∇u||∇ 3 ρ| L 2 + ∇ 4 u L 2 ≤C ∇ 3 u L 2 ∇ρ H 2 + ∇ 2 u L 3 ∇ 2 ρ L 6 + ∇u L ∞ ∇ 3 ρ L 2 + ∇ 4 u L 2 ≤C(1 + ∇ 3 ρ L 2 + ∇ 2 u t 2 L 2 + ∇ 3 θ 2 L 2 ), where we have used (5.9) and Lemma 4.1. Combining this with (5.10) and Grönwall's inequality yields sup 0≤t≤T ∇ 3 ρ L 2 ≤C, which together with (4.4) gives (5.3). The proof of Proposition 5.1 is completed. With Proposition 5.1 at hand, we are now in a position to prove Theorem 1.1. Proof of Theorem 1.1. Let (ρ 0 , u 0 , θ 0 ) satisfying (1.9)-(1.11) be the initial data in Theorem 1.1. Assume that C 0 satisfies (1.12) with ε ε 0 /2,(5.11) where ε 0 is given in Proposition 3.1. First, we construct the approximate initial data (ρ m,η 0 , u m,η 0 , θ m,η 0 ) as follows. For constants m ∈ Z + , η ∈ (0, η 0 ) , η 0 min 1, 1 2 (ρ − sup x∈Ω ρ 0 (x)) ,(5.12) we define Therefore, there exists an η 1 ∈ (0, η 0 ) such that, for any η ∈ (0, η 1 ), we can find some m 2 (η) ≥ m 1 (η) such that C m,η ρ m,η 0 = ρ m 0 + η, u m,η 0 = u m 0 1 + η , θ m,η 0 = θ m 0 + η 1 + 2η , where ρ m 0 satisfies 0 ≤ ρ m 0 ∈ C ∞ , lim m→∞ ρ m 0 − ρ 0 W 2,q = 0,0 ≤ C 0 + ε 0 /2 ≤ ε 0 ,(5.15) provided that 0 < η < η 1 , m ≥ m 2 (η). (5.16) We assume that m, η satisfy (5.16). Proposition 5.1 together with (5.15) and (5.13) thus yields that there exists a smooth solution (ρ m,η , u m,η , θ m,η ) of problem (1.4)-(1.6) with initial data (ρ m,η 0 , u m,η 0 , θ m,η 0 ) on Ω × (0, T ] for all T > 0. Moreover, one has (1.13), (3.6), (3.7), (3.10), (3.105), and (3.106) with (ρ, u, θ) being replaced by (ρ m,η , u m,η , θ m,η ). Next, for the initial data (ρ m,η 0 , u m,η 0 , θ m,η 0 ), the functiong in (4.1) is g (ρ m,η 0 ) −1/2 (−µ∆u m,η 0 − (µ + λ)∇divu m,η 0 + R∇(ρ m,η 0 θ m,η 0 )) = (ρ m,η 0 ) −1/2 √ ρ 0 g + µ(ρ m,η 0 ) −1/2 ∆(u 0 − u m,η 0 ) + (µ + λ)(ρ m,η 0 ) −1/2 ∇div(u 0 − u m,η 0 ) + R(ρ m,η 0 ) −1/2 ∇(ρ m,η 0 θ m,η 0 − ρ 0 θ 0 ), (5.17) where in the second equality we have used (1.11). Since g ∈ L 2 , one deduces from (5.17), (5.13), (5.14), and (1.9) that for any η ∈ (0, η 1 ), there exist some m 3 (η) ≥ m 2 (η) and a positive constant C independent of m and η such that g L 2 ≤ g L 2 + Cη −1/2 δ(m) + Cη 1/2 , We thus obtain from (5.18) and (5.19) that there exists some positive constant C independent of m and η such that (3.6). Hence, (ρ, u, θ) satisfying (1.13) and (1.14) refers to [14] for the detailed proof. Moreover, one deduces from Proposition 3.1 that the desired exponential decay property (1.15). g L 2 ≤ g L 2 + C,(5. Finally, the proof of the uniqueness of (ρ, u, θ) is similar to that of [7, Theorem 1] and will be omitted here for simplicity. The proof of Theorem 1.1 is completed. Proof of Theorem 1.2. We will prove Theorem 1.2 in two steps. Step 1. Construction of approximate solutions. Assume (ρ 0 , u 0 , θ 0 ) satisfying (1.10) and (1.19) is the initial data in Theorem 1.2 and C 0 satisfies (1.12) with ε as in (5.11). For j m −1 (x) being the standard mollifying kernel of width m −1 , we construct owing to (1.10) and (1.12). Now, we claim that the initial normĈ m,η 0 for (ρ m,η 0 ,û m,η 0 ,θ m,η 0 ), i.e., the right hand side of (1.8) with (ρ 0 , u 0 , θ 0 ) replaced by (ρ m,η 0 ,û m,η 0 ,θ m,η 0 ), satisfies lim η→0 lim m→∞Ĉ m,η 0 ≤ C 0 ,(5.24) which leads to that there exists anη ∈ (0, η 0 ) such that, for any η ∈ (0,η), there exists somem(η) ≥ m(η) such thatĈ m,η 0 = (ρ 0 + η) ρ 0 θ 0 + η ρ 0 + η − log ρ 0 θ 0 + η ρ 0 + η − 1 dx = (ρ 0 θ 0 − ρ 0 + (ρ 0 + η) log(ρ 0 + η)) dx − (ρ 0 log(ρ 0 θ 0 + η) + η log(ρ 0 θ 0 + η)) dx ≤ (ρ 0 θ 0 − ρ 0 + (ρ 0 + η) log(ρ 0 + η)) dx − (ρ 0 log(ρ 0 θ 0 ) + η log η) dx → ρ 0 (θ 0 − log θ 0 − 1) dx, as η → 0. ≤ C 0 + ε 0 /2 ≤ ε 0 ,(5. It thus gives (5.27). Step 2. Compactness results. With the approximate solutions (ρ m,η ,û m,η ,θ m,η ) obtained in the previous step at hand, we can derive the global existence of weak solutions by passing to the limit first m → ∞, then η → 0. Since the two steps are similar, we will only sketch the arguments for m → ∞. For any fixed η ∈ (0,η), we simply denote (ρ m,η ,û m,η ,θ m,η ) by (ρ m , u m , θ m ). Then the combination of Aubin-Lions Lemma with (3.6), (3.10), (3.11), (3.106), and Lemma 2.5 yields that there exists some appropriate subsequence m j → ∞ of m → ∞ such that, for any 0 < τ < T < ∞, p ∈ [1, ∞), and p ∈ [1, 6), u m j ⇀ u weakly star in L ∞ (0, T ; H 1 ), (5.28) θ m j ⇀ θ weakly in L 2 (0, T ; H 1 ), [14]. The proof of Theorem 1.2 is finished. ρ ∈ C([0, T ]; W 2,q ), u ∈ C([0, T ]; W 1,p ) ∩ L ∞ (0, T ; H 2 ) ∩ L ∞ (τ, T ; W 3,q ), θ ∈ L ∞ (τ, T ; H 4 ) ∩ C([τ, T ]; W 3,p ), u t ∈ L 2 (0, T ; H 1 ) ∩ L ∞ (τ, T ; H 2 ) ∩ H 1 (τ, T ; H 1 ), θ t ∈ L ∞ (τ, T ; H 2 ) ∩ H 1 (τ, T ; H 1 ),(1.14) ( 2 . 24 ) 224Lemma 2.9. [11, Theorem III.3.1] There exists a linear operator B = [B 1 , B 2 , B 3 ] enjoying the properties: Lemma 3 . 1 . 31Under the conditions of Proposition 3.1, there exists a positive constant C depending only on µ, R, andρ such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying Lemma 3. 2 . 2Under the conditions of Proposition 3.1, there exist positive constants K and ε 1 both depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying substituting (3.27), (3.28), and (3.30) into (3.21), one obtains after choosing δ suitably small that Lemma 3 . 3 . 33Under the conditions of Proposition 3.1, let (ρ, u, θ) be a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying (3.5) with K as in Lemma 3.2. submitting (3.43), (3.49), and (3.50) into (3.41), one obtains after using (3.45), (3.46), (3.5), (3.51), and (3.52) that 2.20) to (3.53) and choosing δ small enough infer (3.36) directly. 3.55) and (3.58)-(3.60) into (3.54), we obtain (3.37) after using (1.2) and choosing δ suitably small. The proof of Lemma 3.3 is completed. With the estimates (3.35)-(3.37) (see Lemma 3.3) at hand, we are now in a position to prove the following estimate on A 3 (T ). Lemma 3 . 4 . 34Under the conditions of Proposition 3.1, there exists a positive constant ε 2 depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1 (3.11), (3.22), (3.24), and (3.5). Next, it follows from (3.11), (3.22), (3.5), and (3.24) that integrating (3.70) over (0, T ), one obtains after using (3.62), (3.65), (3.71), (3.72), and (3.5 Lemma 3. 5 . 5Under the conditions of Proposition 3.1, there exist positive constants C and ε 3,1 depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying (3.5) with K as in Lemma 3.2, the following estimates hold: sup 0≤t≤σ(T ) θ L ∞ dt ≤ C(ρ, M ). (3.78) Combining this with (3.77), (3.16), and Grönwall inequality implies (3.73) directly. Finally, it remains to prove (3.78). Taking m 86) provided C 0 ≤ ε 3,1 min 1, (2C(ρ, M )) −5/2 . (3.87) Lemma 3. 6 . 6Under the conditions of Proposition 3.1, there exists a positive constant ε 3 depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying (3.5) with K as in Lemma 3.2, the following estimate holds: C 0 0(3.73) and(3.72). Thus, one deduces from (3.97), (2.12), and (3.11) thatA 2 (T ) ≤ C(ρ,θ, ≤ ε 3 min ε 3,1 , (C(ρ,θ, M )) −24 , Lemma 3 . 7 . 37Under the conditions of Proposition 3.1, there exists a positive constant ε 4 depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying (3.5) with K as in Lemma 3.2, the following estimate holds: we thus deduce from (3.104), (3.82), (3.101)-(3.103), (2.23), and (3.11) that ρ ≤ρ + 1 + C g L 1 (0,σ(T )) + g L 2 (σ(T ),T ) ≤ρ + 1 + C(ρ, gives (3.100) providedC 0 ≤ ε 4 min 1, ρ − 2 2C(ρ, M ) 48 . Lemma 3 . 8 . 38Under the conditions of Proposition 3.1, there exists a positive constant C depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying (3.5) with K as in Lemma 3.2, the following estimate holds: 4 L 4 + σ ∇θ 2 H 1 + ρ − 1 2 L 2 dt ≤ C(ρ,M ), (3.112) which along with (3.5), (3.79), (3.80), (3.110), (3.111), and (3. 106) is derived from (3.111)-(3.115) immediately. The proof of Lemma 3.8 is finished.Finally, we end this section by establishing the exponential decay-in-time for the classical solutions. Lemma 3 . 9 . 39Under the conditions of Proposition 3.1, there exist positive constants ε 0 , C * , α, and θ ∞ depending only on µ, λ, κ, R, γ,ρ,θ, Ω, and M such that if (ρ, u, θ) is a smooth solution to the problem (1.4)-(1.6) on Ω × (0, T ] satisfying (3.5) with K as in Lemma 3.2, (3.7) holds for any t ≥ 1, provided C 0 ≤ ε 0 . Proof. First, it follows from (3.91), (3.5), (3.11), (3.16), (3.29), (3.22), (3.81), and (3.106) that for any t ≥ 1, ) t + div(ρu ⊗ u)= µ∆u + (µ + λ)∇(divu) − ∇(ρ(Rθ − P )) − P ∇(ρ − 1),multiplying this by B[ρ − 1] and using Lemma 2.9, (3.5), and(3.22), one gets that for any t ≥ 1, 128) which together with (3.126), (3.127), and (2.12) gives for any 1 ≤ t ≤ T < ∞, choosing m = 0 in (3.63), it follows from (3.67), (3.68), and (3.5) that for any t ≥ 1, 3.64). Multiplying this by e αt along with (3.39), (3.64), (3.65), (3.44), (3.125), (3.126), (3.129), (2.13), and (3.5) yields that for any 1 ≤ t ≤ T < ∞, sup 1≤t≤T 133) where we have used (3.22), (3.130), and (3.132). Therefore, the combination of (3.125), (3.129)-(3.131), (3.133), (2.18), and (3.23) concludes (3.124) and finishs the proof of Lemma 3.9. ), and (3.67). Taking into account the compatibility condition (1.11), we can define √ ρu(x, t = 0) −g, which along with (3.39), (3.66), and (4.2) yields that with (3.5), (4.23), (4.24), (4.17)-(4.19), and (4.3) yields (4.4). The proof of Lemma 4.1 is finished. Lemma 4. 2 . 2The following estimates hold: H −1 dt ≤ C, which combined with (4.34) implies (4.27). The proof of Lemma 4.2 is completed. Lemma 4. 3 . 3The following estimate holds: u t · n = 0, curlu t × n = 0, on ∂Ω × [0, T ]. i (i = 0, · · · , 6) can be estimated as follows: First, it follows from simple calculations, (1.4) 1 , (4.26), (4.4), (4.3), and (4.28) that by virtue of (3.25), (4.26), (4.33), and Lemma 4.1, it holds 4.45) by σ and integrating the resulting inequality over (0, T ), one thus deduces from (2.10), (4.3), (4.4), (4.28), (4.38), (4.46), and Grönwall' Lemma 4. 4 . 4For q ∈ (3, 6) as in Theorem 1.1, it holds that sup 0≤t≤T ( ρ L q ≤ C, which together with Lemma 4.1, (4.53), and (4.50) gives (4.48). We finish the proof of Lemma 4.4. Lemma 4 . 5 . 45For q ∈ (3, 6) as in Theorem 1.1, the following estimate holds: derive (4.54) from (4.70), (4.71), (4.28), (4.30), and (4.4). The proof of Lemma 4.5 is completed. Proposition 5 . 1 . 51For given numbers M > 0 (not necessarily small),ρ > 2, and θ > 1, assume that (ρ 0 , u 0 , θ 0 ) satisfies (2.1), (3.4), and (3.8). Then there exists a unique classical solution (ρ, u, θ) of problem (1.4)-(1.6) in Ω × (0, ∞) satisfying (2.3)-(2.5) with T 0 replaced by any T ∈ (0, ∞). Moreover, (3.6), (3.10), (3.105), and (3.106) hold for any T ∈ (0, ∞) and (3.7) holds for any t ≥ 1. Proof. First, by the standard local existence result (Lemma 2.1), there exists a T 0 > 0 which may depend on inf x∈Ω ρ 0 (x), such that the problem (1.4)-(1.6) with initial data (ρ 0 , u 0 , θ 0 ) has a unique classical solution (ρ, u, θ) on Ω × (0, T 0 ] satisfyinng (2.2)-(2.5). It follows from (3.1)-(3.4) and (3.8) thatA 1 (0) ≤ M 2 , A 2 (0) A 3 (0) = 0, ρ 0 <ρ, θ 0 ≤θ,which implies there exists a T 1 ∈ (0, T 0 ] such that (3.5) holds for T = T 1 . We setT * = sup T sup t∈[0,T ] (ρ, u, θ) H 3 < ∞ , and T * = sup{T ≤ T * | (3.5) holds}. (5.1)Then T * ≥ T * ≥ T 1 > 0. Next, we claim that T * = ∞.(5.2) Otherwise, T * < ∞. Proposition 3.1 shows (3.6) holds for all 0 < T < T * , which together with (3.8) yields Lemmas 4.1-4.6 still hold for all 0 < T < T * . Note here that all constants C in Lemmas 4.1-4.6 depend on T * and inf x∈Ω ρ 0 (x), and are in fact independent of T . Then, we claim that there exists a positive constantC which may depend on T * and inf x∈Ω ρ 0 (x) such that, for all 0 < T < T * , sup 0≤t≤T ρ H 3 ≤C, 2 and (2.2), we can defineu t (·, 0) −u 0 · ∇u 0 + ρ −1 0 (µ∆u 0 + (µ + λ)∇divu 0 − R∇(ρ 0 θ 0 )) ,which along with (2.1) gives ∇u t (·, 0) L 2 ≤C. u m 0 0is the unique smooth solution to the following elliptic equation:∆u m 0 = ∆ũ m 0 , in Ω, u m 0 · n = 0, curlu m 0 × n = 0, on ∂Ω, withũ m 0 ∈ C ∞ satisfying lim m→∞ ũ m 0 − u 0 H 2 = 0, and θ m 0 satisfying Ω θ m 0 dx = Ω θ 0 dx isthe unique smooth solution to the following Poisson equation: ∆θ m 0 = ∆θ m 0 − ∆θ m 0 , in Ω, ∇θ m 0 · n = 0, on ∂Ω, with 0 ≤θ m 0 ∈ C ∞ satisfying lim m→∞ θ m 0 − θ 0 H 2 = 0.Then for any η ∈ (0, η 0 ), there exists m 1 (η) ≥ 1 such that for m ≥ m 1 ≤ δ(m) → 0 as m → ∞. Hence, for any η ∈ (0, η 1 ), there exists some m 4 (η) ≥ m 3 (η) such that for any m ≥ m 4 (η), δ(m) < η.(5.19) we assume that m, η satisfy (5.21). It thus follows from (5.13)-(5.15), (5.20), Proposition 3.1, and Lemmas 3.8, 4.1-4.6 that for any T > 0, there exists some positive constant C independent of m and η such that (1.13), (3.6), (3.10), (3.105), (3.106), (4.3), (4.4), (4.26), (4.27), (4.48), (4.54), and (4.72) hold for (ρ m,η , u m,η , θ m,η ). Then passing to the limit first m → ∞, then η → 0, together with standard arguments yields that there exists a solution (ρ, u, θ) of the problem (1.4)-(1.6) on Ω×(0, T ] for all T > 0, such that the solution (ρ, u, θ) satisfies (1.13), (3.10), (3.105), (3.106), (4.3), (4.4), (4.26), (4.27), (4.48), (4.54), (4.72), and the estimates of A i (T ) (i = 1, 2, 3) in θ 0 1 0Ωm ) * j m −1 + η (ρ 0 1 Ωm ) * j m −1 + η ,where Ω m = {x ∈ Ω|dist(x, ∂Ω) > 2/m} and u m 0 satisfiesu m 0 ∈ C ∞ ∩ H 1 ω and lim m→∞ u m 0 − u 0 H 1 = 0.Then for any η ∈ (0, η 0 ) with η 0 as in (5.12), there exists m(η) > 1 such that for m ≥ m(η), the approximate initial data (ρ 0 that there exists a classical solution (ρ m,η ,û m,η ,θ m,η ) of problem (1.4)-(1.6) with initial data (ρ m,η 0 ,û m,η 0 ,θ m,η 0 ) on Ω×(0, T ] for all T > 0. Furthermore, (ρ m,η ,û m,η ,θ m,η ) satisfies (1.13), (3.6), (3.10), (3.11), (3.105), (3.106), and (3.7) respectively for any T > 0 and t ≥ 1 with (ρ, u, θ) replaced by (ρ m,η ,û m,η ,θ m,η ).It remains to prove (5.24). Indeed, we just need dx ≤ ρ 0 (θ 0 − log θ 0 − 1) dx,(5.27) since the other terms in (5.24) can be proved in a similar and even simpler way.(j m −1 * (ρ 0 (θ 0 − 1)1 Ωm )) 2 j m −1 * (ρ 0 1 Ωm ) m −1 * (ρ 0 (θ 0 − 1)1 Ωm ) + j m −1 * (ρ 0 1 Ωm ) + η dα ∈ 0,ρη −2 (j m −1 * (ρ 0 (θ 0 − 1)1 Ωm )) 2 ,which combined with Lebesgue's dominated convergence theorem yields 29) ρ m j → ρ in C([0, T ]; L p -weak) ∩ C([0, T ]; H −1 ),(5.30)ρ m j u m j → ρu, ρ m j θ m j → ρθ in C([0, T ]; L 2 -weak) ∩ C([0, T ]; H −1 ), (5.31) ρ m j |u m j | 2 → ρ|u| 2 in C([0, T ]; L 3 -weak) ∩ C([0, T ]; H −1 ), (5.32) G m j → G, curlu m j → curlu in C([τ, T ]; H 1 -weak) ∩ C([τ, T ]; Lp), (5.33) u m j → u in C([τ, T ]; W 1,6 -weak) ∩ C(Ω × [τ, T ]),(5.34)andθ m j → θ in C([τ, T ]; H 2 -weak) ∩ C([τ, T ]; W 1,p ),(5.35)referring to[14] for the detailed proof. Now we consider the approximate solutions(ρ m j , u m j , θ m j ) in the weak forms, i.e. (1.16)-(1.18), then take appropriate limits. Standard arguments as well as (5.23) and (5.28)-(5.35) thus conclude that the limit (ρ, u, θ) is a weak solution of (1.1) (1.5) (1.6) in the sense of Definition 1.1 and satisfies (1.20)-(1.23) and the exponential decay property (1.15). Moreover, we obtain the estimates (1.24)-(1.26) with the aid of (3.6), (3.10), (3.106), and (5.28)-(5.35). Finally, (1.27) shall be obtained by adopting the same way as in 1.18)Then we state our second main result as follows:Theorem 1.2. Under the conditions of Theorem 1.1 except (1.11), where the condition (1.9) is replaced by u 0 ∈ H 1 ω , (1.19) assume further that C 0 as in (1.8) satisfies (1.12) with ε as in Theorem 1.1. Then there exists a global weak solution (ρ, u, E = 1 2 |u| 2 + R γ−1 θ) to the problem (1.1) (1.5) (1.6) satisfying Remark 1.2. It seems that Theorem 1.1, which extends the global existence result of the barotropic flows studied in[6] to the full compressible Navier-Stokes system, is the first result concerning the global existence of classical solutions with initial vacuum to (1.1) in general bounded domains. Although its energy is small, the oscillations could be arbitrarily large.29) which together with (1.4) 1 and (1.28) shows ρ t ∈ C(Ω × [τ, T ]). (1.30) Analogously, we have (u t , θ t ) ∈ C(Ω × [τ, T ]), which along with (1.28)-(1.30) arrives at the solution (ρ, u, θ) obtained in Theorem 1.1 is a classical one to the problem (1.4)-(1.6) in Ω × (0, ∞). Remark 1.3. To obtain the global existence and uniqueness of classical solutions with vacuum, we only need the compatibility condition on the velocity (1.11) as in with G defined in (1.39). For m ≥ 0, operating σ muj [∂/∂t + div(u·)] to (3.40) j and integrating the resulting equality over Ω by parts lead to39) Proof. 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Speech separation is a problem in the field of speech processing that has been studied in full swing recently. However, there has not been much work studying a multi-accent speech separation scenario. Unseen speakers with new accents and noise aroused the domain mismatch problem which cannot be easily solved by conventional joint training methods. Thus, we applied MAML and FOMAML to tackle this problem and obtained higher average Si-SNRi values than joint training on almost all the unseen accents. This proved that these two methods do have the ability to generate well-trained parameters for adapting to speech mixtures of new speakers and accents. Furthermore, we found out that FOMAML obtains similar performance compared to MAML while saving a lot of time.

10.18653/v1/2021.metanlp-1.7

2106.11713

bb8b5ed11fbbe5f4eb925a710a5c9c5fce20d791

Multi-accent Speech Separation with One Shot Learning MetaNLP 2021. August 5, 2021 Kuan Po Huang National Taiwan University Yuan-Kuei Wu National Taiwan University Hung-Yi Lee [email protected]* National Taiwan University Multi-accent Speech Separation with One Shot Learning Proceedings of the 1st Workshop on Meta Learning and Its Applications to Natural Language Processing the 1st Workshop on Meta Learning and Its Applications to Natural Language ProcessingBangkok, ThailandMetaNLP 2021. August 5, 202159 1 Graduate Institute of Computer Science and Information Engineering 23 Graduate Institute of Communication Engineering Speech separation is a problem in the field of speech processing that has been studied in full swing recently. However, there has not been much work studying a multi-accent speech separation scenario. Unseen speakers with new accents and noise aroused the domain mismatch problem which cannot be easily solved by conventional joint training methods. Thus, we applied MAML and FOMAML to tackle this problem and obtained higher average Si-SNRi values than joint training on almost all the unseen accents. This proved that these two methods do have the ability to generate well-trained parameters for adapting to speech mixtures of new speakers and accents. Furthermore, we found out that FOMAML obtains similar performance compared to MAML while saving a lot of time. Introduction Speech separation has been a well-known task to solve in the speech processing field. Many model architectures mentioned in Section 2 have been proposed and achieved high performance. This suggests that deep learning based methods are suitable for the speech separation task. Despite having promising results, the generalizability of these models is still questionable. The performance of switching to different datasets or environments is not guaranteed. A straightforward solution is to exhaustively collect data under all kinds of environment settings and train a model with these data jointly. Although this may sound reasonable, it is difficult to always consider every situation during training. To make sure that models can be quickly adapted to mixtures spoken by new speakers with not many samples, metalearning comes to the rescue. Meta-learning has * The two first authors made equal contributions. been widely applied on different speech tasks, especially on speech recognition mentioned in Section 2. Nonetheless, there is not much work that applied meta-learning on the speech separation task. In our previous work, (Wu et al., 2020), we first proposed to solve the speech separation problem with meta-learning. Their setting is viewing utterance mixtures of two different speakers as a meta task. These speakers have the same accents. However, we hope that a speech separation model can have the ability to adapt to mixtures with accents never seen before. Thus, besides the setting of two different speakers forming a meta task, we also added a setting that meta tasks with speakers of same accents form an accent task set. Section 4 and 5.1 describe more about the dataset and task construction procedure. Our contributions are listed below: • To our best knowledge, we are the first to conduct speech separation experiments on a multi-accent dataset. • We applied meta-learning to help improve the multi-accent speech recognition task. The remaining sections of this paper are organized as follows. In Section 2, we give a brief overview of existing works related to speech separation and meta-learning. In Section 3, we elaborate the problem formulation of speech separation in detail. In Section 4, we list out the two phases of MAML, including the meta training phase and meta testing phase. Additionally, we show how FO-MAML is modified from MAML. The experimental setup, dataset, and model we used are presented in Section 5. Finally, results and conclusions are given in Section 6 and 7. Figure 1: Illustration of joint training and meta-learning for multi-accent speech separation. The oval area is the accent task sets. Each accent task set contains multiple meta tasks. The solid lines are the pretraining process, joint training on the left, and meta-learning on the right. The dashed lines represent the adaptation paths from parameters θ to the unseen accents of unseen speakers. This figure is modified from Gu et al. (2018) and our previous work Wu et al. (2020). Related Work Speech Separation End-to-end separation models have shown great success in separating speech mixtures of the WSJ0-2mix dataset designed by (Hershey et al., 2016) which is generated from the WSJ0 corpus (Paul and Baker, 1992). (Luo and Mesgarani, 2018) came up with a time-domain audio separation network (TasNet) that takes waveforms as input to alleviate the separation model from dealing with time-frequency representations. They further proposed convolutional TasNet (Luo and Mesgarani, 2019) which substitutes the LSTM layers in TasNet with convolutional layers. This overcame the problem of long temporal dependencies of LSTM and reduced the model size. Before long, they came up will the Dual-path RNN model, which used intra-and inter-blocks to capture local and global information dependencies within the speech mixtures. (Nachmani et al., 2020) utilized the idea of Dual-path RNN and added a speaker identity loss to improve performance on separating mixtures with an unknown number of speakers. (Tzinis et al., 2020) proposed to use a separator constructed with U-ConvBlocks which can not only reduce the number of layers while still having high performance but also require less computational resources and time. This helped the model to more likely be used in real-time speech separation. (Zeghidour and Grangier, 2020) integrated speaker identity information into the separating process, and obtained state-of-the-art performance. Meta-learning Meta-learning has recently become a trend when it comes to solving multitask problems. This training method has been widely applied in the computer vision field, for instance, (Vinyals et al., 2016;Rusu et al., 2018;Sun et al., 2019). Meta-learning is also used in the natural language processing field. (Gu et al., 2018) used MAML (Finn et al., 2017) for low-resource neural machine translation (NMT). Moreover, in the speech processing domain, some speech-related problems are solved with metalearning, too. (Winata et al., 2020) applied metatransfer learning on code-switched speech recognition. (Xiao et al., 2020; applied meta-learning to solve the multilingual lowresource speech recognition problem. (Winata et al., 2019) also used MAML to adapt models to unseen accents on speech recognition. (Indurthi et al., 2019) adopted meta-learning algorithms to perform speech translation on speech-transcript paired low-resource data. (Chen et al., 2021) came up with some improvements of meta-learning to help the speaker verification task. Speech Separation In this work, we perform single channel speech separation. Given a mixture x = C c=1 s c(1) where C is the number of speakers in mixture x ∈ R T and s c ∈ R T are the ground truth sources. For speech separation, the goal is to estimate C sources {ŝ 1 , · · · ,ŝ C } ∈ R T such that the estimates sources are as similar as the ground truth sources. The model we used in this work is Conv-TasNet (Luo and Mesgarani, 2019). In their work, the similarity of the estimated sources and ground truth sources are measured by scaleinvariant signal-to-noise ratio (Si-SNR) shown in Eq. (4): s proj = s ·ŝ s 2 s (2) error =ŝ − s proj(3) Si-SNR = 10 log 10 s proj 2 error 2 (4) The Conv-TasNet model is a mask-based model which consists of an encoder, separator, and decoder. The encoder encodes the mixture x to a latent space as shown in Eq.(5). x enc = enc(x)(5) x enc ∈ R H×T is the encoder output, where H is the dimension of the latent space and T is the length of x enc . The separator then calculates C masks m i ∈ R H×T , i ∈ {1, · · · , C} based on x enc shown in Eq. (6). m i = sep(x enc )(6) The masks are then multiplied with the encoder output, forming separated features d i shown in Eq. (7), d i = x enc m i(7) where is the element-wise multiplication. The separated features d i can be viewed as source representations, and are further input to a decoder to estimate separated sources shown in Eq. (8). s i = dec(d i )(8) At this point, before measuring the estimated sources with Si-SNR, there is a label permutation problem. An align between {ŝ 1 , · · · ,ŝ C } and {s 1 , · · · , s C } needs to be decided. We used the utterance-level permutation invariant training(uPIT) method described in (Kolbaek et al., 2017) to solve this problem. MAML The procedure of MAML (Finn et al., 2017) is stated as follows. Given a set of multi-accent tasks T = {{T i 1 } tq 1 i=1 , · · · , {T i K } tq K i=1 }, where K is the number of accents. T k = {T i k } tq k i=1 is the accent task set containing tasks only with the k th accent and tq k denotes the task quantity of the k th accent task set. The set of tasks T is split into the source task set T source and the target task set T target . The model denoted as f , will be trained on the source task set T source in the hope of having the ability to quickly adapt to the target task set T target . Meta Training Phase During the meta training phase, the MAML algorithm aims to find initialized parameters θ that can further be quickly adapted to new tasks. Moreover, these initialized parameters should be sensitive to the difference between two different tasks, such that adaptation of the initialized parameters can significantly improve the performance on new tasks sampled from the source task set T source . This is achieved by the inner loop and outer loop optimization. A batch of tasks τ source = {τ 1 , · · · , τ b } is sampled from T proportional to the task quantity of every accent task set, e.g., for an accent task set T k , the larger tq k is, the more likely a task is to be sampled from it. Each task in τ source is further split into a support set τ sup and a query set τ qry . The support set is used to adapt the model parameters by performing a one-step gradient decent, which is known as the inner loop shown in Eq.(9). θ j ← θ − α∇ θ L τ sup j (f θ )(9) where α is the learning rate. The goal of the inner loop is to minimize the loss of τ sup j with respect to f θ . More concisely, θ j = arg min θ L τ sup j (f θ )(10) At this point, the sum of the query loss of each query set in τ source is calculated by L qry = b j=1 L τ qry j (f θ j )(11) The goal of the meta training phase is to minimize the total loss of the query sets. This is also performed by a one-step gradient decent, known as the outer loop shown in Eq. (12). θ ← θ − β∇ θ L qry(12) Meta Testing Phase During the meta testing phase, we perform a procedure (see Eq. (13)) similar to the inner loop in the meta training phase. This procedure adapts the parameters θ obtained in the meta training phase to the target tasks τ target = {τ 1 , · · · , τ b }. θ j ← θ − β∇ θ L τ sup j (f θ )(13) First-order MAML (FOMAML) Eq. (14) is the calculation of the gradient in the outer loop, where L τ qry j is denoted as L j for simplicity. ∇ θ L qry = ∇ θ b j=1 L j (f θ j ) = b j=1 ∇ θ L j (f θ j ) (14) When performing the outer loop during the meta training phase, high computational cost is needed to calculate the second-order derivatives with backpropagation. Eq.(15) is the first-order approximation of the second-order derivative, ∂L j (f θ j ) ∂θ d = D i=1 ∂L j (f θ j ) ∂θ i j ∂θ i j ∂θ d ≈ ∂L j (f θ j ) ∂θ d j(15) where θ is a D dimensional parameter, θ d is the dth dimension of θ and θ i j is the i-th dimension of θ j . The difference between FOMAML and MAML is that this approximation is used instead of the second-order derivatives. Thus, compared to MAML, FOMAML can save a lot of computational time, resulting in a faster gradient calculation. Experiments Dataset The multi-accent speech utterances are collected from the speech accent archive (Weinberger, 2014). This archive currently has more than 200 kinds of accents and 2939 samples. Each native or nonnative speaker speaks the same English paragraph. We selected 123 accents that contain more than one speaker since we need utterances of two different speakers to generate mixtures. We split these accents into three sets, 85 accents for generating the training tasks and 19 accents each for generating the developing and testing tasks. The utterance of each speaker is split into segments with a duration of 4 seconds. For each accent, we construct meta tasks by following the task construction method Figure 2: Illustration of a meta task. For two different speakers with the same accent, we sample 3 utterance segments to form a meta task. Thus, there will be 9 mixtures. However, during training, we only sample one mixture to form the support set since our setting is one shot learning. The other 4 mixtures that do not contain the utterance segments in the support set are selected to form the query set. described in (Wu et al., 2020). We select at most 12 speakers for each accent and generate speech mixtures for each pair of speakers with the same accents. Thus, there will be at most 12 2 = 66 meta tasks and at least 2 2 = 1 meta task for each accent. In each meta task, 3 utterance segments are selected from each speaker and mixed with an SNR level randomly selected between 0 to 5 dB and resampled at an 8kHz sample rate. This results in 3 × 3 = 9 speech mixtures in one meta task. Fig.(2) is an illustration describing the support set and query set of a meta task. Finally, for the training, developing, and testing set, 22.4, 3.8, and 3.9 hours of speech mixtures are generated. Model The model we used is Conv-TasNet (Luo and Mesgarani, 2019). It consists of an encoder, separator, and a decoder. The encoder is a 1-dim convolution, which transforms the input mixture into a representation. The separator then calculates two masks based on the encoder output. More specifically, it consists of R stacks of temporal convolutional networks (TCN). Each TCN layer consists of M 1-dim exponentially increasing dilated convolutional blocks. These M blocks each have a residual connection and a skip connection. The residual connection is the input of the next block and the skip connection of all blocks are summed together, passing a parametric relu, linear projection, and a sigmoid function to produce two masks. The two masks are multiplied with the representation output from the encoder respectively and further input into the decoder to generate two separate waveforms of the two speakers. The decoder is also a 1-dim convolution. The configuration that we used is the one that obtained the best performance reported in (Luo and Mesgarani, 2019). Joint Training and Transfer Learning There are many other works such as Tong et al., 2017), that try to solve the domain mismatch problem, where the source domain and target domain datasets do not have a similar distribution. Joint training refers to pretraining a model with different source domain data together. Transfer learning refers to adapting the pretrained model to some partial target domain data and testing the fine-tuned model on the target domain data. The most common adaptation method is fine-tuning. Moreover, the domain mismatch scenario has a low-resource problem if the target domain has only fewer data compared to the scale of the source domain data. There are also several works that tried to solve this problem, such as (Chen and Mak, 2015;Zoph et al., 2016;. Our jointly trained model is also based on this low-resource scenario. MAML and FOMAML To deal with the domain mismatch and lowresource problem, we applied MAML as our training method in the hope of performing better than joint training. We set the number of the support set in each task as 1, meaning that the model needs to have the ability to adapt to a new task by only seeing one speech mixture of two new different speakers with a new accent never seen before. We also trained our model with FOMAML in order to know whether calculating gradients with firstorder approximation still obtains relatively good performance compared to training with MAML. Experiment Settings For both the joint training and MAML methods, we trained the model from randomly initialized parameters for 100 epochs with the Adam optimizer of 0.001 learning rate and 0.00001 weight decay. For the MAML methods, during the meta training phase, we set α = 0.01. For joint training, we also fine-tuned the model parameters with the method in Eq.(13). We tested the fine-tuning learning rate β on the testing set, reported it in section 6, and used the learning rates that obtained the best performance for joint training as our baseline. However, for the models trained with MAML methods, the fine-tuning learning rate β is fixed at 0.01 since other values lead to significant performance degradation. Results Joint Training For joint training, we tested the fine-tuning learning rate β on the testing set as shown in Fig.(3), and found out that β = 5e−4 obtained the best performance on the clean testing set, while β = 1e−3 obtained the best performance on the testing set with noise. We use these two experiment settings as our baseline. h a u s a l i t h u a n i a n b a r i q u e c h u a y i d d i s h k u r d i s h s y n t h e s i z e d t a m i l r u s s i a n t h a i i t a l i a n e w e m e n d e m a l a y b a s q u e a l b a n i a n g a e s t o n i a n r o t u m a n on all accents when there is no noise involved and performs better on most of the accents when there is noise in the mixtures. MAML and FOMAML By comparing models (d) and (f), we found out that these two training methods have similar performance. Model (d) has a slightly higher performance than model (f) under the circumstances that the mixtures are clean in the testing tasks, while model (d) has a slightly lower performance than model (f) under the circumstances that there is noise in the testing tasks. However, MAML requires more than 10 times the training time compared to FOMAML, indicating that the first-order approximation takes advantage over calculating the second-order derivatives by saving a lot of time while still obtaining similar performance. Moreover, FOMAML without fine-tuning (model (c)) has similar performance compared to the baseline model, and yet somehow, initialized parameters obtained by MAML (model (e)) do not have the ability to perform speech separation. Conclusion Our results show that MAML and FOMAML training methods are effective on multi-accent speech separation. More specifically, it is confirmed that these two methods are better than joint training when adapting to new speakers with new accents and even noisy environments. Besides, FOMAML is shown to be sufficient for dealing with the multiaccent speech separation task and can reduce a large amount of training time. Despite the fact that FOMAML outperforms joint training on the testing set, we can still see that the performance of each accent task set varies a lot from Fig.(4). This is probably due to the task-difficulty imbalance issue described in (Xiao et al., 2020), perhaps some speakers with special accents may be hard to separate. Thus, in the future, we will try to solve this problem with meta sampling methods mentioned in (Xiao et al., 2020). Figure 3 : 3For fine-tuning after joint training, we evaluated the performance by adjusting the learning rate β in the range of 10 −5 to 10 −1 . Figure 4 : 4Evaluation results of each testing accent task set for model (b) and (d) in table 1. This suggests that the initial model parameters obtained by MAML and FOMAML have the better potential to be adapted to new unseen tasks. Besides, the standard deviation of the testing accent task sets of models (d) and (f) are both less than model (b). This implies that the performance of the models trained with MAML and FOMAML have small dispersion with respect to the mean Si-SNRi value of all the accents compared to the model jointly trained. FromFig.(4), we can see that model (b) performs betterTable 1: Evaluation results of joint training and MAML methods on the testing accent task sets with and without noise. The two numbers in a cell denote the average Si-SNRi of all the testing tasks and the standard deviation of all the testing accent task sets.Comparing models (d), (f) with model (b), we can see that MAML and FOMAML perform better than the joint training baseline. method fine-tune test w/o noise test w/ noise (a) before 8.40 ± 2.25 6.67 ± 2.10 (b) Joint Training after 8.52 ± 2.20 6.89 ± 1.84 (c) before 8.45 ± 3.19 6.66 ± 2.59 (d) FOMAML after 10.13 ± 2.12 8.19 ± 1.62 (e) before -6.19 ± 1.38 -6.85 ± 1.31 (f) MAML after 10.11 ± 1.86 8.26 ± 1.52 Multitask learning of deep neural networks for lowresource speech recognition. 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It is shown that the M 2+ ion yield in the double photoionization of the aromatic hydrocarbons, pyrene and coronene, can be expressed as a superposition of a contribution from a resonance involving carbon atoms on the perimeter and coherent contributions from carbon atoms inside the perimeter. In the case of pyrene, the two interior atoms are associated with a resonance peak at 10 eV and linear behavior above 75 eV. The resonance peak is an optically excited state of the interior carbon pair. The linear behavior arises from the coherent emission of two electrons with equal energy and opposite momenta, as occurs in pyrrole. Coronene has a low energy peak along with two pairing resonances, however, the linear region as in the case of pyrene is absent. The low energy resonance is associated with the atoms on the perimeter and the high energy resonance is associated with the hexagonal array of six carbon atoms at the center of the molecule. It is proposed that the quasi-independence of the contributions from the perimeter and interior atoms is related to Hückel's Rule for the stability of aromatic hydrocarbons.

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Independent-subsystem interpretation of the double photoionization of pyrene and coronene 29 May 2019 D L Huber University of Wisconsin-Madison 53706MadisonWisconsinUSA R Wehlitz University of Wisconsin-Madison 53706MadisonWisconsinUSA Independent-subsystem interpretation of the double photoionization of pyrene and coronene 29 May 2019(Dated: May 31, 2019) It is shown that the M 2+ ion yield in the double photoionization of the aromatic hydrocarbons, pyrene and coronene, can be expressed as a superposition of a contribution from a resonance involving carbon atoms on the perimeter and coherent contributions from carbon atoms inside the perimeter. In the case of pyrene, the two interior atoms are associated with a resonance peak at 10 eV and linear behavior above 75 eV. The resonance peak is an optically excited state of the interior carbon pair. The linear behavior arises from the coherent emission of two electrons with equal energy and opposite momenta, as occurs in pyrrole. Coronene has a low energy peak along with two pairing resonances, however, the linear region as in the case of pyrene is absent. The low energy resonance is associated with the atoms on the perimeter and the high energy resonance is associated with the hexagonal array of six carbon atoms at the center of the molecule. It is proposed that the quasi-independence of the contributions from the perimeter and interior atoms is related to Hückel's Rule for the stability of aromatic hydrocarbons. I. INTRODUCTION The photon-energy dependence of the simultaneous removal of two electrons from an atom or molecule by a single photon has been studied for several cases (see, e.g., [1]). In recent publications [2,3] the double photoionization (DPI) results for the aromatic hydrocarbons pyrene and coronene using monochromatized synchrotron radiation over a wide range of energies have been reported. The general goal of these investigations is to elucidate the role of the physical molecular structure on the DPI process. A unique feature among the hydrocarbons studied using DPI is that pyrene and coronene have carbon atoms located inside their molecular perimeter as well as on their perimeter [4]. The chemical formula of pyrene is C 16 H 10 with 14 carbon atoms on the perimeter and two carbon atoms in the interior, positioned as nearest-neighbors. Coronene has the formula C 24 H 12 with 18 carbon atoms on the perimeter and six carbon atoms in the interior that are arranged in a benzenelike ring. The focus in this paper is the role played by carbon atoms inside the perimeters of the two molecules on DPI. We will discuss that the perimeter carbon atoms and the interior carbon atoms can be regarded as independent subsystems in the DPI process. The current analysis of the experimental data was undertaken in light of the previous theoretical studies of DPI in aromatic hydrocarbons [5,6]. Our results are presented in plots of the DPI ratio vs excess energy relative to the DPI threshold. Here, the DPI ratio is defined by the expression R = M 2+ /M tot − K,(1) where M 2+ and M tot denote the yields of doubly charged and singly plus doubly charged parent ions, respectively. K is the knock-out contribution [4] to the DPI process, in which the first electron on its way out of the molecules knocks out a second electron. The contribution from the knock-out mechanism is modelled by the double-to-total photoionization ratio of helium, which is justified for photon energies as low as in this study. We begin our analysis with pyrene and coronene, and will then test our model of independent subsystems on anthracene. Corresponding error bars are given in [4]. See text for details. II. PYRENE In pyrene, after subtracting the contribution from the knock-out mechanism to the DPI ratio as discussed in [3], there are three regions in the photon energy spectrum with distinctly different behaviors in the DPI: (i) a low-energy region, with a comparatively sharp resonance peak at 10 eV above the DPI threshold, (ii) a mid-range region with a 25 eV onset and a broad peak at 51 eV, and (iii) a quasi-linear region extending to more than 250 eV above threshold ( Fig. 1). We associate the low-energy and linear regions with the two carbon atoms inside the perimeter of the molecule, while the mid-range region is linked to the atoms on the perimeter. In the 10-eV resonance, the energy of the photon that has been absorbed is equal to the sum of the resonance energy and the energy required to promote two electrons to the DPI threshold. A theory for the origin of the low-energy resonance in pyrene is discussed in [6]. The approach is based on the Hubbard model [7], sometimes referred to as the Pariser-Parr-Pople model. The optical absorption associated with the two-site Hubbard Hamiltonian characterizes a process where a π electron is transferred to the site that is already occupied by a π electron with opposite spin. It consists of a single peak at the energy E OA where [8] E OA = U/2 + [(U/2) 2 + 4t 2 ] 1/2 .(2) Here t denotes the transfer integral characterizing the interaction between π electrons on nearest-neighbor sites and U is the electrostatic interaction between two π electrons on the same carbon site. In order to calculate the location of the DPI peak, we take parameter values appropriate for benzene: U = 10 eV and t = 2.5 eV [9] and obtain the result E OA = 12 eV, in reasonable agreement with experiment. An alternative interpretation of the low-energy resonances in pyrene and coronene is outlined in [2]. As discussed there, the peak is identified as a pairing resonance having a de Broglie wavelength of 2.8Å, i.e. twice the C-C distance. The linear behavior of the DPI shown in Fig. 1 The mid-range region in pyrene reflects a continuum resonance that is likely to be related to the pairing of two mobile electrons in a one-dimensional periodic potential [12]. In our analysis, the π electrons involved in the resonance are associated with the 14 carbon atoms on the perimeter of the molecule. In Fig. 1, there is a region where the resonance and the linear DPI-mechanism overlap. Assuming the contributions from the two processes superpose incoherently, we can extrapolate the linear curve to zero, thus obtaining an onset energy of 61 eV. Subtracting the linear response from the DPI data we obtain a plot of the perimeter resonance structure with an onset at 30 eV, a peak at 50 eV, and the upper cut-off at 75 eV (Fig. 2). It is useful to compare the DPI results for coronene [3] with the corresponding pyrene data. Like pyrene, coronene has a 10-eV peak. which we attribute to an excitation involving the six carbon atoms in the interior. Unlike pyrene, there is no linear behavior at high energies. Instead, coronene has two regions of resonance structures, 30 to 70 eV and 80 to 125 eV (Fig. 3). The low-energy region qualitatively resembles what is found for the perimeter resonance in pyrene. We attribute the high-energy region to a pairing resonance involving the six interior carbon atoms. In Fig. 4 resonance in benzene, 40 eV [10]. In Figs. 2 and 3, the contribution to the DPI from the perimeter pairing resonance is approximately symmetric. There is, however, a significant asymmetry in the high-energy pairing resonance. These results suggest that the asymmetry is related to the number of carbon atoms in the closed loop. When the number is comparatively large, as in the perimeters of pyrene and coronene, the resonance is nearly symmetric with respect to the mid-point; when the number is small, as in the interior resonance in coronene, there is greater spectral weight below the midpoint. IV. DISCUSSION In order to test our model of independent subsystems, we turn now to anthracene (C 14 H 10 ), which has the same number of perimeter atoms as pyrene but no interior atoms. In Fig. 5 we display the reduced resonance peak in anthracene that is obtained after subtracting the knock-out as well as the linear contribution to the DPI ratio. This resonance resembles the benzene resonance extending from 30 eV to approximately 90 eV above threshold with a peak at ca. 45 eV. It also resembles the mid-range resonance in pyrene (cf. Fig. 2) and is similar to the mid-range resonance in coronene (cf. Fig. 3). Taken together, these results support the hypothesis of independent DPI contributions from interior and perimeter carbon atoms. Finally, we comment on treating the perimeter carbon atoms and the interior carbon atoms as independent subsystems. The justification for this approximation is associated with Hückel's Rule, which predicts a high stability of an aromatic hydrocarbons for N = 2 + 4n, n = 0, 1, 2, ... where N is the number of π electrons (i.e. carbon atoms) in the planar ring [14]. Although neither pyrene nor coronene satisfy the rule when N is the total number of carbon atoms, the rule is satisfied for the perimeter subsystem and the interior subsystem in both molecules. The increased stability associated with Hückels Rule suggests the subsystems are likely to act quasi-independently in DPI processes. V. SUMMARY We have analyzed the DPI yield in the aromatic hydrocarbons pyrene and coronene. In both molecules, there are perimeter and interior carbon atoms. The results of our analysis show that the dominant features in the yield are independent contributions from the perimeter and interior subsystems. It is proposed that the quasi-independence of the two subsystems is due to the interior and perimeter subsystems independently obeying Hückel's Rule. 7 2 FIG. 1 : 21Relative DPI yield R as defined in Eq. (1) in pyrene as a function of excess energy. The solid line connects the data points. resembles the linear behavior in pyrrole and the related molecules furan and selenophene[4]. These molecules consist of a fivemember ring with four carbon atoms and an impurity site occupied by an oxygen atom (furan) or selenium atom (selenophene), and in the case of pyrrole, a nitrogen-hydrogen complex. The effect of the impurity atom in these systems is to interrupt the periodicity associated with a ring of carbon atoms by replacing the ring with a linear array. In pyrene, the linear region is associated with the two interior carbon atoms. The linear behavior of the DPI in pyrrole and similar molecules is characteristic of a coherent process where the two photoelectrons are emitted simultaneously with equal kinetic energies and oppositely directed momenta[5,10,11]. According to our interpretation, a similar process takes place in pyrene.FIG. 2: The relative perimeter pairing yield R in pyrene vs excess energy. The data are obtained by subtracting the linear part of the ratio curve R shown in Fig. 1 extrapolated down to zero. FIG. 3 : 3Relative DPI yield R as defined in Eq. (1) in coronene vs excess energy. Note the peak at 10 eV and the two pairing resonances at higher energies. we display the upper pairing resonance on a larger scale. The width of the resonance, 45 eV, is comparable to the width of the FIG. 4: Relative high-energy pairing yield R in coronene vs excess energy. The resonance is associated with the six interior carbon atoms. 6 FIG. 5 : 65Relative DPI yield R of anthracene[13] as defined in Eq. (1) after subtraction of the linear increase of the ratio by a straight line extrapolated down to the abcissa. ACKNOWLEDGMENTSThe authors thank Dr. Narayan Appathurai for critical reading of the manuscript and helpful comments, R Wehlitz, Advances in Atomic, Molecular, and Optical Physics. E. Arimondo, P. R. Berman, C.C. LinNew YorkAcademic Press58R. Wehlitz, in Advances in Atomic, Molecular, and Optical Physics edited by E. Arimondo, P. R. Berman, C.C. Lin (Academic Press, New York, 2010), Vol. 58, pp. 1-76. . R Wehlitz, T Hartman, J. Phys.: Conf. Ser. 48812013R. Wehlitz and T. Hartman, J. Phys.: Conf. Ser. 488, 012013 (2014). . T Hartman, P N Juranić, K Collins, B Reilly, E Makoutz, N Appathurai, R Wehlitz, Phys. Rev. A. 8763403T. Hartman, P. N. Juranić, K. Collins, B. Reilly, E. Makoutz, N. Appathurai, and R. Wehlitz, Phys. Rev. A 87, 063403 (2013). . R Wehlitz, J. Phys. B: At. Mol. Opt. Phys. 49222004R. Wehlitz, J. Phys. B: At. Mol. Opt. Phys. 49, 222004 (2016). . D L Huber, Phys. Rev. A. 8951403D. L. Huber, Phys. Rev. A 89, 051403 (2014). . D L Huber, Mod. Phys. Lett. B. 321850083D. L. Huber, Mod. Phys. Lett. B 32, 1850083 (2018). . J Hubbard, Proc. Roy. Soc. Lond. Ser. A. 276238J. Hubbard, Proc. Roy. Soc. Lond. Ser. A 276, 238 (1963). . N Maeshima, K Yonemitsu, Phys. Rev. B. 74155105N. Maeshima and K. Yonemitsu, Phys. Rev. B 74, 155105 (2006). . R J Bursill, C Castleton, W Barford, Chem. Phys. Lett. 294305R. J, Bursill, C. Castleton, and W. Barford, Chem. Phys. Lett. 294, 305 (1998). . K Jänkälä, P Lablanquie, F Penent, J Palaudoux, L Andric, M Huttula, Phys. Rev. Lett. 112143005K. Jänkälä, P. Lablanquie, F. Penent, J. Palaudoux, L. Andric, and M. Huttula, Phys. Rev. Lett. 112, 143005 (2014). . D L Huber, arXiv:1805.00891v3D. L. Huber, arXiv:1805.00891v3 (2019). . S M Mahajan, A Thyagaraja, J. Phys. A: Math. Gen. 39667S. M. Mahajan and A. Thyagaraja, J. Phys. A: Math. Gen. 39, L667 (2006). . R Wehlitz, P N Juranić, K Collins, B Reilly, E Makoutz, T Hartman, N Appathurai, S B Whitfield, Phys. Rev. Lett. 109193001R. Wehlitz, P. N. Juranić, K. Collins, B. Reilly, E. Makoutz, T. Hartman, N. Appathurai, and S. B. Whitfield, Phys. Rev. Lett. 109, 193001 (2012). . E Hückel, Z. Phys. 70204E. Hückel, Z. Phys. 70, 204 (1931); . Z. Phys. 72310Z. Phys. 72, 310 (1931); . Z. Phys. 76628Z. Phys. 76, 628 (1932).

In these proceedings, we explore the possible reach of the LHCb dataset in the area of charm semileptonic decays. Specifically, we give prospects for the measurement of |V cs |/|V cd | usingwith Run I data. Preliminary projections show that the LHCb Run I dataset would give a relative statistical uncertainty of ∼ 0.2% on this ratio. We also motivate the search for lepton non-universality in the charm sector.9th International Workshop on the CKM Unitarity Triangle

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e9d6760a25027d04ae470bbd65fd41a6294ef2dc

Charm semileptonic decays at LHCb Charm semileptonic decays at LHCb 28 November -3 December 2016 30 Mar 2017 Adam C S Davis [email protected] Speaker Adam C S Davis Center for High Energy Physics Tata Institute for Fundamental Research (TIFR) Tsinghua University 100084Beijing, MumbaiP. R. China, India Charm semileptonic decays at LHCb Charm semileptonic decays at LHCb 28 November -3 December 2016 30 Mar 2017† On behalf of the LHCb Collaboration c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). In these proceedings, we explore the possible reach of the LHCb dataset in the area of charm semileptonic decays. Specifically, we give prospects for the measurement of |V cs |/|V cd | usingwith Run I data. Preliminary projections show that the LHCb Run I dataset would give a relative statistical uncertainty of ∼ 0.2% on this ratio. We also motivate the search for lepton non-universality in the charm sector.9th International Workshop on the CKM Unitarity Triangle Introduction The LHCb detector at CERN [1] is a single arm forward spectrometer designed for precision studies of b and c hadrons in collisions at the Large Hadron Collider (LHC). During Run I of the LHC, the LHCb detector collected approximately 3 fb −1 of data at √ s =7 and 8 TeV, and in Run II, approximately 2 fb −1 to date. Between 2011 and 2016, LHCb has reconstructed approximately 1.8 billion charm hadrons. In these proceedings, we explore the physics reach of the LHCb dataset with respect to semileptonic D meson decay. These proceedings are broken into four sections: First, a brief review of the formalism of semileptonic decays in charm, including the relevant differential decay rate. Second, we explore the experimental challenges present for LHCb and address specific concerns for neutrino reconstruction and q 2 resolution, where q is the momentum transfer to the lepton and neutrino. Third, we present sensitivity estimates for the measurement of |V cs |/|V cd | or form-factors from the ratio of decays D 0 → K − µ + ν µ and D 0 → π − µ + ν µ 1 . Finally we present the motivation for the search for lepton non-universality in the charm system. Theoretical Overview Differential decay rates The differential decay rate of the D 0 meson to a pseudoscalar final state P via semileptonic decay at leading order can be written in the following well-known form [2] dΓ (D 0 → P − + ν ) dq 2 = |V cQ | 2 G 2 F 24π 3 (q 2 − m 2 ) 2 E 2 P − m 2 P q 4 m 2 D 0 × 1 + m 2 2q 2 m D 0 (E 2 P − m 2 P )| f + (q 2 )| 2 + 3m 2 8q 2 (m 2 D 0 − m 2 P ) 2 | f 0 (q 2 )| 2 . (2.1) Here, Q represents the outgoing quark from the weak vertex, the terms f + (q 2 ) and f 0 (q 2 ) are the vector and scalar form factors, respectively, used to parameterize the hadronic current. The differential decay rate measured at experiments can be broken down into two pieces of interest. First, on the right hand side of Equation 2.1 starts with CKM factors V cQ , which are of interest in testing unitarity of the CKM matrix. Second the right hand side depends on the vector form factor f + (q 2 ) and scalar form factor f 0 (q 2 ). Measurements of the form factor dependence provides useful input to Lattice QCD calculations. The interesting measurements which could be made by LHCb are: 1. Measure the differential branching fractions for D 0 → h − µ + ν µ as a function of q 2 . This helps constrain the quantity | f + (q 2 )| 2 |V cQ | 2 and | f 0 (q 2 )| 2 |V cQ | 2 2. Using external constraints on the form factors from Lattice QCD, measure |V cQ | directly relative to a normalization channel 3. Using external constraints on |V cQ |, measure the form factor dependence, relative to a normalization channel 4. Test lepton universality using decays which differ only by the lepton in the final state Experimental Challenges One of the major challenges of measuring semileptonic decays at LHCb is the partially reconstructed final state. Unlike at e + e − machines, the hadron collider environment does not allow reconstruction of the missing neutrino using beam energy constraints, nor separation of the decays of interest into hemispheres. However, a host of experimental techniques exist for reconstruction of the neutrino momentum and the q 2 distributions. From kinematics alone, the neutrino momentum is completely constrained perpendicular to the D 0 flight direction. We label this transverse momentum p T (ν) = −(p T (P) + p T ( )). A sketch is given in Figure 1. Using energy and momentum conservation to solve for the remaining component of the neutrino momentum leaves a two-fold ambiguity. While possible, simply choosing one of the solutions of the neutrino momentum can lead to biases in the distributions of interest, specifically with respect to q 2 . There are several well-known methods for calculating the missing neutrino momentum component, relying on a variety of different techniques: • k-factor method: Using simulated events, determine the factor k as a function of visible daughter mass which predicts the true D 0 momentum. Such an approach usually determines an average value k written as p(D 0 ) = p(K ) k(m(K )) . This approach has been used previously in many analyses, one such example being the measurement of the B 0 meson oscillation frequency ∆m d [3]. • Corrected D 0 mass: One can use as an approximation to the true D 0 mass the quantity m Corrected = m 2 (K ) + |p T | 2 + p T . This quantity has the usefulness that it will peak at the nominal mother mass for true decays and have a long tail that extends to lower m corrected . Decays of multibody final states will then peak more strongly towards lower m corrected allowing for good separation between signal and background. Such a method has been used successfully in the measurement of |V ub | from Λ b → pµν decays [4]. • Cone-closure: By enforcing that the D 0 be the daughter of a D * + → D 0 π + decay, the additional mass constraint of the D * + breaks the ambiguity of the momentum of the neutrino. The method relies on the fact that in the K rest frame, p(D 0 ) = p(ν), and the slow pion forms the axis of a cone around which the neutrino momentum lies. The solution of the momentum is then the one that most closely aligns the D 0 momentum to the D 0 flight direction. Such a procedure was used in the E687 experiment at Fermilab [5]. • Recently, a new method using multivariate regression to help choose the correct neutrino momentum of b-hadron has been presented [6]. While the use case presented is for b-hadron decays, the algorithm should be easily extendible to c-hadron decays. Each of these methods has its shortcomings, but provides an estimate of the momentum of the D 0 and thus the calculation of q 2 . Sensitivity for CKM matrix element measurements As an example of an accessible measurement at LHCb, we estimate the sensitivity of the measurement of the ratio of CKM matrix elements |V cs |/|V cd |. The measurement can be made by comparing the branching ratios of D * + → D 0 π with D 0 → K − µ + ν m u and D 0 → π − µ + ν µ . Such a measurement is tractable at LHCb due to the high efficiency of reconstructing muons. By enforcing that the D 0 originate from a D * + decay, the q 2 dependence of both channels can be made unambiguously. This is a very similar measurement to the measurement in [4]. The advantage of such a ratio measurement is that the majority of the trigger, selection and detection efficiencies cancel in the ratio. Additionally, the corrected mass can provide a stable handle on missing neutral backgrounds. The sensitivity estimate can be made in the following way: as the branching ratio of the decay D → K − µ + ν µ is similar to that of D 0 → K − π + , an initial estimate of yields can be made for the Run I dataset by taking number of D 0 → K − π + decays from the CPV search in charm using the same decay [7]. This gives an initial estimate of ∼56 million D 0 → K − µ + ν µ decays, and roughly one order of magnitude fewer decays of D 0 → π − µ + ν µ decays. The software trigger to use for such a decay at the LHCb experiment is one which searches for inclusive D * + → D 0 π + decays. Such a trigger was only operational during two-thirds of the 2012 Run I operation, limiting the statistics to roughly 60% of the nominal value. Assuming that the remaining efficiency differences are at the order of 20%, as the major differences will be reconstructing the muon and forming a good vertex, this leaves roughly 4.4 million signal candidates. This yield would directly relate to a relative systematic uncertainty of 0.2%. These numbers are not simply out of thin air. A validation of the tracking efficiency in the analysis of a s sl [8] used the sample D 0 → K − µ + ν µ to calculate the tracking asymmetry of the muonpion pair. With loose selections, the analysis reconstructed 5M signal candidates. The total fit was performed simultaneously between D 0 → K − µ + ν µ and D 0 → K + µ − ν µ samples in individual bins of visible mass m(Kµ), with the signal shape being derived empirically. An example plot of the visible mass difference m(Kµπ) − m(Kµ) for the 2012 dataset using the magnet down polarity fit in the range of visible mass 1600 ≤ m(Kµ) < 1700 MeV/c 2 is shown in Figure 2. Lepton non-universality in semileptonic D decays Lepton non-universality has been a hot topic in the past few years mainly driven by possible anomalies in the semileptonic decays of B → D * ν with a τ in the final state compared to a µ, as well as possible differences seen in decays of the form b → s . Such tensions with the standard model beg the question whether or not there is a similar measurement to be made in other modes. No such measurement of lepton non-universality has been made in the charm sector, but each of the individual relevant branching ratios has been measured [9]. By taking the simple ratio of relevant decays, and comparing with the q 2 integrated standard model prediction from [10] shows a consistent trend towards higher ratios than expected. This is illustrated in Figure 3. All of the modes presented in Figure 3 are worth pursuing. Recent theory activity [10] shows that the measurement of the ratio B(D 0 → K − e + ν e )/B(D 0 → K − µ + ν µ ), for example, specifically as a function of q 2 , allows for a direct probe of current bounds on allowed scalar Wilson coefficients. By using the same statistics quoted above for D 0 → K − + ν , the measurement of the ratio of the electron to muon modes would reduce the error on the bottom point in Figure 3 by an order Figure 3: Comparison of the ratio of measured branching fractions for D 0 → h − e + ν and D 0 → h − µ + ν. The term h denotes the hadron in question, either K − , π − or K * − (892). Each point represents the ratio from the measurements reported in [9] compared to the standard mode prediction (red dashed line) and its error(green band) provided by S. Fajfer using [11]. The central values of each point lie systematically to one side of the standard model prediction. of magnitude. While the measurement of the electron mode is certainly difficult at LHCb, due to the fact that bremsstrahlung recovery is not easy, and the measurement of the tracking efficiencies of the electron candidate is non-trivial, the measurement is not impossible; LHCb has already measured many channels with electrons, including the angular analysis of B 0 → K * 0 e + e − [12] and the search for lepton flavor violation in the decay of D 0 → e ± µ ∓ [13]. Conclusion We present preliminary estimates on the reach of LHCb in the field of semileptonic D meson decay. We find that a measurement of |V cs |/|V cd | would give a relative statistical uncertainty of ∼0.2% using the Run I dataset. We also motivate the first search for lepton non-universality in the charm sector. It is important to note that all estimates are using the Run I dataset, and LHCb continues to take its Run II dataset with more statistics and improved triggering strategies. Figure 1 : 1Cartoon of the D 0 meson semileptonic decay, with the definition of the momentum component p T (ν). The red and green dots represent the origin and decay vertex of the D 0 , respectively, the arrowed black lines represent the momentum components of the final state particles, the purple line indicates the initial flight direction of the D 0 , and the red and blue dashed lines represent the component of the momentum p T perpendicular to the D 0 flight direction. Figure 2 : 2Fit to the visible difference in mass m(Kµπ) − m(Kµ) of the decay D 0 → Kµν. The points correspond to the sample of D 0 → Kµν decays in the 2012 magnet down dataset lying in the mass range 1600 ≤ m(Kµ) < 1700 MeV/c 2 . The total fit is shown in magenta, with the signal component shown in dashed blue, and the combinatorial background in dashed red. Unless explicitly stated otherwise, charge-conjugate decays are implied . R Aaij, LHCb Collaboration10.1142/S0217751X15300227arXiv:1412.6352Int. J. Mod. Phys. A. 30071530022hep-exR. Aaij et al. [LHCb Collaboration], Int. J. Mod. Phys. A 30, no. 07, 1530022 (2015) doi:10.1142/S0217751X15300227 [arXiv:1412.6352 [hep-ex]]. . S Aoki, 10.1140/epjc/s10052-016-4509-7arXiv:1607.00299Eur. Phys. J. C. 772112hep-latS. Aoki et al., Eur. Phys. J. C 77, no. 2, 112 (2017) doi:10.1140/epjc/s10052-016-4509-7 [arXiv:1607.00299 [hep-lat]]. . R Aaij, LHCb Collaboration10.1140/epjc/s10052-016-4250-2arXiv:1604.03475Eur. Phys. J. C. 767412hep-exR. Aaij et al. [LHCb Collaboration], Eur. Phys. J. C 76, no. 7, 412 (2016) doi:10.1140/epjc/s10052-016-4250-2 [arXiv:1604.03475 [hep-ex]]. . R Aaij, LHCb Collaboration10.1038/nphys3415arXiv:1504.01568Nature Phys. 11hep-exR. Aaij et al. [LHCb Collaboration], Nature Phys. 11, 743 (2015) doi:10.1038/nphys3415 [arXiv:1504.01568 [hep-ex]]. . W E Johns, UMI-96-02371FERMILAB-THESIS-1995-05W. E. Johns, FERMILAB-THESIS-1995-05, UMI-96-02371. . G Ciezarek, A Lupato, M Rotondo, M Vesterinen, 10.1007/JHEP02(2017)021arXiv:1611.08522JHEP. 170221hep-exG. Ciezarek, A. Lupato, M. Rotondo and M. Vesterinen, JHEP 1702, 021 (2017) doi:10.1007/JHEP02(2017)021 [arXiv:1611.08522 [hep-ex]]. . R Aaij, LHCb Collaboration10.1103/PhysRevLett.111.251801arXiv:1309.6534Phys. Rev. Lett. 11125251801hep-exR. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 111, no. 25, 251801 (2013) doi:10.1103/PhysRevLett.111.251801 [arXiv:1309.6534 [hep-ex]]. . R Aaij, LHCb Collaboration10.1103/PhysRevLett.117.061803arXiv:1605.09768Phys. Rev. Lett. 117661803hep-exR. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 117, no. 6, 061803 (2016) doi:10.1103/PhysRevLett.117.061803 [arXiv:1605.09768 [hep-ex]]. . C Patrignani, 10.1088/1674-1137/40/10/100001Chin. Phys. C. 4010100001Particle Data GroupC. Patrignani et al. [Particle Data Group], Chin. Phys. C 40, no. 10, 100001 (2016). doi:10.1088/1674-1137/40/10/100001 . S Fajfer, I Nisandzic, U Rojec, 10.1103/PhysRevD.91.094009arXiv:1502.07488Phys. Rev. D. 91994009hep-phS. Fajfer, I. Nisandzic and U. Rojec, Phys. Rev. D 91, no. 9, 094009 (2015) doi:10.1103/PhysRevD.91.094009 [arXiv:1502.07488 [hep-ph]]. . J Koponen, C T H Davies, G C Donald, E Follana, G P Lepage, H Na, J Shigemitsu, arXiv:1305.1462hep-latJ. Koponen, C. T. H. Davies, G. C. Donald, E. Follana, G. P. Lepage, H. Na and J. Shigemitsu, arXiv:1305.1462 [hep-lat]. . R Aaij, LHCb Collaboration10.1007/JHEP04(2015)064arXiv:1501.03038JHEP. 150464hep-exR. Aaij et al. [LHCb Collaboration], JHEP 1504, 064 (2015) doi:10.1007/JHEP04(2015)064 [arXiv:1501.03038 [hep-ex]]. . R Aaij, LHCb Collaboration10.1016/j.physletb.2016.01.029arXiv:1512.00322Phys. Lett. B. 754167hep-exR. Aaij et al. [LHCb Collaboration], Phys. Lett. B 754, 167 (2016) doi:10.1016/j.physletb.2016.01.029 [arXiv:1512.00322 [hep-ex]].

Black holes emit high energy particles which induce a finite density potential for any scalar field φ coupling to the emitted quanta. Due to energetic considerations, φ evolves locally to minimize the effective masses of the outgoing states. In theories where φ resides at a metastable minimum, this effect can drive φ over its potential barrier and classically catalyze the decay of the vacuum. Because this is not a tunneling process, the decay rate is not exponentially suppressed and a single black hole in our past light cone may be sufficient to activate the decay. Moreover, decaying black holes radiate at ever higher temperatures, so they eventually probe the full spectrum of particles coupling to φ. We present a detailed analysis of vacuum decay catalyzed by a single particle, as well as by a black hole. The former is possible provided large couplings or a weak potential barrier. In contrast, the latter occurs much more easily and places new stringent limits on theories with hierarchical spectra. Finally, we comment on how these constraints apply to the standard model and its extensions, e.g. metastable supersymmetry breaking.

10.1103/physrevd.89.104035

1309.0530

327432efe6c158eb50d6288ced66558be2d2ae0e

Limits on New Physics from Black Holes Clifford Cheung California Institute of Technology 91125PasadenaCA Stefan Leichenauer California Institute of Technology 91125PasadenaCA Limits on New Physics from Black Holes Black holes emit high energy particles which induce a finite density potential for any scalar field φ coupling to the emitted quanta. Due to energetic considerations, φ evolves locally to minimize the effective masses of the outgoing states. In theories where φ resides at a metastable minimum, this effect can drive φ over its potential barrier and classically catalyze the decay of the vacuum. Because this is not a tunneling process, the decay rate is not exponentially suppressed and a single black hole in our past light cone may be sufficient to activate the decay. Moreover, decaying black holes radiate at ever higher temperatures, so they eventually probe the full spectrum of particles coupling to φ. We present a detailed analysis of vacuum decay catalyzed by a single particle, as well as by a black hole. The former is possible provided large couplings or a weak potential barrier. In contrast, the latter occurs much more easily and places new stringent limits on theories with hierarchical spectra. Finally, we comment on how these constraints apply to the standard model and its extensions, e.g. metastable supersymmetry breaking. Introduction Black holes are a naturally occurring source of high energy particles. During evaporation, a black hole emits a continuous flux of Hawking radiation which in steady state forms a halo of free-streaming particles. If this distribution is sufficiently dense it will influence the dynamics of a scalar field coupled to the outgoing states. Away from the horizon the emitted quanta are out of thermal equilibrium, so they induce a finite density, zero temperature potential for the scalar. While the precise form of this potential depends on the microscopic dynamics, the scalar field always moves to minimize the effective masses of the emitted quanta. This phenomenon is reminiscent of the bag mechanism discussed in [1,2,3], only here it is viable in weakly coupled theories, provided there is a sufficiently large flux of particles created by the black hole. If the scalar field resides in a metastable vacuum, then it is possible for the finite density potential to overcome the potential barrier and catalyze the decay of the vacuum. In this paper we present a detailed analysis of this mechanism and its implications for new physics. While this idea has been studied in the context of weakly coupled string moduli [4], we believe it has much broader applications to physics beyond the standard model (SM). As we will see, black hole catalyzed vacuum decay can place powerful new limits on theories which have in the past been deemed safe by conventional stability bounds. This is true because: • Vacuum decay is classical. This mechanism utilizes the classical activation of a scalar field over its potential barrier. Unlike for quantum tunneling, the associated decay rate is not exponentially suppressed. Consequently, a single black hole in our past light cone can be sufficient to destabilize the vacuum. This constrains many models which in empty space are metastable with lifetimes longer than the age of the universe. • Black holes get very hot. As a black hole decays, its temperature scans adiabatically up to the Planck scale. There is no kinematic limit for high energy particle production because every mass threshold is accessible to a sufficiently hot black hole. Contrast this with limits from thermally assisted vacuum decay in the early universe, which depend on the initial reheating temperature. Catalyzed vacuum decay remains a relatively unexplored topic. The authors of [5,6] considered catalysis by individual particles, concluding that the decay rate is only modestly enhanced. Meanwhile, [7,8] analyzed vacuum decay in the presence of a black hole, incorporating the effects of the metric but neglecting the effects of the emitted Hawking radiation. Critically, past examples have emphasized quantum mechanical tunneling rather than the mechanism of classical activation discussed in this paper and [4]. The outline of this paper is as follows. In Sec. 2 we compute the phase space density of Hawking radiation emitted by a black hole. We then derive a simple formula for the finite density potential. Afterwards, in Sec. 3 we determine the conditions under which vacuum decay is catalyzed by a point particle, and by the halo of quanta surrounding a black hole. Finally, in Sec. 4 we discuss the implications of catalysis for a concrete model, as well as for the SM and its extensions. We summarize our conclusions and discuss prospects for future work in Sec. 5. Finite Density Potential In this section we compute the phase space density of Hawking radiation far from the black hole horizon. We also demonstrate that these particles are effectively free-streaming and out of thermal equilibrium. Finally, we derive a simple formula for the finite density potential, using independent methods from classical and quantum mechanics. Hawking Radiation Distribution Black holes emit Hawking radiation at a rate of [9,10,11,12] Γ = d 3 k ξ(k) , ξ(k) = 1 (2π) 3 σv e w/T ∓ 1 ,(1) where T is the temperature of the black hole and σ is the absorption cross-section of the black hole with respect to the emitted particle. As is well-known, the spectrum of Hawking radiation is only approximately thermal due to grey body factors encoding the dependence on particle species in the absorption cross-section [11,12]. For example, the emission rate is lower for particles of higher spin. We characterize the outgoing flux of particles with a phase space density, n(k, x), describing an ensemble of classical particles with well-defined momentum and position. We can crudely estimate n(k, x) by equating the number of particles in a given frequency band and infinitesimal volume region with the flux of Hawking radiation emitted at the horizon in an infinitesimal time interval, dN = n(k, x) d 3 k d 3 x = Γ dt = ξ(k) d 3 k dt,(2) Here we ignore metric effects because we are interested in particles far from the horizon. Of course, gravitational redshift should still be included in the calculation of the initial black hole emission spectrum. If we assume that the emitted particles free stream in the radial direction, then the resulting phase space density is spherically symmetric and n(k, r) = ξ(k) 4πr 2 ω k ,(3) where we have assumed that the distribution is in steady state so that we can set dr/dt equal to the velocity of the emitted relativistic massive particle, v = k/ω. Naively n(k, r) diverges at low velocities, but this spurious singularity is cancelled when computing physical quantities like the spectrally integrated number density, n(x) = d 3 k n(k, x). Eq. (3) can be derived in a less heuristic way. The phase space distribution is equal to a spectrally weighted sum of delta functions, each localized to the position of an emitted particle. Consider a time period t ∈ [0, τ ], where τ is a fiducial time interval over which the emission spectrum can be treated as constant. We find that n(k, r) = ξ(k) τ 0 dt δ 3 (r − vt) = ξ(k) 4πr 2 ω k × θ k ω − r τ ,(4) which for τ → ∞ asymptotes to the more crudely derived estimate in Eq. (3). Note that the large τ approximation is justified because black holes decay on extremely long time scales relative to their characteristic size. Concretely, the black hole lifetime is τ BH = 5120πG 2 M 3 BH , which in units of the Schwarzschild radius is τ BH R BH = 640πG −1 R 2 BH 1.(5) We will return to the issue of the black hole lifetime later on in Sec. 3. Our derivation of the phase space density has made use of several critical assumptions: 1) the mean field approximation is valid, 2) the emitted particles are free-streaming, and 3) they are sufficiently long-lived that the number density reaches a steady state. Let us scrutinize each of these assumptions in turn. First of all, the mean field description is only justified if n(k, x) describes a large number of particles. Concretely, we require that within a region of size r R BH , the total number of particles is N = d 3 k d 3 x n(k, x) = rΓ ∼ r/R BH (4π) 4 1,(6) where in the last line we have inserted Γ ∼ 1/R BH (4π) 4 , which is a parametric estimate of Eq. (1) taking into account 4π factors. Thus, as long as we restrict to distances very far from the event horizon and only consider dynamics on length scales much larger than R BH , the mean field approximation applies. Second, we must ascertain whether the emitted particles can be treated as free particles after they are emitted. The scattering rate amongst relativistic emitted particles is Γ scatt = n(x)σ scatt = Γ × σ scatt 4πr 2 ,(7) where n(x) is the spectrally integrated phase space density. The scattering rate scales with the number density, which decreases at large radii. Assuming a perturbative cross-section of order σ scatt ∼ 4πα 2 /T 2 , we find that Γ scatt = Γ × (4πα) 2 R BH r 2 .(8) The number of times an emitted particle scatters before it escapes to infinity is N scatt = ∞ R dr Γ scatt = ΓR BH × (4πα) 2 ∼ (α/4π) 2 < 1.(9) Hence, the flux of Hawking radiation does not thermalize. These estimates are consistent with [13], which reached a similar conclusion. Third, let us consider the lifetime of the emitted particles. For simplicity, we assume a perturbative decay rate, Γ dec ∼ αM,(10) where M is the mass of the Hawking radiation. Crucially, the associated decay length is enhanced by a substantial boost factor, γ, because the particles are relativistic. For the phase space density of emitted particles to reach a steady state, the decay length is bounded by R dec = Γ −1 dec γ Γ −1 so that the rate of particle depletion is overcome by the rate of production by Hawking radiation. This implies the parametric condition, γ √ α, which is easily satisfied for relativistic particles. Classical Derivation Consider a scalar field, φ, which couples to the outgoing Hawking radiation, χ. We now present a classical derivation of the finite density potential for φ induced the ambient χ particles. To begin, consider a microscopic Lagrangian density, L, containing arbitrary interactions between φ and χ. In a regime where φ is slowly varying relative to χ we can define a φ-dependent, effective mass squared for χ, µ 2 [φ] = − ∂ 2 L ∂χ 2 ∂ 2 L ∂χ 2 −1/2 .(11) We ignore metric effects because we are interested physics far from the horizon, where gravitational redshift is negligible. 1 The first factor in Eq. (11) denotes the φ-dependent "bare mass" induced by non-derivative couplings such as φχ 2 . The second factor denotes the φ-dependent "wavefunction renormalization" induced by derivative couplings such as φ∂χ 2 . While Eq. (11) applies to a real scalar χ, the generalization to complex or higher spin fields is obvious. Thus far we have only considered terms quadratic in χ, but a general theory will also include couplings that mediate χ self-interactions and decays. However, since the emitted χ particles are freely propagating in our regime of interest, χ particle number is effectively conserved and these interactions can be ignored. We will return to the issue of χ decays later on. Consider a regime in which φ is slowly varying relative to the momenta of the background χ particles. The ensemble of χ particles is described by a worldline action, S = dt L, where S = − i ds i µ[φ(t, x i )] = − i dt µ[φ(t, x i )] 1 −ẋ 2 i ,(12) where i labels each χ particle and (t, x i ) its worldline trajectory. The canonical momentum for each particle is k i = ∂L/∂x i = µ[φ(t, x i )]ẋ i / 1 −ẋ 2 i and the associated Hamiltonian is H = i k 2 i + µ 2 [φ(t, x i )] = d 3 x i k 2 i + µ 2 [φ(t, x)] × δ 3 (x − x i ),(13) which is the finite density analog of the Coleman-Weinberg potential. We will only consider situations where the distance between χ particles is much less than the length scales in the φ potential. Then it is valid to take the continuum limit, H = d 3 x d 3 k k 2 + µ 2 [φ(t, x)] × n(k, x) = d 3 x µ 2 [φ(t, x)] × 1 2 d 3 k n(k, x) 1 k + O(1/k 3 ) ,(14) where n(k, x) is the phase space density and we have expanded around the relativistic limit. Plugging in our expression for n(k, x) from Eq. (3) into Eq. (14), we obtain a simple expression for the finite density potential induced by a black hole, ∆V BH = f r 2 × µ 2 [φ],(15) which is valid in the limit of relativistic χ particles. Here the dimensionless number f encodes the effects of grey body factors, f = 1 8π d 3 k ξ(k) k = 1 16π 3 dk kσ(k, T ) e k/T ∓ 1 ,(16) and we have set ω = k throughout. Eq. (15) makes sense physically: φ is driven towards field values that minimize the effective masses of ambient χ particles. However, this effect falls off at large distances due to the dissipation of the flux of Hawking radiation. By dimensional analysis f is independent of T , but it varies with the spin of the emitted particle. Employing the approach of [14], we have calculated f from black hole grey body factors determined by the absorption cross-section of a particle incident on the black hole. Specifically, we solved numerically for the reflection and transmission coefficients of an incoming wave of a Klein-Gordon field propagating in a Schwarzschild background. Our results closely match those of [14], and are consistent with the approximate analytic expressions in [15]. We find that f 10 −4 ×            3.9 , s = 0 0.7 , s = 1/2 0.1 , s = 1(17) As is well known, higher spin particles have a suppressed rate of emission. Quantum Derivation Eq. (15) can also be derived quantum mechanically. In the approximation that φ is a slowly varying background for χ, the microscopic Hamiltonian density is H = 1 2χ 2 + 1 2 ∇χ 2 + 1 2 µ 2 [φ]χ 2 ,(18) where the kinetic terms have been canonically normalized. Near a black hole, the exiting χ quanta are relativistic and their total energy is dominated by their φ-independent kinetic energy. Consequently, the leading order φ dependence arises from the effective mass term in Eq. (18). The finite density potential is then determined by the usual rules of quantum mechanical perturbation theory, so ∆V BH = 1 2 µ 2 [φ] χ 2 ,(19) where the expectation value is evaluated on the wavefunction characterizing the outgoing radiation. Classically, we know that the number density of outgoing particles reaches an approximate steady-state, and so the expectation value of Eq. (19) must be constant in time. In the interaction picture, the wavefunction itself for free-streaming particles is also constant. There is a trivial sort of time-dependence coming from the continual emission of new particles from the black hole, but this only serves to replace the particles which are free-streaming away. Therefore we are justified in approximating the wavefunction in Eq. (19) as time-independent. As we will see, the precise form of this wavefunction is unimportant, provided the true wavefunction describes classical, free-streaming particles far from the black hole. Note that we are explicitly neglecting any transient effects from the black hole formation, as well as slow changes occurring on the timescale of the black hole lifetime and all nontrivial metric effects (including interactions with the Newtonian gravitational potential). These should all be good approximations over the length, time, and energy scales we are interested in. In particular, though we are not computing the effective potential in the near-horizon region, we are assuming that its net effect on vacuum decay over much larger distances is negligible. Let us compute the expectation value of χ 2 on a single particle state, |ψ = d 3 k ψ(k)|k , using the normalizations k |k = δ 3 (k − k ) and d 3 k |ψ(k)| 2 = 1. A short calculation yields ψ|χ(t, x) 2 |ψ = d 3 k 1 d 3 k 2 (2π) 3 ψ * (k 1 )ψ(k 2 ) √ ω 1 ω 2 e i(ω 1 −ω 2 )t e −i(k 1 −k 2 )x ,(20) where ω 1 k 1 and ω 2 k 2 because the initial particle is relativistic. Since we are interested in length and time scales much longer the Compton wavelength of the relativistic particle, the momentum transfer is minute, so k 1 k 2 . Expanding the denominator of the integrand of Eq. (20) as √ ω 1 ω 2 = (ω 1 + ω 2 )/2 + O(ω 1 − ω 2 ), we obtain ψ|χ(t, x) 2 |ψ d 3 k W (k, x) k ,(21) where we have defined the quantum mechanical Wigner function, W (k, x) = d 3 k 1 d 3 k 2 (2π) 3 ψ * (k 1 )ψ(k 2 )δ 3 (k − (k 1 + k 2 )/2)e i(ω 1 −ω 2 )t e −i(k 1 −k 2 )x .(22) As is well known, W (k, x) is a Fourier transform of the density matrix which faithfully encodes all of the information of the wavefunction. Expectation values are computed from the moments of W (k, x) distribution. Moreover, the Wigner function has the remarkable property that it asymptotes to the classical phase space distribution in the classical limit. This limit applies in regions far from the event horizon, where the emitted quanta are free-streaming. While Eq. (21) applies to the case of a single particle, we can straightforwardly accommodate the effects of N independent particles by including a multiplicity factor. We can then relate the Wigner function to the classical phase space density according to W (k, x) = n(k, x)/N , yielding the result of Eq. (14). The quantum and classical derivations yield the same expression for the finite density potential. Catalyzed Vacuum Decay In this section we present a detailed analysis of catalyzed vacuum decay. Parameterizing the metastable vacuum with a simple scalar field theory, we derive precise criteria for catalyzed decay by a point particle and by a black hole. For the sake of generality, we express our results in terms the general finite density potential rather than any specific model. Scalar Potential Consider a scalar field φ with a generic potential. Without loss of generality, we choose the origin of the field space to be the location of the vacuum. Expanding around the vacuum, the leading renormalizable potential is V = m 2 φ 2 2 − aφ 3 3! + λφ 4 4! ,(23) where m 2 > 0 so the origin is a local minimum, and λ > 0 so the potential is bounded from below. For later convenience, we go to "hatted" dimensionless variables, x = xm φ = φ √ λ/m a = a/ √ λm ,V = V λ/m 4 =φ 2 2 −âφ 3 3! +φ 4 4!(24) The origin is metastable provided the cubic coupling is sufficiently large, a > √ 3,(25) in which case the potential has local minima at φ false = 0 φ true = 3 2 â 2 − 8/3 +â .(26) Since the minima are separated by a potential barrier, vacuum decay is mediated by quantum mechanical tunneling. Occasionally, rare quantum fluctuations will nucleate a bubble of true vacuum. Driven by the energy differential between minima, the bubble walls quickly accelerate to near the speed of light and convert the entire universe to true vacuum. As is well-known, the decay rate is exponentially suppressed by the Euclidean action evaluated at the saddle point associated with the vacuum decay process [17,18]. When Eq. (25) approaches saturation, a → √ 3, the minima are approximately degenerate and the nucleated bubble is thin wall. In the opposite regime,â √ 3, the potential barrier is weak and the nucleated bubble is thick wall. In either case, quantum mechanical vacuum decay is exponentially slow because it requires tunneling from the metastable vacuum into a coherent field configuration. Point Particle Instability In empty space, vacuum decay is exponentially slow. Can it be accelerated in the presence of matter? For concreteness, we introduce interactions between φ and a Dirac fermion χ, − L int = (M − yφ)χχ.(27) It is illuminating to understand the most extreme possibility: vacuum decay catalyzed by a single χ particle. At first glance this sounds impossible. On the other hand, if χ is much heavier than the characteristic mass scale of the φ potential, then this prospect becomes less outlandish. When M m, a single χ particle can be treated as a point particle source for φ, At quadratic order in the Lagrangian, this induces a Yukawa potential, ∆V PP = −yφδ 3 (x).(28)φ = ye −mr 4πr .(29) The 1/r divergence is regulated near r ∼ M −1 , the Compton wavelength of χ. At small radii, φ can be much greater than than m, signifying a coherent state comprised of a large number of φ quanta. This is precisely analogous to the huge number of photon quanta that comprise the classical electric field around an electron. The Yukawa profile for φ can easily attain field values at or beyond the true vacuum. In principle, this can drive φ over its potential barrier and catalyze the decay of the vacuum. There are important subtleties, however. First of all, the source term must overcome the gradient energy cost of nucleating a bubble of true vacuum. Second, the Yukawa potential screens at r m −1 , so it is unclear that φ has support at sufficiently large radii to catalyze vacuum decay. Naively, these effects may be compensated for by larger values of y, but the viability of catalyzed vacuum decay remains a detailed question. A definitive answer requires an analysis of the φ equations of motion. In particular, vacuum decay is catalyzed if the equations of motion do not admit a stable solution that interpolates to the metastable vacuum far from the black hole. This proposition is equivalent to saying that every solution which connects to the false vacuum is unstable. Because the system is rotationally symmetric and the metastable vacuum carries zero angular momentum, we restrict to radially symmetric solutions. For our boundary conditions we impose φ(r → ∞) = 0 so that φ asymptotes to the metastable vacuum at infinity, and φ(r → 0) = y/4πr so that φ interpolates correctly onto the Yukawa potential at short distances. In dimensionless variables, the equation of motion isˆ φ +φ −âφ 2 2 +φ 3 3! −ŷδ 3 (x) = 0,(30) whereŷ = y √ λ. While Eq. (30) is not analytically solvable, it is straightforwardly integrated via numerical methods 2 . Doing so over a range values of the two model parameters,â andŷ, we have determined when the system permits a radially symmetric ground state that asymptotes to the metastable vacuum at infinity. Our results are presented in Fig. (1), which is a phase diagram depicting the value ofŷ above which the decay is destabilized. Fig. (1) implies that point particle catalyzed vacuum decay is difficult. Near the thin wall limit, the critical coupling is non-perturbative, so our perturbative analysis cannot be trusted. While catalysis may be viable in the very thick wall limit, it requires very large values of the coupling. Obviously, catalyzed vacuum decay is more efficient in the presence of multiple χ particles. Such a situation can arise in the early universe if the reheating temperature is greater than the mass of χ. However, the associated limits depend sensitively on the reheating temperature after inflation. On the other hand, a black hole is a source of χ quanta whose temperature is essentially independent of the cosmological history of the universe. Black Hole Instability In this section we determine when vacuum decay is catalyzed by a black hole. Expanding ∆V BH around the metastable vacuum, we find ∆V BH = f r 2 × µ 2 + µ 2 φ + 1 2 µ 2 φ 2 + . . . ,(31) introducing a shorthand notation where µ 2 , µ 2 , and µ 2 denote µ 2 [φ] and its derivatives evaluated at φ = 0. Note that µ is equal to the mass of χ in the metastable vacuum. Neglecting φ-independent terms, there are two possibilities for a leading instability: • Tadpole (∆V BH ∼ φ). For generic couplings, µ 2 = 0 and the leading potential term is a tadpole. The basin of attraction of ∆V BH is misaligned from the metastable vacuum. For the appropriate sign of µ 2 , the field is driven towards the potential barrier. • Tachyon (∆V BH ∼ φ 2 ). For certain theories with additional symmetry, µ 2 = 0 and the leading potential term is quadratic. If µ 2 < 0, then ∆V BH induces a local tachyon. In Sec. 3.3.1 and Sec. 3.3.2 we analyze each of these possibilities in turn. As we will see, catalyzed vacuum decay is viable for the case of tadpoles, but not tachyons. Tadpole Instability Consider a scenario where the leading potential term of the finite density potential is a tadpole. This is generic, absent special symmetries restricting the couplings of the theory. Without loss of generality, this scenario is described by the potential in Eq. (23) together with the induced tadpole from the finite density potential. In dimensionless units, the equations of motion arê φ +φ −âφ 2 2 +φ 3 3! −b r 2 = 0,(32) whereb parameterizes the effect of ambient χ particles on the metastable vacuum, b = − f √ λµ 2 m .(33) Catalysis depends sensitively on the sign of µ 2 because this quantity controls how the effective mass of χ varies with φ. In particular,b > 0 drives φ positive and towards the true vacuum, while vice versa forb < 0. When µ m, the Compton wavelength of φ is the longest length scale of the problem. Thus, it is valid to treat φ as a classical field driven by a localized source of χ particles. In analogy with the analysis of Sec. 3.2, we can solve the equations of motion in search of radially symmetric, stable solutions which interpolate to the metastable vacuum. As before we impose Dirichlet boundary conditions at large radii so that the field interpolates to the metastable vacuum. We also fix Dirichlet boundary conditions at the origin. This choice is physically motivated: due to gravitational redshift, the field is tethered to its metastable value at the horizon of the black hole. That said, our results will be insensitive to the choice of boundary conditions at the origin. 3 We can understand the underlying physics by analyzing the shape of the potential near the black hole. In dimensionless units, the full potential isV + ∆V BH , where of the gradient energy compared to the finite density potential. For example, for the model parameters in in Fig. (2), the equations of motion support a stable solution forφ which is depicted by the solid red line. Asb is increased, however, this stationary solution is slowly dragged upwards to larger field values. Past a certain critical value ofb, the solution "snaps" and the equations of motion can no longer support a stable solution. In this case vacuum decay is classically catalyzed by the black hole. ∆V BH = ∆V BH λ/m 4 .(34) To determine the critical value ofb let us study this system in various simplifying limits. For example, consider the theory asâ √ 3, corresponding to a weak potential barrier. In empty space, the associated vacuum tunneling transition is mediated by a thick wall bubble. In the largeâ limit, the dynamics are independent of the quartic stabilization term in the potential. Dropping theφ 3 term in Eq. (32), it is clear that the equations of motion depend on the model parameters in the specific combinationâb. Thus, at largeâ, catalyzed vacuum decay will occur above a critical valueb > constant a ,(35) with a positive proportionality constant which is difficult to compute analytically. This result is physically reasonable: larger values ofâ imply a smaller potential barrier and larger values of b imply stronger finite density effects. Alternatively, consider the opposite limit,â → √ 3, in which the metastable and stable vacua are approximately degenerate. In empty space, the vacuum decay transition is mediated by a thin wall bubble. We can crudely characterize the strength of catalysis by computing the effective potential for a collective coordinate labeling the radius of a nucleated thin wall bubble. In particular, consider a thin wall ansatz of the form φ = 0 ,r >R φ true ,r <R(36) whereR = Rm is the bubble wall radius in dimensionless units andφ true = 2 √ 3 is the true vacuum in the thin wall limit. Integrating the induced tadpole −b/r 2 over the bubble ansatz, we obtain the finite density contribution to the effective potential for the bubble wall, V bubble = V bubble λ/m 4 = 4πR 2σ − 4πR 3ˆ 3 − 4πRbφ true ,(37)whereσ = φ true 0 dφ (2V ) 1/2 = 2 andˆ = 4 √ 3(â− √ 3) are the surface tension and vacuum energy difference, respectively, in the thin wall limit. The potential barrier in Eq. (37) disappears when b >σ 2 φ true = 1 6(â − √ 3) ,(38) which diverges in the thin wall limit. Thus, very largeb is needed to catalyze vacuum decay in this regime. Physically, this is reasonable because a thin wall bubble nucleates with a radius parametrically greater than the microphysical scale of the φ potential. Hence, to catalyze vacuum decay, the total flux of emitted quanta must be commensurately higher to induce a sizable finite density potential at such large radii. Our analytic results are consistent with a numerical analysis of the equations of motion. We have scanned the parameter space, identifying all points that support stable, radially symmetric solutions. The resulting phase diagram is presented in Fig. (3). The red line indicatesb crit , the critical value ofb above which the vacuum is classically unstable. As predicted, forâ √ 3, the critical boundary asymptotes to the form in Eq. (35); forâ → √ 3, the critical boundary diverges as suggested by Eq. (38). To compute the position of the red line in Fig. (3) we have assumed that χ is stable. This result is robust, provided the decay length of the emitted χ particles, R dec = Γ −1 dec γ, is much greater than m −1 , the characteristic length scale of the φ potential. This is often the case because the emitted quanta are relativistic and γ 1. However, for sufficiently heavy χ particles, the decay length may be of order or shorter than the Compton wavelength of φ. In this scenario, the profile of χ flux is modified by the replacement and the finite density potential is suppressed at large radii. As a result, catalyzed vacuum decay is more difficult to achieve. To illustrate the effect of χ decays, we have computedb crit assuming R dec = m −1 , which is depicted by the blue line in Fig. (3). Interestingly, even if χ decays, black hole catalyzed vacuum decay can still occur. That said, for R dec m −1 ,b crit diverges and catalysis shuts off. 1 r 2 → e −r/R dec r 2 ,(39) Finally, let us comment on the effects of the φ quanta emitted by the black hole. Setting χ = φ in the above analysis, we find that b = f a √ λ m .(40) Sinceb > 0, φ is driven toward the true vacuum. However, because f 1 and a ∼ m, we always findb 1. It is clear from Fig. (3) that we needb ∼ O(1) to catalyze vacuum decay, so φ quanta are negligible for catalyzed vacuum decay. Tachyon Instability Next, we consider a scenario where the leading instability of the finite density potential is a tachyon. This requires that µ 2 = 0 and µ 2 < 0, so the tadpole contribution vanishes. This occurs if the dynamics are invariant under a parity of the scalar field, φ → −φ, which is automatic when φ is a component of a charged multiplet. This parity constrains the equations of the motion to the formˆ φ + 1 −ĉ r 2 φ + O(φ 3 ) = 0,(41) whereĉ > 0 controls the strength of the induced tachyon, c = −f µ 2 . (42) As we will see, the potential beyond quadratic order will be unimportant for the coming discussion. Naively, Eq. (41) suggests that small fluctuations of φ are unstable to a localized tachyon in a picture reminiscent of the localized tadpole in Eq. (32). To understand whether this instability catalyzes vacuum decay, let us compute the general solution of Eq. (41) in the linearized limit. We assume a radially symmetric ansatz, φ(t,r) = e −iωt r U (r),(43) whereω is the frequency in dimensionless units. The equation of motion can be massaged into the form of a Schrodinger equation, ω 2 U (r) = − ∂ 2 ∂r 2 + 1 −ĉ r 2 U (r).(44) Applying the quantum mechanical analogy, we identify the quantity in square brackets as the "Hamiltonian". If this Hamiltonian supports negative energy bound state solutions, thenω 2 < 0 and the "ground state" of the system corresponds to a configuration with imaginary frequency. This signals a true tachyonic instability in the theory. Eq. (44) has a general analytic solution which is real and vanishing at infinity, U (r) ∝ √ εrK β (εr),(45) where β = 1/4 −ĉ and ε 2 = 1 −ω 2 . Because U (r) is a bound state solution, ε should be a quantized. However, it appears as a continuous parameter labeling the eigenmodes of Eq. (44). This is a sign of the underlying conformal symmetry of the potential, whereby space and time rescale uniformly. With no dimensionful parameters to provide a gap for the discretuum, the spectrum of eigenmodes is continuous and unbounded from below. However, in any realistic physical system, the potential is regulated at small radii by a physical short distance scale. For the realistic system of a black hole, this regulator is the Schwarzschild radius. More generally, fixing a boundary condition at some small radius will discretize ε, provided that U (r) has nodes. This only happens if the solution is oscillatory, which is only possible if β is imaginary, sô c > 1/4.(46) This condition is well known in the context of conformal quantum mechanical systems [16]. Eq. (46) is reasonable because the localized tachyon must overpower the gradient energy cost required to destabilize the field. However, this inequality is difficult to satisfy because according to Eq. (17), f is small, and µ 2 is proportional to perturbative couplings. Thus, Eq. (46) is never satisfied and the theory does not support an unstable mode; vacuum decay is not catalyzed by black hole induced tachyons. There is no catalysis in the white region because the finite density potential induced by emitted χ particles cannot overcome the potential barrier for φ. There is no catalysis in the black region because black holes that are sufficiently hot to produce χ particles decay too fast to influence the φ dynamics. Phenomenological Implications In this section we formulate the necessary conditions for black hole catalyzed vacuum decay. For a scalar φ sitting at a metastable vacuum, decay is catalyzed by Hawking radiated χ particles if the following criteria are simultaneously satisfied: i) At least one black hole has decayed in our past light cone. ii) This black hole has a lifetime longer than the Compton wavelength of φ. iii) The emitted χ quanta have a decay length longer than the Compton wavelength of φ. iv) These particles couple to φ appropriately to drive φ over its potential barrier. For our analysis, we simply assume condition i). Crucially, the Hawking temperature of the decaying black hole scans up to the Planck scale, inevitably crossing the threshold for χ particle production, T M . To influence the evolution of φ, however, this black hole must survive for a period longer than the characteristic wavelength set by the φ potential. This is condition ii), which is often satisfied because the black hole lifetime scales inversely with the strength of gravity. Given that the black hole is sufficiently long-lived, a cloud of Hawking radiation forms around it. For stable χ particles, this halo forms a 1/r 2 profile, while for unstable χ particles the distribution dissipates with an extra factor of e −r/R dec , where R dec is the χ decay length. Condition iii) guarantees that the phase space distribution of χ is non-zero at scales of order the Compton wavelength of φ. Otherwise, the finite density potential induced by the χ quanta will be too weak to affect the φ dynamics. Since the emitted χ particles are relativistic, R dec is enhanced by a substantial boost factor which makes this condition more easily satisfied. Lastly, condition iv) says that the couplings of φ to χ must have the appropriate sign and magnitude to drive φ over its potential barrier. Example Model Safeguarding the vacuum from catalyzed decay implies new constraints on particle physics models. For concreteness, consider an explicit model defined by the scalar potential in Eq. (23), together with the interaction term in Eq. (27). Presented all together, the Lagrangian is: L = ∂φ 2 2 − m 2 φ 2 2 − aφ 3 3! + λφ 4 4! +χi/ ∂χ −χ(M − yφ)χ,(47) where a > √ 3λm so that the origin is metastable; the true vacuum is at positive values of φ. If condition i) is true, then χ particles start to be emitted by the black hole as soon as T M . However, hotter black holes decay faster, which is in tension with condition ii), the criterion that τ BH m −1 where τ BH is the black hole lifetime defined in Eq. (5). Plugging T M into condition ii), we find that M 10 m m 2 Pl π 2 n * 1/3 ,(48) which is a necessary condition for black hole catalyzed vacuum decay. Here n * is the effective number of massless degrees of freedom emitted by the black hole, accounting for the differences in grey body factors between particles of different spin [12]. Without knowledge of the full spectrum beyond the standard model, we cannot say for certain how large n * is. However, for n * 10 3 it makes little difference. Physically, Eq. (48) is reasonable: if M is too large, then black holes that are hot enough to produce χ will decay too quickly to influence the evolution of φ. Given the Lagrangian in Eq. (47), χ is stable and condition iii) is thus satisfied. We will return to the possibility of unstable χ shortly. If y is positive, then φ is driven to positive values in order to decrease the mass of the emitted χ particles. According to Eq. (11), the effective mass is µ 2 [φ] = (M − yφ) 2 . Plugging into Eq. (33), we find that condition iv) is satisfied if y > 0 and M > m 2f δ .(49) Here we have defined a quantity δ = y √ λ/b crit which is order one or smaller, whereb crit is indicated by the red line in Fig. (3). Since f 1, catalysis requires a hierarchy between M and m. Note that in this regime, Eq. (6) is safely satisfied when T M at radii r ∼ m −1 , so the mean field approximation for the χ phase space density is justified. Interestingly, black hole catalyzed vacuum decay places a stringent constraint on large hierarchies among interacting states: a stable vacuum requires M to sit outside a window bounded from above by Eq. (48) and from below by Eq. (49). In Fig. (4), the large shaded red region satisfies conditions i) -iv) and is subject to catalyzed vacuum decay. The black region violates Eq. (48) because black holes sufficiently hot to produce χ particles decay too fast to affect the φ field evolution. The white region violates Eq. (49) because the χ mass is too small to overcome the potential barrier. While χ is stable for the theory defined in Eq. (47), it decays in many realistic models where condition iii ) is not automatic. For unstable χ particles, the phase space distribution of χ dissipates according to the replacement in Eq. (39). The decay length of χ is R dec = Γ −1 dec γ, which can be significantly enhanced by the boost factor γ ∼ T /M , as discussed in detail in Sec. 3.3.1. To see how χ decays weaken our limits, we have included contours of red bands in Fig. (4) indicating critical values of Γ dec /M δ, above which R dec ≤ m −1 , so the decay length is shorter than the Compton wavelength of φ and catalysis shuts off. For prompt decays, the instability region is smaller, but still viable, especially at smaller values of m. On the other hand, as the decays become longer-lived-say if they are mediated through higher dimension operators-then a greater portion of the parameter space falls victim to catalyzed vacuum decay. Standard Model At tree-level, the SM Higgs potential supports a unique minimum: the electroweak symmetry breaking vacuum. However, as is well-known the vacuum structure is enriched at loop-level [21,22]. Taken at face value, the observed Higgs mass [23,24] implies that the Higgs quartic runs negative at an intermediate scale of order 10 10 GeV [25,26]. Strictly speaking, the Higgs potential is unbounded from below at this scale and the quantum theory does not have a ground state. Of course, in any well-defined ultraviolet completion, this unbounded field direction is lifted by higher dimension operators. In fact, negative quartic couplings are a natural byproduct of integrating out additional scalar fields. Naively, this instability is quite severe. The mass is tachyonic and the quartic coupling is negative. However, [27] famously showed that even in the absence of a potential barrier, fluctuations of the Higgs are classically stable, on account of the substantial gradient energy cost of nucleating a bubble of true vacuum. Instead, the Higgs must quantum mechanically tunnel from the electroweak vacuum. The resulting decay rate is exponentially suppressed and thus the lifetime of the electroweak vacuum is longer than the age of the universe [25,26]. Hence, the SM Higgs lies in an apparent region of metastability, but in a way that is consistent with observation. Note that this conclusion is subject to important experimental uncertainties on the top quark mass and strong gauge coupling. If the electroweak symmetry breaking minimum is indeed metastable, then it is reasonable to ask whether vacuum decay can be catalyzed by a black hole. Since φ is a component of an electroweak doublet, the scalar potential preserves a φ → −φ parity. Hence, the finite density potential is even in φ, so µ 2 = 0 and the leading contribution is quadratic. In the SM, the dominant contributions to the finite density potential come from the top quark, the electroweak bosons, and the Higgs itself. Since these particles acquire mass entirely from electroweak symmetry breaking, larger values of the Higgs will increase their effective mass. Thus, µ 2 > 0 and the induced potential tends to push φ towards the origin of field space. While this phenomenon may have interesting implications for restoration of electroweak symmetry, e.g. for the purposes of electroweak baryogenesis [28], it does not drive the field in the direction of the quartic instability. Black holes do not catalyze decay of the electroweak vacuum in the SM. Beyond the Standard Model In Sec. 4.1 we derived new stability limits on a simple scalar model. We then argued in Sec. 4.2 that catalyzed decay does not constrain the SM. What about constraints on motivated extensions of the SM? Trivially, our limits apply to the SM augmented by a singlet scalar φ with the potential in Eq. (23). However, similar constraints also apply to metastable vacua in the singletextended SM [31,32] and the next-to-minimal supersymmetric standard model (NMSSM) [33,34,35]. Depending on the precise couplings between the Higgs and the singlet, a black hole can drive the Higgs from the electroweak symmetry breaking vacuum. A proper treatment of this phenomenon will likely require a multi-field analysis beyond the scope of the present work. We have seen that black hole catalyzed vacuum decay places stringent limits on light scalars coupled to heavy particles. As emphasized in [4] this scenario automatically arises when φ is the pseudo-Goldstone boson of a spontaneously broken symmetry. Here χ will be parametrically heavier than φ, provided its mass is unprotected by the preserved symmetry group. For example, we could identify φ with the radion field parameterizing a de-compactification transition to a higher dimensional vacuum, and χ with the associated Kaluza-Klein particles. Such mass hierarchies also arise in models of metastable supersymmetry (SUSY) breaking. As discussed in [36,37], at tree-level, the SUSY breaking modulus is generically a flat direction in field space. It is then natural to identify φ with the SUSY breaking modulus and χ with heavier messenger states. Without loss of generality, we can shift φ so that its vacuum expectation value is at the origin; in analogy with Eq. (27), its couplings become W int = (M − yφ)χχ.(50) An evaporating black hole will emit a flux of χ messenger particles which induces a finite density potential for the SUSY breaking modulus φ. If conditions i) -iv) are satisfied, then the metastable SUSY breaking vacuum is unstable to black hole catalyzed vacuum decay. It remains to be seen whether these conditions are consistent with detailed model building constraints such as R-symmetry breaking. We leave a more detailed analysis for future work. Conclusions We have presented a systematic analysis of vacuum decay induced by ambient matter. For perturbative theories, it is difficult for a single particle χ to drive φ over its potential barrier. As shown in Fig. (1), catalysis only occurs if these states are strongly coupled or if the barrier is very weak. On the other hand, catalysis is much easier in the presence of many χ quanta, for example as would result from the Hawking radiation of a black hole. In Eq. (15), we have presented the finite density potential induced for φ by the χ quanta emitted by a black hole. Because we are interested in distances far from the event horizon, subtleties about the information paradox are irrelevant to our analysis-the black hole is simply a source of high energy particles, much like star. If the basin of attraction of the finite density potential is sufficiently misaligned from the metastable vacuum, then φ can be driven over its potential barrier, catalyzing decay. The critical values of couplings at which catalysis occurs are shown in Fig. (3). Finally, we have summarized the necessary conditions for black hole catalyzed vacuum decay in the beginning of Sec. 4. Demanding vacuum stability implies new constraints on theories with hierarchical spectra, e.g. as shown in Fig. (4) for the model defined Eq. (47). This work leaves many avenues for future work. As noted in Sec. 4.3, first and foremost is a comprehensive analysis of new limits on beyond the SM theories, e.g. singlet extensions of the SM, the NMSSM, the radion, and metastable SUSY breaking models. Stability bounds will have the most significance for theories with hierarchical spectra. Moreover, our findings could have more general implications for the landscape: vacua with larger mass hierarchies are in greater danger of catalyzed vacuum decay. Throughout, our discussion has assumed the decay of at least one black hole in our past light cone. As is well-known, primordial black holes can be produced by over-densities of curvature perturbations after inflation, as well as during first order phase transitions. It would be interesting to understand the likelihood that exactly zero primordial black holes were produced in our past light cone. Lastly, it may be fruitful to consider other applications of the finite density potential induced by a black hole. In principle, this potential can drive the φ potential into a more symmetric phase. Effectively, this forms a domain wall surrounding the black hole. This field configuration could accommodate a novel mechanism for baryogenesis, e.g. if electroweak symmetry or a grand unified symmetry is restored. Theoretical Cosmology and Physics. C.C is supported by a DOE Early Career Award #DE-SC0010255 and S.L. is supported by a John A. McCone Postdoctoral Fellowship. C.C. would also like to thank the Aspen Center for Physics and the Kavli Institute for Theoretical Physics in Santa Barbara, where part of this work was completed. Figure 1 : 1Phase diagram indicating when a single point particle classically catalyzes the decay of the vacuum. Destabilization requires a very large coupling (ŷ 1) or a very weak potential barrier (â √ 3). Fig. ( 2 Figure 2 : 22) depicts contours ofV + ∆V BH as a function ofφ andr, fixingâ = 2 andb = 1.3. The finite density potential dominates atr = 0 and is negligible asr → ∞. The dashed blue curves label critical points at which ∂(V + ∆V BH )/∂φ = 0. At large radii there are three such lines, indicating the positions of the metastable vacuum, stable vacuum, and potential barrier. Near the black hole, two of these lines merge at a critical radius at which the metastable vacuum and the potential barrier are coincident. At this inflection point the barrier disappears completely. Naively, small field fluctuations near this critical radius are unstable to exponential growth, suggesting the onset of catalyzed vacuum decay. This conclusion is incorrect, however-while the potential drivesφ towards the true vacuum, this is counteracted by the gradient energy cost of spatial variations ofφ. Hence, the stability of the vacuum depends on the relative strength Contours of the full potential,V + ∆V BH , as a function of the radius,r, and the field value,φ, fixingâ = 2 andb = 1.3. The dashed blue lines label the positions of the metastable vacuum, stable vacuum, and potential barrier. At finite radius the barrier merges with the metastable vacuum and disappears. The solid red line depicts a stableφ profile for this model. Figure 3 : 3Critical values of model parameters, above which vacuum decay of φ is classically catalyzed by a black hole. The red (blue) line corresponds to emitted χ particles which have a decay length much longer than (equal to) the Compton wavelength of φ. Large values of b >b crit typically arise in theories with hierarchical masses. The thick wall (â √ 3) and thin wall (â → √ 3) limits are consistent with the analytic predictions of Eq. (35) and Eq. (38). Figure 4 : 4Phase diagram of the parameter space constrained by black hole catalyzed vacuum decay for the explicit model in Eq. (47). It is possible that a large correction to the effective potential in the near-horizon region could alter our conclusions, but since the near-horizon region is parametrically small compared to our region of interest we do not believe this will be the case. Eq. (30) can be numerically solved by substituting φ = ∆φ + y/4πr to eliminate the delta function term. 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We introduce a calculus for parameter-dependent singular Green operators on the half-space R n + that combines both elements of Grubb's calculus for boundary value problems of finite regularity and techniques of Schulze's calculus for pseudodifferential operators on manifolds with edges.

1812.07661

6ca9e99a6f053b506cf4e4022b24cb4802d87c8e

Singular Green Operators of Finite Regularity in the Edge Algebra Formalism 17 Dec 2018 Jörg Seiler Singular Green Operators of Finite Regularity in the Edge Algebra Formalism 17 Dec 2018 We introduce a calculus for parameter-dependent singular Green operators on the half-space R n + that combines both elements of Grubb's calculus for boundary value problems of finite regularity and techniques of Schulze's calculus for pseudodifferential operators on manifolds with edges. Introduction Since the seminal papers of Seeley [14], [15] on complex powers of elliptic operators and resolvents of boundary value problems, respectively, it became a broadly applied technique in partial differential equations and geometric analysis to exploit the pseudodifferential structure of resolvents or other parameter-dependent families of operators. Applications are quite manifold, including resolvent estimates and resolvent trace asymptotics, complex and imaginary powers, H ∞ -calculus, maximal L p -regularity, heat kernels and heat trace asymptotics, index theory, spectral asymptotics, ζ-functions and regularized determinants. In [3], Boutet de Monvel introduced a calculus for operators on manifolds with boundary. It contains all classical differential boundary value problems and, in case of Shapiro-Lopatinskii ellipticity, Fredholm inverses can be constructed in the calculus. If we locally model the manifold as a half-plane R n + with boundary R n−1 , one considers operators of the form A + + G K T Q : S (R n + , C L ) ⊕ S (R n−1 , C M ) −→ S (R n + , C L ′ ) ⊕ S (R n−1 , C M ′ ) , (1.1) acting between spaces of rapidly decreasing functions (extendable to Sobolev and Besov spaces); here L, M, L ′ , M ′ are non-negative integers, modelling involved vector bundles over the manifold (these numbers are allowed to be zero). Moreover, A + is the restriction of a pseudodifferential operator to the half-space, K is a so-called potential or Poisson operator, T is a trace operator, and Q is a pseudodifferential operator on the boundary. In the terminology of [3], such matrices are called Green operators, while those with A + = 0 are termed singular Green operators. In the light of what has been said above, it is natural to seek a corresponding calculus for parameter-dependent boundary value problems, where parameter-elliptic elements possess an inverse (for large parameter) in the calculus. Typically, the parameter ranges in a conical subset of the Euclidean space; in what follows we focus for simplicity on a real parameter µ ≥ 0. Such a calculus has been realized by Grubb in the middle 1980's. A complete exposition one can find in [5], where also applications as mentioned above are described. In particular, the calculus allows the construction of resolvents of pseudodifferential (and non-differential) operators, subject to boundary conditions. One of the key features in this calculus is to start out from operators A(µ) = a(x, D, µ) build upon symbols of finite regularity, i.e., satisfying uniform estimates of the form |D β x D α ξ D j µ a(x, ξ, µ)| ≤ C αβj ξ ν−|α| + ξ, µ ν−|α| ξ, µ d−ν−j for arbitrary orders of derivatives. Here, d is the order of the symbol, and ν its socalled regularity (note that regularity in this context does not refer to smoothness of the symbols in the x-variable). This structure induces corresponding notions of order and regularity in the class of singular Green operators. In the early 1980's Schulze initiated his programme on calculi of pseudodifferential operators on manifolds with singularities, cf. [8], [11], [12]. In his approach, manifolds with boundary can be interpreted as a special case of a manifold with edge. Near the edge, a manifold with edge has the structure of a fibre-bundle over a smooth manifold with fibre being a cone over some base manifold; if the base is a single point one obtains a manifold with boundary. The operators considered on a manifold with edge are more general than those on a manifold with boundary, in particular, there is no concept of transmission condition involved. The resulting pseudodifferential calculus, the so-called edge algebra, again consists of blockmatrices similar to (1.1), but with A + replaced by more general operators, and a different class of singular Green operators. We cannot go into details here, but only want to point out one key ingredient, which is significant for the present paper, namely the systematic use of operator-valued pseudodifferential symbols acting in Hilbert spaces equipped with a strongly continuous group action. As it turns out, singular Green operators have a very elegant and transparent description in this set-up. In the end, the calculus for such operator-valued symbols is nearly identical with that of usual scalar-valued pseudodifferential symbols in Euclidean space (as laid out, for example, in the text-book of Kumano-go [7]). Schulze also introduced a parameter-dependent version of the edge algebra. It allows the construction of resolvents of differential operators, subject to edge conditions. However, pseudodifferential (non-differential) operators are not covered in this setting. A parameterdependent version of Boutet de Monvel's calculus using the formalism of the edge algebra is described in Schrohe, Schulze [10]. In the present paper we introduce a calculus of singular Green operators on the half-space R n + , combining both Grubb's concept of operators of finite regularity and Schulze's concept of operator-valued pseudodifferential symbols. Grubb's class of singular Green operators contains two natural sub-classes of operators: those of infinite regularity (which we refer to as strongly parameter-dependent operators) and so-called weakly parameter-dependent operators. One of our key observations is that weakly parameter-dependent operators can be characterized in the framework of operator-valued symbols in the spirit of Schulze's theory (this is also true for strongly parameter-dependent operators, but this was already known, cf. [9] for example). In Section 4 we show this result in detail for the class of Poisson operators. We then implement the rsulting structure of operator-valued symbols in our class of singular Green operators which, by definition, consists in the sum of the weakly and strongly parameter-dependent operators. Doing so, it does not coincide with Grubb's class but is a subclass of it. We show that our class admits a full calculus and do recover many results stated in [5] in an alternative way. In particular, this concerns the parametrix construction discussed in Section 6. We hope that our findings could prove useful in a possible generalization of the concept of finite regularity from boundary value problems to the edge algebra, thus enabling the treatment of resolvents of pseudodifferential operators on manifolds with edges. We plan to address this in future research. The paper is structured as follows: In Section 2 we introduce some classes of operator-valued symbols in Hilbert spaces with group actions, called strongly and weakly parameter-dependent symbols, and discuss their calculus. The sum of such symbols leads to a new class whose calculus is discussed in Section 3. After the above mentioned analysis of Poisson operators in Section 4 we introduce our class of singular Green operators in Section 5. Section 6 is devoted to the paramtrix construction of operators of the form "identity plus singular Green operator" of positive regularity. Section 7 introduces singular Green operators of positive type and Section 8 concerns the action of singular Green operators in Sobolev spaces. In the first section of the appendix, Section 9, we prove a characterization of pseudodifferential operators with operator-valued symbols by iterated commutators in the sense of the classical result of Beals [1], [2], which we need in the parametrix construction. In the final Section 10 we recall the definition of certain scales of Sobolev spaces on the half-axis which will be used frequently throughout the text. Important notation: For x ∈ R n and y ∈ R m we shall write x = (1 + |x| 2 ) 1/2 , x, y = (x, y) = (1 + |x| 2 + |y| 2 ) 1/2 . For two functions f, g defined on some set Ω the notation f g or f (y) g(y) means that there exists a constant C ≥ 0 such that f (y) ≤ Cg(y) for all y ∈ Ω. In a chain like f g h, the constant may be different in any inequality Operator-valued symbols and their calculus Let E be a Hilbert space. A group-action is a map κ : (0, +∞) → L (E) which is continuous in the strong operator topology and satisfies κ 1 = 1 as well as κ λ κ σ = κ λσ for every λ, σ > 0. A standard result from semigroup theory ensures the existence of a constant M = M (κ) ≥ 0 such that (2.1) κ λ L (E) max(λ, λ −1 ) M . Given such a group-action, we shall write κ(y) = κ y , κ −1 (y) = κ −1 y = κ 1/ y (y ∈ R m ). For the present paper, the following example of a group-action is fundamental. Example 2.1. For smooth scalar-valued functions φ defined on R + [or R] with compact support we define (κ λ φ)(t) = λ 1/2 φ(λt), λ > 0. We extend κ λ to act on the space of distributions D ′ (R + ) [or D ′ (R)] by κ λ T, φ = T, κ −1 λ φ . We shall call this the "standard group-action" on R + [or R]; its restriction to various function spaces defines a group-action in the above sense, for example, the restriction to the Sobolev spaces H s (R + ) [or H s (R)]. Note that it defines a group of unitary operators on the standard L 2 -spaces. In the following let E 0 , E 1 , E 2 be Hilbert spaces equipped with group actions κ 0 , κ 1 , and κ 2 , respectively. 2.1. Strongly parameter-dependent symbols. We recall the definition of parameter-dependent symbols in the sense of Schulze. Definition 2.2. Let d ∈ R. Then S d 1,0 (R n−1 ×R + ; E 0 , E 1 ) is the space of all smooth functions a(x, ξ; µ) : R n × R n → L (E 0 , E 1 ) satisfying (2.2) κ −1 1 (ξ, µ) D α ξ D β x D j µ a(x, ξ; µ) κ 0 (ξ, µ) L (E0,E1) ξ, µ d−|α|−j . for every order of derivatives. Due to (2.1), the associated space of regularizing symbols S −∞ (R n × R + ; E 0 , E 1 ), defined by taking the intersection over all d ∈ R, consists of those symbols a satisfying D α ξ D β x D j µ a(x, ξ; µ) L (E0,E1) ξ, µ −N for every N ≥ 0 and every order of derivatives. Obviously, it does not depend on the involved semi-groups. In other terms, S −∞ (R n × R + ; E 0 , E 1 ) = C ∞ b (R n x , L (E 0 , E 1 )) ⊗ π S (R n × R + ), the completed projective tensor-product of the space of smooth L (E 0 , E 1 )-valued functions with bounded derivatives of arbitrary order and the space of rapidly decreasing functions. 2.1.1. Homogeneous functions. Let V ⊆ (R n × R + ) \ {0} be a conical set. A function a : R n × V → L (E 0 , E 1 ) is called κ-homogeneous (or also twisted homogeneous) of degree d ∈ R if a(x, λξ; λµ) = λ d κ 1,λ a(x, ξ; µ)κ −1 0,λ ∀ (ξ, µ) ∈ V ∀ λ > 0. Definition 2.3. Let us denote by S (d) (R n × R + ; E 0 , E 1 ) the space of all smooth functions a : R n ×V → L (E 0 , E 1 ) with V = (R n ×R + )\{0} that are κ-homogeneous of degree d and satisfy, for every order of derivatives, (2.3) κ −1 1,|ξ,µ| D α ξ D β x D j µ a(x, ξ; µ) κ 0,|ξ,µ| L (E0,E1) |ξ, µ| d−|α|−j In particular, if a ∈ S (d) (R n × R + ; E 0 , E 1 ) then a(x, ξ; µ) = |ξ, µ| d κ 1,|ξ,µ| a x, ξ |ξ, µ| ; µ |ξ, µ| κ −1 0,|ξ,µ| . If χ(ξ, µ) is an arbitrary 0-excision function and if a ∈ S (d) (R n × R + ; E 0 , E 1 ) then χa ∈ S d 1,0 (R n × R + ; E 0 , E 1 ). Classical symbols. Classical symbols (also called poly-homogeneous symbols) have an asymptotic expansion in homogeneous components. Definition 2.4. S d (R n ×R + ; E 0 , E 1 ) denotes the space of all symbols a ∈ S d 1,0 (R n × R + ; E 0 , E 1 ) for which exists a sequence a (d−j) ∈ S (d−j) (R n × R + ; E 0 , E 1 ), j ∈ N 0 , such that, for every N ∈ N, a − N −1 j=0 χ(ξ, µ)a (d−j) ∈ S d−N 1,0 (R n × R + ; E 0 , E 1 ). If a ∈ S d (R n × R + ; E 0 , E 1 ) is as in the previous definition, then a (d) is called the homogeneous principal symbol of a. It is well defined, since a (d) (x, ξ; µ) = lim λ→+∞ λ −d κ −1 1,λ a(x, λξ; λµ)κ 0,λ , (ξ, µ) = 0. 2.2. Weakly parameter-dependent symbols. Roughly speaking, strong parameter-dependence means considering the parameter µ together with the covariable ξ as a new joint co-variable η := (ξ, µ) and then to impose the standard estimates of pseudodifferential symbols in η. We now introduce a class where the behaviour is different: differentiation with respect to ξ only improves the decay in ξ; still, differentiation with respect to µ improves the decay in η = (ξ, µ). We shall use the terminology "weak" parameter-dependence. Definition 2.5. Let d, ν ∈ R. Then S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ denotes the space of all symbols satisfying the estimates κ −1 1 (ξ, µ) D α ξ D β x D j µ a(x, ξ; µ) κ 0 (ξ, µ) L (E0,E1) ξ ν−|α| ξ, µ d−ν−j . The reason for using the subscript "κ, κ" will become clear later on, cf. Section 5.1. The corresponding class of regularizing symbols is defined by S d−∞,ν−∞ (R n × R + ; E 0 , E 1 ) κ,κ := ∩ N ∈R S d−N,ν−N 1,0 (R n × R + ; E 0 , E 1 ) κ,κ . It consists of those symbols with κ −1 1 (ξ, µ) D α ξ D β x D j µ a(x, ξ; µ) κ 0 (ξ, µ) L (E0,E1) ξ −N ξ, µ d−ν−j for arbitrary N ≥ 0. Due to (2.1), since 1 µ ξ, µ −1 ξ and since N is arbitrary, we may equivalently use the estimates κ −1 1 (µ) D α ξ D β x D j µ a(x, ξ; µ) κ 0 (µ) L (E0,E1) ξ −N µ d−ν−j for arbitrary N ≥ 0. Note that this space generally depends on the involved groupactions. Remark 2.6. Asymptotic summation of a sequence of symbols a k of order d − k and regularity ν − k is well-defined. In fact, if χ(ξ) is a 0-excision function and 0 < ε k k→+∞ − −−−− → 0 sufficiently fast, then the series a(x, ξ; µ) = +∞ k=0 χ(ε k ξ)a k (x, ξ; µ) converges in S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ and a − N −1 k=0 a k belongs to S d−N,ν−N 1,0 (R n × R + ; E 0 , E 1 ) κ,κ for every N . 2.2.1. Classical symbols. Also in case of weak parameter-dependence we can introduce the subclass of classical symbols. Definition 2.7. Let S (d,ν) (R n−1 × R + ; E 0 , E 1 ) κ,κ be the space of all smooth func- tions a : R n × V → L (E 0 , E 1 ) with V = (R n \ {0}) × R + that are κ-homogeneous of degree d, and satisfy the estimates (2.4) κ −1 1,|ξ,µ| D α ξ D β x D j µ a(x, ξ; µ) κ 0,|ξ,µ| L (E0,E1) |ξ| ν−|α| |ξ, µ| d−ν−j . Note that homogeneous functions in the present context are only defined for ξ = 0 while, in case of strong parameter-dependence, they were defined whenever (ξ, µ) = 0. If χ(ξ) is an arbitrary 0-excision function and if a ∈ S (d,ν) (R n × R + ; E 0 , E 1 ) κ,κ then it is easy to verify that χ a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ . Definition 2.8. S d,ν (R n × R + ; E 0 , E 1 ) κ,κ denotes the space of all a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ which admit a sequence a (d−j,ν−j) ∈ S (d−j,ν−j) (R n ×R + ; E 0 , E 1 ) κ,κ , j ∈ N 0 , such that a − N −1 j=0 χ(ξ) a (d−j,ν−j) ∈ S d−N,ν−N 1,0 (R n × R + ; E 0 , E 1 ) κ,κ for every N ∈ N 0 . If a ∈ S d,ν (R n × R + ; E 0 , E 1 ) κ,κ is as in the previous definition, then a (ν,d) is called the homogeneous principal symbol of a. It is well defined, since a (d,ν) (x, ξ; µ) = lim λ→+∞ λ −d κ −1 1,λ a(x, λξ; λµ)κ 0,λ , ξ = 0. To this end note that if r belongs to S d−1,ν−1 with some M ≥ 0, cf. (2.1); for fixed (ξ, µ) with ξ = 0, the right-hand side behaves like λ d−1 for λ → +∞. Remark 2.9. Let a ∈ S (d,ν) (R n × R + ; E 0 , E 1 ) κ,κ . In a small neighborhood of a point (0, µ 0 ) with µ 0 > 0 we have that a(x, ξ, µ) L (E0,E1) |ξ| ν . Thus if ν > 0, it follows that lim ξ→0 a(x, ξ; µ) = 0, µ > 0. 1,0 (R n × R + ; E 0 , E 1 ) κ, Hence, in this case, a extends to a κ-homogeneous function on R n × V with V = (R n × R + ) \ {0} by setting a(x, 0, µ) = 0 for µ > 0. We shall implicitly identify a with its extension. Let us also remark that (2.5) S d (R n × R + ; E 0 , E 1 ) ⊆ S d,0 (R n × R + ; E 0 , E 1 ) κ,κ . Oscillatory integrals and calculus. With a parameter-dependent operator valued symbol a from one of the spaces introduced above, we associate the µ-dependent family of continuous operators op(a)(µ) = a(x, D; µ) : S (R n , E 0 ) −→ S (R n , E 1 ) defined by [op(a)(µ)u](x) = e ixξ a(x, ξ; µ) u(ξ) dξ, u ∈ S (R n , E 0 ). The most efficient way to describe the calculus for such operator-families is based on the concept of oscillatory integrals in the spirit of [7], but extended to Frèchet space valued amplitude functions. For convenience of the reader and for applications in the sequel we give a short summary of this concept; for more details see [16], [4]. Let E be a Fréchet space whose topology is described by a system of semi-norms p n , n ∈ N. A smooth function q(y, η) : R m × R m → E is called an amplitude function with values in E, provided there exist sequences (m n ) and (τ n ) such that p n D α η D β y q(y, η) y τn η mn for all n and for all orders of derivatives. If χ(y, η) denotes a cut-off function with χ(0, 0) = 1, the so-called oscillatory integral Os − e −iyη q(y, η) dydη := lim ε→0 R n ×R n e −iyη χ(εy, εη)q(y, η) dydη exists and is independent on the choice of χ. Note that for a continuous, E-valued function f with compact support, f (y, η) dydη is the unique element e ∈ E such that e ′ , e = e ′ , f (y, η) dydη for every functional e ′ ∈ E. For simplicity of notation we shall simply write rather than Os − . Example 2.10. Given two parameter-dependent symbols a 0 and a 1 , consider q(y, η) = (x, ξ) → a 1 (x + y, ξ; µ)a 0 (x, ξ + η; µ) , y, η ∈ R n . It is then straight-forward to verify the following: i) If a j ∈ S dj 1,0 (R n × R + ; E j , E j+1 ), j = 0, 1, then q is an amplitude function with values in S d0+d1 1,0 (R n × R + ; E 0 , E 2 ). ii) If a j ∈ S dj,νj 1,0 (R n × R + ; E j , E j+1 ) κ,κ , j = 0, 1, then q is an amplitude function with values in S d0+d1,ν0+ν1 1,0 (R n × R + ; E 0 , E 2 ) κ,κ . Correspondingly, in each of these cases, we can define the so-called Leibniz-product of a 0 and a 1 by (a 1 #a 0 )(x, ξ; µ) = e −iyη a 1 (x + y, ξ; µ)a 0 (x, ξ + η; µ) dydη. The Leibniz-product corresponds to composition of operators, i.e., (2.6) op(a 1 #a 0 )(µ) = op(a 1 )(µ)op(a 0 )(µ). Another important operation for pseudo-differential operators is the so-called formal adjoint. (H s,δ (R + ), L 2 (R + ), H −s,−δ 0 (R + )), (K s,γ (R + ) δ , L 2 (R + ), K −s,−γ (R + ) −δ ) together with the standard group-action of Example 2.1 are Hilbert-triples. If (E 0 , H 0 , E 0 ) and (E 1 , H 1 , E 1 ) are two Hilbert-triples and A : S (R n , E 0 ) → S (R n , E 1 ), then denote by A * the operator S (R n , E 1 ) → S (R n , E 0 ) defined by (Au, v) L 2 (R n ,H1) = (u, A * v) L 2 (R n ,H0) , u ∈ S (R n , E 0 ), v ∈ S (R n , E 1 ). This is the so-called formal adjoint of A. Example 2.13. Let (E 0 , H 0 , E 0 ) and (E 1 , H 1 , E 1 ) be two Hilbert-triples. Given a symbol a, consider q(y, η) = (x, ξ) → a(x + y, ξ + η; µ) * , y, η ∈ R n . It is then straight-forward to verify the following: i) If a ∈ S d 1,0 (R n × R + ; E 0 , E 1 ) then q is an amplitude function with values in S d 1,0 (R n × R + ; E 1 , E 0 ). ii) If a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ then q is an amplitude function with values in S d0,ν0 1,0 (R n × R + ; E 1 , E 0 ) κ,κ . Correspondingly, in each of these cases, we can define the adjoint symbol of a by a ( * ) (x, ξ; µ) = e −iyη a(x + y, ξ + η; µ) * dydη. The adjoint symbol corresponds to the formally adjoint operator, i.e., (2.7) op(a)(µ) * = op(a ( * ) )(µ). To keep the exposition short we have focused on so-called left-symbols a(x, ξ; µ), only depending on the variable x. As in the standard theories of pseudodifferential operators, one can also introduce the corresponding classes of double-symbols a(x, y, ξ; µ) by substituting in the above definitions x by (x, y). The associated pseudodifferential operator is then given by (2.8) op(a)(µ)u](x) = e −iyη a(x, x + y, η; µ)u(x + y) dydη. Here, for fixed µ, the integrand a(x, y, ξ; µ)u(x + y) can be viewed as an amplitude function with values in S (R n , E 1 ). Actually, in this representation, one can also admit functions u ∈ C ∞ b (R n , E 0 ), since then the integrand is an amplitude function with values in C ∞ b (R n , E 1 ). Starting out from a double-symbol a(x, y, ξ; µ) there exist unique left-and right-symbols a L (x, ξ; µ) and a R (y, ξ; µ) such that op(a)(µ) = op(a R )(µ) = op(a L )(µ). There are explicit oscillatory-integral formulas for a L and a R ; for details we refer the reader to the literature. Parameter-dependent symbols of finite regularity The following definition is motivated by Grubb's calculus of pseudodifferential operators with finite regularity. Definition 3.1. With d, ν ∈ R set S d,ν 1,0 (R n × R + ; E 0 , E 1 ) := S d 1,0 (R n × R + ; E 0 , E 1 ) + S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ . Analogously define S d,ν (R n ×R + ; E 0 , E 1 ) . The corresponding spaces of κ-homogeneous symbols is S (d,ν) (R n × R + ; E 0 , E 1 ) :=S (d) (R n × R + ; E 0 , E 1 )+ + S (d,ν) (R n × R + ; E 0 , E 1 ) κ,κ . All spaces in the previous definition are a non-direct sum of Fréchet spaces, hence a Frèchet space itself. In general, we shall use notations like a = a 0 + a if we represent an arbitrary symbol as the sum of two symbols. We shall say that such symbols have order d and regularity ν. Remark 3.2. A symbol a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) satisfies the estimates κ −1 1 (ξ, µ) D β x D α ξ D j µ a(x, ξ; µ) κ 0 (ξ, µ) L (E0,E1) ξ ξ, µ ν−|α| + 1 ξ, µ d−|α|−j for every order of derivatives. Analogously for the space of κ-homogeneous functions where one needs to replace ξ , ξ, µ by |ξ| and |ξ, µ|, respectively. Note that S d,ν 1,0 (R n × R + ; E 0 , E 1 ) = S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ , ν ≤ 0 and S d,+∞ 1,0 (R n × R + ; E 0 , E 1 ) := ∩ ν∈R S d,ν (cl) (R n × R + ; E 0 , E 1 ) =S d 1,0 (R n × R + ; E 0 , E 1 ); in particular, infinite regularity corresponds to strong parameter-dependence. The corresponding fact is true for the classes of classical symbols. From now on, for convenience of notation, we shall drop R n × R + from notation and will use the short-hand notations S d 1,0 (E 0 , E 1 ) := S d 1,0 (R n × R + ; E 0 , E 1 ), S d,ν 1,0 (E 0 , E 1 ) := S d,ν 1,0 (R n × R + ; E 0 , E 1 ), S d,ν 1,0 (E 0 , E 1 ) κ,κ := S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ and similarly for the spaces of classical and κ-homogeneous functions. 3.1. Calculus: composition and formal adjoint. Putting together the results of Section 2.3 and the embedding (2.5), the following theorems are easily verified: Theorem 3.3. The Leibniz-product of symbols, respectively the composition of pseudodifferential operators, induces a bilinear continuous map (a 1 , a 0 ) → a 1 #a 0 : S d1,ν1 1,0 (E 1 , E 2 ) × S d0,ν0 1,0 (E 0 , E 1 ) −→ S d,ν 1,0 (E 0 , E 2 ) and analogously for classical symbols, where d = d 0 + d 1 , ν = min(ν 0 , ν 1 , ν 0 + ν 1 ). In case a j ∈ S dj,νj 1,0 (E j , E j+1 ) κ,κ for at least one of the values j = 0 or j = 1, then a 1 #a 0 ∈ S d,ν 1,0 (E 0 , E 2 ) κ,κ (analogously for classical symbols). In this sense, the S-class is a two-sided ideal. Using the explicit formula for the Leibniz-product and the standard technique of Taylor expansion, one sees that, for every N ∈ N 0 , (3.1) r N (a 1 , a 0 ) := a 1 #a 0 − N −1 |α|=0 1 α! (∂ α ξ a 1 )(D α x a 0 ) ∈ S d−N,ν−N 1,0 (E 0 , E 2 ) with continuous dependence on a 0 and a 1 (analogously for classical symbols). Theorem 3.4. Let (E 0 , H 0 , E 0 ) and (E 1 , H 1 , E 1 ) be two Hilbert-triples. Taking the formal adjoint induces a linear continuous map a → a ( * ) : S d,ν 1,0 (E 0 , E 1 ) −→ S d,ν 1,0 ( E 1 , E 0 ). Moreover, for every N ∈ N 0 and with continuous dependence on a, r ( * ) N (a) := a ( * ) − N −1 |α|=0 1 α! ∂ α ξ D α x a * ∈ S d−N,ν−N 1,0 ( E 1 , E 0 ). The analogous statements hold true for classical symbols. The homogeneous principal symbol. Given a = a 0 + a ∈ S d,ν (E 0 , E 1 ) we define σ d,ν (a)(x, ξ; µ) = a (d) 0 (x, ξ; µ) + a (d,ν) (x, ξ; µ) = lim λ→+∞ λ −d κ −1 1,λ a(x, λξ; λµ)κ 0,λ , ξ = 0. (3.2) (recall that in case of positive regularity ν > 0, the principal symbol extends by continuity to all (ξ, µ) = 0, cf. Remark 2.9). Due to (3.1), the principal symbol behaves multiplicatively under composition, σ d,ν (a 1 #a 0 ) = σ d1,ν1 (a 1 )σ d0,ν0 (a 0 ). Proposition 3.5. Let χ(ξ, µ) and χ(ξ) be two 0-excision functions. Then a (d) 0 + a (d,ν) → χa (d) 0 + χ a (d,ν) induces a well-defined and surjective map S (d,ν) (E 0 , E 1 ) −→ S d,ν (E 0 , E 1 ) S d−∞,ν−∞ (E 0 , E 1 ). Proof. After multiplication with |ξ, µ| −d we may assume w.l.o.g. that d = 0. Then it suffices to show that χa (0) 0 + χ a (0,ν) ∈ S −∞,ν−∞ (E 0 , E 1 ) κ,κ provided a (0) 0 + a (0,ν) = 0. Since changing the cut-off functions results in a smoothing remainder, we may assume that χ = 0 on |ξ, µ| ≤ 1 and χ = 1 on |ξ, µ| ≥ 2 as well as χ = 0 on |ξ| ≤ 1/2 and χ = 1 on |ξ| ≥ 1. Let χ 1 (ξ, µ) be another 0-excision function, vanishing for |ξ, µ| ≤ 2 and being 1 for |ξ, µ| ≥ 3. Since a (0,ν) = −a (0) 0 we have χa (0) 0 + χ a (0,ν) = (χ − χ)a (0) 0 = (1 − χ 1 )(χ − χ)a (0) 0 + χ 1 (χ − χ)a (0) 0 . The first term on the right-hand side is a smooth compactly supported function, hence is regularizing. The second term is supported in the strip-like region Σ : = {(ξ, µ) | |ξ, µ| ≥ 2, |ξ| ≤ 1}. In case ν ≤ 0 it is clear that the second term is regularizing. So we assume ν > 0. Let N be the largest integer strictly smaller than ν. Using the symbol estimates in S (0,ν) (E 0 , E 1 ) κ,κ and (2.1) we find that D α ξ a (0) 0 (x, ξ; µ) L (E0,E1) |ξ| ν−|α| |ξ, µ| −ν max{|ξ, µ|, |ξ, µ| −1 } M , ξ = 0, with a suitable M ≥ 0. Thus these derivatives vanish for ξ = 0 and µ > 0 provided |α| ≤ N . By Taylor expansion we conclude a (0) 0 (x, ξ; µ) = |α|=N +1 N + 1 α! ξ α 1 0 (1 − t) N (∂ α ξ a (0) 0 )(x, tξ; µ) dt. Now observe that 1 2 ≤ µ µ ≤ µ |ξ, µ| ≤ |tξ, µ| |ξ, µ| ≤ 1, ∀ (ξ, µ) ∈ Σ, 0 ≤ t ≤ 1. Hence κ ±1 ℓ,|tξ,µ|/|ξ,µ| L (E ℓ ) 1, ∀ (ξ, µ) ∈ Σ, 0 ≤ t ≤ 1, and we can estimate, for (ξ, µ) ∈ Σ, κ −1 1 (ξ, µ) D β x D α ξ D j µ a (0) 0 (x, ξ; µ) κ 0 (ξ, µ) L (E0,E1) ξ −L 1 0 |tξ, µ| −(N +1)−j dt ξ −L µ −ν−j with arbitrary L ≥ 0 (recall that |ξ| ≤ 1 in Σ). It follows that χ 1 (χ − χ)a (0) 0 belongs to S −∞,ν−∞ (E 0 , E 1 ) κ,κ . This completes the proof. Corollary 3.6. Let a ∈ S d,ν (E 0 , E 1 ) with σ d,ν (a) = 0. Then a ∈ S d−1,ν−1 (E 0 , E 1 ). In other words, we have the short exact sequence 0 −→ S d−1,ν−1 (E 0 , E 1 ) ֒→ S d,ν (E 0 , E 1 ) σ d,ν − −− → S (d,ν) (E 0 , E 1 ) −→ 0. Ellipticity and parametrix construction. In the present set-up, one can introduce a concept of ellipticity only in case ν > 0. Again, after multiplication by [ξ, µ] −d , we can focus on the case of symbols of order d = 0. Below we shall frequently use the following fact: If u : Ω y → L (E 0 , E 1 ) is a smooth function which is pointwise invertible on Ω ⊂ R m , then D α y u(y) −1 is a linear combination of terms of the form (3.3) [u −1 D α1 y u] · · · [u −1 D α ℓ y u]u −1 , ℓ ≥ 1, α 1 + . . . + α ℓ = α. Lemma 3.7. Let ν > 0 and a ∈ S (0,ν) (E 0 , E 1 ) κ,κ . Let χ ∈ C ∞ (S n + ) with 0 ≤ χ ≤ 1 and define a χ (x, ξ; µ) = χ (ξ, µ) |ξ, µ| a(x, ξ; µ). Then there exists a neighborhood K ⊂ S n + of the north-pole (0, 1) such that if χ is supported in K then (1 + a χ )(x, ξ; µ) is invertible whenever (ξ, µ) = 0. Moreover, there exists a b ∈ S (0,ν) (E 0 , E 1 ) κ,κ such that (1 + a χ ) −1 = 1 + b. Proof. Note that a χ ∈ S (0,ν) (E 0 , E 1 ) κ,κ . Choose K in such a way that a(x, ξ; µ) |ξ| ν ≤ 1/2 for all x and all (ξ, µ) ∈ K. Hence 1 + a χ is invertible on R n × S n and (1 + a χ ) −1 is uniformly bounded. By κ-homogeneity it follows that (1 + a χ ) −1 exists everywhere with κ −1 0,|ξ,µ| (1 + a χ (x, ξ; µ)) −1 κ 1,|ξ,µ| L (E1,E0) 1 whenever (ξ, µ) = 0. From (3.3) it follows that (1 + a χ ) −1 ∈ S (0,0) (E 0 , E 1 ) κ,κ . But then (1 + a χ ) −1 = 1 − a χ + a χ (1 + a χ ) −1 a χ =: 1 + b gives the desired result. Proposition 3.8. Let ν > 0 and a ∈ S (0,ν) (E 0 , E 1 ). Assume that a(x, ξ; µ) is invertible whenever (ξ, µ) = 0 and that a(x, ξ; µ) −1 L (E1,E0) 1 ∀ x ∈ R n ∀ |ξ, µ| = 1. Then a −1 ∈ S (0,ν) (E 1 , E 0 ). Proof. Write a = a 0 + a. Obviously, a −1 is κ-homogeneous of degree 0. Thus the assumed estimate is equivalent to κ −1 0,|ξ,µ| a(x, ξ; µ) −1 κ 1,|ξ,µ| L (E1,E0) 1 ∀ x ∈ R n ∀ (ξ, µ) = 0. By chain rule (cf. (3.3)) it follows that κ −1 0,|ξ,µ| {D β x D α ξ ′ D j µ a −1 (x, ξ; µ)}κ 1,|ξ,µ| L (E1,E0) |ξ| |ξ, µ| ν−|α| + 1 |ξ, µ| −|α|−j Hence, if χ(ξ, µ) = χ((ξ, µ)/|ξ, µ|) denotes the homogeneous extension of degree 0 of an arbitrary smooth function χ ∈ C ∞ (S n + ) which is identically to 1 in a neighborhood of the north-pole, then (1 − χ)a −1 belongs to S (0) (E 1 , E 0 ), since |ξ| ∼ |ξ, µ| on the support of 1 − χ. Since a(x, 0; 1) = a 0 (x, 0; 1), there exists a C ≥ 0 such that a 0 (x, 0; 1) −1 L (E1,E0)) ≤ C for every x. By Taylor formula, a 0 (x, rφ; 1 − r 2 ) − a 0 (x, 0; 1) L (E0,E1)) r uniformly in x and φ ∈ S n−1 . Hence there exists a neighborhood K ⊂ S n + of the north-pole (0, 1) such that a 0 (x, ξ; µ) −1 exists and is uniformly bounded in R n × K. Thus if we choose χ to be compactly supported in K and use chain-rule (cf. (3.3)), we obtain that χa −1 0 belongs to S (0) (E 1 , E 0 ). Now let χ 1 ∈ C ∞ (S n + ) be also supported in K, 0 ≤ χ 1 ≤ 1, and χ 1 ≡ 1 in a neighborhood of the support of χ. Then we can write χa −1 = χ(a 0 + a) −1 = χa −1 0 (1 + χ 1 a −1 0 a) −1 , where χ 1 is the extension of χ 1 by homogeneity of degree 0. Since χ 1 a −1 0 a belongs to S (0,ν) (E 0 , E 1 ) κ,κ , we may choose χ 1 (and χ) to be supported so closely to the north-pole that Lemma 3.7 applies, i.e., (1 + χ 1 a −1 0 a) −1 = 1 + b with a symbol S (0,ν) (E 1 , E 0 ) κ,κ . Altogether we find a −1 = [(1 − χ)a −1 + χa −1 0 ] + χa −1 0 b. This is the desired representation of a −1 . Definition 3.9. A symbol a ∈ S 0,ν (E 0 , E 1 ), ν > 0, is called elliptic if its principal symbol σ (0,ν) (a)(x, ξ; µ) is invertible whenever (ξ, µ) = 0 and σ (0,ν) (a)(x, ξ; µ) −1 L (E1,E0) 1 ∀ x ∈ R n ∀ |ξ, µ| = 1. Theorem 3.10. Let a ∈ S 0,ν (E 0 , E 1 ), ν > 0, be elliptic. Then a has a parametrix b ∈ S 0,ν (E 1 , E 0 ), i.e., 1 − a#b ∈ S −∞,ν−∞ (E 1 , E 1 ), 1 − b#a ∈ S −∞,ν−∞ (E 0 , E 0 ). Proof. Due to the previous proposition, we can write σ (0,ν) (a) −1 = b (0) + b (0,ν) with the obvious meaning of notation. Define b 0 = χ 0 (ξ, µ)b (0) + χ 1 (ξ) b (0,ν) with 0-excision functions χ 0 and χ 1 . Then b 0 ∈ S 0,ν (E 1 , E 0 ) and r 0 := 1 − ab 0 ∈ S −1,ν−1 cl (E 1 , E 1 ), since r 0 has vanishing principal symbol. By the standard Neumann series argument we find a b ∈ S 0,ν (E 1 , E 0 ) which is a right-parametrix of a. Analogously, we find a left-parametrix. It differs from b by a regularizing term, hence b is both left-and right-parametrix. Poisson operators of Grubb's class In this section we discuss the main example that underlies our approach to describe singular Green operators, as it will be presented in the following Section 5.2. To this end, let us recall the definition of Poisson operators due to [5]. A parameterdependent Poisson-operator of order d + 1 2 and regularity ν is of the form (4.1) [K(µ)u](x ′ , x n ) = e ix ′ ξ ′ k(x ′ , x n , ξ ′ ; µ) u(ξ ′ ) dξ ′ , u ∈ S (R n−1 ), where the so-called symbol-kernel k(x ′ , x n , ξ ′ ; µ) is a smooth function satisfying the estimates D β x ′ D α ξ ′ D j µ x ℓ n D ℓ ′ xn k L2(R+) ξ ′ ξ ′ , µ ν−[ℓ−ℓ ′ ]+−|α| + 1 ξ ′ , µ d−ℓ+ℓ ′ −|α|−j (4.2) for every order of derivatives and every ℓ ∈ N 0 . Here, for arbitrary r ∈ R, r + = max{0, r}, r − = max{0, −r}, r = r + − r − . Obviously, for every fixed µ, a Poisson operator induces a map K(µ) : S (R n−1 ) −→ S (R n + ) := S (R n )| R n + , i.e., maps functions defined on the boundary of the half-plane R n + to functions defined on the half-space. We shall need the following definition: Definition 4.1. Let E be a Fréchet space. Then S d 1,0 (R n−1 × R + ; E) =: S d 1,0 (E) consists of all smooth E-valued functions a(x ′ , ξ ′ ; µ) satisfying the uniform estimates p(D β x ′ D α ξ ′ D j µ a(x ′ , ξ ′ ; µ)) ξ ′ , µ d−|α|−j for every continuous semi-norm p(·) of E and every order of derivatives. Instead, the estimates p(D β x ′ D α ξ ′ D j µ a(x ′ , ξ ′ ; µ)) ξ ′ ν−|α| ξ ′ , µ d−ν−j define the space S d,ν 1,0 (R n−1 × R + ; E) =: S d,ν 1,0 (E). Strongly parameter-dependent Poisson operators. First let us discuss those Poisson operators that depend strongly on the parameter, i.e., have infinite regularity. In case of order d + 1 2 , the corresponding symbol-kernels are characterized by the estimates D β x ′ D α ξ ′ D j µ x ℓ n D ℓ ′ xn k L 2 (R+) ξ ′ , µ d−ℓ+ℓ ′ −|α|−j . For such operators the following characterization is known, cf. equations (3) and (6) k(x ′ , x n , ξ ′ ; µ) = ξ ′ , µ 1/2 k(x ′ , ξ ′ , ξ ′ , µ x n ; µ), with k(x ′ , ξ ′ , x n ; µ) ∈ S d 1,0 (R n−1 × R + ; S (R + )). Observe, for purposes below, that in this case we can also write k(x ′ , x n , ξ ′ ; µ) = ξ ′ 1/2 k(x ′ , ξ ′ , ξ ′ x n ; µ) with k(x ′ , ξ ′ , x n ; µ) = ξ ′ , µ ξ ′ 1/2 k x ′ , ξ ′ , ξ ′ , µ ξ ′ x n ; µ ∈ S d,0 1,0 (R n−1 × R + ; S 0 (R + )). Weakly parameter-dependent Poisson operators. Returning to the defining estimate (4.2), strongly parameter-dependent Poisson symbol-kernels correspond to the second summand on the right-hand side. Those corresponding to the first summand, i.e., that satisfy the estimates (4.3) D β x ′ D α ξ ′ D j µ x ℓ n D ℓ ′ xn k L2(R+) ξ ′ ξ ′ , µ ν−[ℓ−ℓ ′ ]+−|α| ξ ′ , µ d−ℓ+ℓ ′ −|α|−j we shall refer to as weakly parameter-dependent Poisson symbol-kernels. They can be characterized as follows: k 1 (x ′ , ξ ′ , µ; x n ) ∈ S d,ν 1,0 (R n−1 × R + ; S 0 (R + )), k 2 (x ′ , ξ ′ , µ; x n ) ∈ S d,ν 1,0 (R n−1 × R + ; H ∞ (R + )) such that k(x ′ , x n , ξ ′ ; µ) = ξ ′ 1/2 k 1 (x ′ , ξ ′ , µ; ξ ′ x n ) = ξ ′ , µ 1/2 k 2 (x ′ , ξ ′ , µ; ξ ′ , µ x n ). Proof. For simplicity of notation let us assume independence of the x ′ -variable; the general case is verified in the same way. By multiplication of k with ξ ′ , µ ν−d ξ ′ −ν we may assume w.l.o.g. that ν = d = 0. Thus let us start out from the estimates D α ξ ′ D j µ x ℓ n D ℓ ′ xn k L 2 (R+)    ξ ′ ℓ ′ −ℓ ξ ′ −|α| ξ ′ , µ −j : ℓ ≥ ℓ ′ ξ ′ , µ ℓ ′ −ℓ ξ ′ −|α| ξ ′ , µ −j : ℓ < ℓ ′ . (4.4) Now let us define k 1 (ξ ′ , µ; x n ) := ξ ′ −1/2 k(ξ ′ , µ; ξ ′ −1 x n ). Using chain rule it is not difficult to see that the estimates (4.4) are equivalent to D α ξ ′ D j µ x ℓ n D ℓ ′ xn k 1 L 2 (R+)    ξ ′ −|α| ξ ′ , µ −j : ℓ ≥ ℓ ′ ξ ′ ,µ ξ ′ ℓ ′ −ℓ ξ ′ −|α| ξ ′ , µ −j : ℓ < ℓ ′ (4.5) (for all choices of indices α, j, ℓ, ℓ ′ ). Now we argue that these estimates in turn are equivalent to D α ξ ′ D j µ x ℓ n D ℓ ′ xn k 1 L 2 (R+) ξ ′ −|α| ξ ′ , µ −j , ℓ ≥ ℓ ′ , (4.6) and D α ξ ′ D j µ D ℓ ′ xn k 1 L 2 (R+) ξ ′ , µ ξ ′ ℓ ′ ξ ′ −|α| ξ ′ , µ −j , ℓ ′ ≥ 0. (4.7) Clearly, (4.5) implies both (4.6) and (4.7). Vice versa, let 0 < ℓ < ℓ ′ be given. Observe that, by Hölder inequality, uv L 2 (R+) ≤ |u| p v 1/p L 2 (R+) v 1/q L 2 (R+) , 1 p + 1 q = 1. We apply this inequality with u = x ℓ n , v = D α ξ ′ D j µ D ℓ ′ xn k 1 and p = ℓ ′ /ℓ and find D α ξ ′ D j µ x ℓ n D ℓ ′ xn k 1 L 2 (R+) ≤ D α ξ ′ D j µ x ℓ ′ n D ℓ ′ xn k 1 ℓ/ℓ ′ L 2 (R+) D α ξ ′ D j µ D ℓ ′ xn k 1 1−ℓ/ℓ ′ L 2 (R+) . Inserting the estimates from (4.6) and (4.7) yields the second estimate in (4.5). If we define k 2 (ξ ′ , µ; x n ) = ξ ′ ξ ′ , µ 1/2 k 1 ξ ′ , µ; ξ ′ ξ ′ , µ x n = ξ ′ , µ −1/2 k(ξ ′ , µ; ξ ′ , µ −1 x n ), then (4.7) is equivalent to D α ξ ′ D j µ D ℓ ′ xn k 2 L 2 (R+) ξ ′ −|α| ξ ′ , µ −j , ℓ ′ ≥ 0. (4.8) Summing up, we have verified that the assumption (4.4) is equivalent to the validity of the estimates (4.6) and (4.8). Now recall that u ∈ S 0 (R + ) ⇐⇒ ωu ∈ H ∞,0 (R + ) and (1 − ω)u ∈ S (R) ⇐⇒ t ℓ D ℓ ′ t u L 2 (R+) < +∞ ∀ ℓ ≥ ℓ ′ ; moreover, the semi-norms t ℓ D ℓ ′ t u L 2 (R+) , ℓ ≥ ℓ ′ ∈ N 0 , define the topology of S 0 (R + ). Hence the estimates in (4.6) are equivalent to k 1 ∈ S 0,0 (R n−1 ×R + ; S 0 (R + )). Similarly, recalling that v ∈ H ∞ (R + ) ⇐⇒ D ℓ ′ t v L 2 (R+) < +∞ ∀ ℓ ′ ∈ N 0 , the estimates in (4.8) are equivalent to k 2 ∈ S 0,0 (R n−1 ×R + ; H ∞ (R + )). This finishes the proof of the theorem. 4.3. Poisson symbol-kernels viewed as operator-valued symbols. We shall identify a Poisson-kernel k(x ′ , x n , ξ ′ ; µ) with the function k : R n−1 x ′ × R n−1 ξ ′ × R + −→ L (C, S (R + )), c → k(x ′ , ·, ξ ′ ; µ)c. Since S (R + ) is embedded both in H s,δ (R + ) and K s,0 (R + ) δ , k ∈ ∩ s,δ∈R C ∞ R n−1 × R n−1 × R + , L (C, H s,δ (R + )) , k ∈ ∩ s,ρ∈R C ∞ R n−1 × R n−1 × R + , L (C, K s,0 (R + ) ρ ) (4.9) Theorem 4.4. A function k from (4.9) corresponds to a strongly parameter dependent Poisson symbol-kernel of order d + 1 2 if, and only if, (4.10) κ −1 (ξ ′ , µ)D α ξ ′ D β x ′ D j µ k(x ′ , ξ ′ ; µ) L (C,H s,δ (R+)) ξ ′ , µ d+ 1 2 −|α|−j for every s, δ ∈ R and all orders of derivatives. In other terms, k ∈ ∩ s,δ∈R S d+ 1 2 1,0 (R n−1 × R + ; C, H s,δ (R + )), where H s,δ (R + ) is equipped with the standard group-action from Example 2.1, while on C we consider the trivial group-action κ ≡ 1. Proof. Let us first start out from a Poisson symbol-kernel. Then, by Theorem 4.2 and chain-rule, D α ξ ′ D β x ′ D j µ k is a linear-combination of terms of the form p i,j ′ ,α ′ (ξ ′ ; µ) ξ ′ , µ 1/2 (D β x ′ (x n D xn ) i D α ′′ ξ ′ D j ′′ µ k)(x ′ , ξ ′ , ξ ′ , µ x n ; µ) with i + j ′ + j ′′ + |α ′ | + |α ′′ | = |α| + j and symbols p i,j ′ ,α ′ ∈ S −i−j ′ −|α ′ | 1,0 (R n−1 × R + ) . Therefore the left-hand side of (4.10) can be estimated by a linear combination of terms ξ ′ , µ −i−j ′ −|α ′ | D β x ′ D i xn D α ′′ ξ ′ D j ′′ µ k H s,δ (R+) . Then (4.10) follows from the fact that k ∈ S d+ 1 2 1,0 (R n−1 × R + ; S (R + )). Now consider k as in (4.9) satisfying (4.10), identified with a function k(x ′ , x n , ξ ′ ; µ). Define k = κ −1 (ξ ′ , µ)k, i.e., k(x ′ , x n , ξ ′ ; µ) = ξ ′ , µ −1/2 k(x ′ , ξ ′ , ξ ′ , µ −1 x n ; µ). Again using chain rule, D α ξ ′ D β x ′ D j µ k is a linear combination of terms p i,j ′ ,α ′ (ξ ′ ; µ)(x n D xn ) i κ −1 (ξ ′ , µ)D β x ′ D i xn D α ′′ ξ ′ D j ′′ µ k with indices and p i,j ′ ,α ′ as before. This implies k ∈ S d+ 1 2 1,0 (R n−1 × R + ; S (R + )), as desired. Similarly we can treat weakly parameter-dependent Poisson operators: Theorem 4.5. A function k from (4.9) corresponds to a weakly parameter-dependent Poisson symbol-kernel of order d + 1 2 and regularity ν if, and only if, κ −1 (ξ ′ )D α ξ ′ D j µ k(ξ ′ ; µ) L (C,K s,0 (R+) ρ ) ξ ′ ν−|α| ξ ′ , ν d+ 1 2 −ν−j and κ −1 (ξ ′ , µ)D α ξ ′ D j µ k(ξ ′ ; µ) L (C,H s (R+)) ξ ′ ν−|α| ξ ′ , ν d+ 1 2 −ν−j for every s, ρ and all orders of derivatives, where κ is the standard group-action from Example 2.1. Homogeneous Poisson symbol-kernels. A Poisson symbol-kernel which is strictly homogeneous of degree d − 1 2 and has regularity ν (in the sense of Grubb) is a kernel k(x ′ , x n , ξ ′ , µ), defined for ξ ′ = 0 only, satisfying (4.11) k(x ′ , x n /λ, λξ ′ ; λµ) = λ d+ 1 2 k(x ′ , x n , ξ ′ ; µ) ∀ λ > 0, and D β x ′ D α ξ ′ D j µ x ℓ n D ℓ ′ xn k L2(R+) |ξ ′ | |ξ ′ , µ| ν−[ℓ−ℓ ′ ]+−|α| + 1 |ξ ′ , µ| d−ℓ+ℓ ′ −|α|−j for every order of derivatives and every ℓ. We can repeat the above discussion and introduce strongly parameter-dependent homogeneous kernels by requiring the estimates D β x ′ D α ξ ′ D j µ x ℓ n D ℓ ′ xn k L2(R+) |ξ ′ , µ| d−ℓ+ℓ ′ −|α|−j (and k being defined for (ξ ′ , µ) = 0) and weakly parameter-dependent homogeneous kernels by requiring D β x ′ D α ξ ′ D j µ x ℓ n D ℓ ′ xn k L2(R+) |ξ ′ | |ξ ′ , µ| ν−[ℓ−ℓ ′ ]+−|α| |ξ ′ , µ| d−ℓ+ℓ ′ −|α|−j . Then Theorem 4.4 and Theorem 4.5 have a corresponding version in the homogeneous setting (using an identification with an operator-valued function as described in the beginning of Section 4.3): Strongly homogeneous kernels are characterized by the estimates κ −1 |ξ ′ ,µ| D α ξ ′ D β x ′ D j µ k L (C,H s,δ (R+)) |ξ ′ , µ| d−|α|−j ; weakly homogeneous kernels are characterized by simultaneous validity of the estimates κ −1 |ξ ′ | D α ξ ′ D j µ k L (C,K s,0 (R+) ρ ) |ξ ′ | ν−|α| |ξ ′ , ν| d−ν−j and κ −1 |ξ ′ ,µ| D α ξ ′ D j µ k L (C,H s (R+)) |ξ ′ | ν−|α| |ξ ′ , ν| d−ν−j . Note that the homogeneity relation (4.11) then can be rephrased as (4.12) k(x ′ , λξ ′ ; λµ) = λ d κ λ k(x ′ , ξ ′ ; µ) ∀ λ > 0. 5. The algebra of singular Green symbols 5.1. Other classes of operator-valued symbols. Let E 0 and E 1 be Hilbert spaces equipped with group actions κ 0 and κ 1 , respectively. Recall from Definition 2.5 that S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ denotes the space of all smooth L (E 0 , E 1 )-valued functions a satisfying the estimates (5.1) κ −1 1 (ξ, µ) D α ξ D β x D j µ a(x, ξ; µ) κ 0 (ξ, µ) L (E0,E1) ξ ν−|α| ξ, µ d−ν−j . However, in Theorem 4.5 we have seen that naturally arise operator-valued symbols involving the group-action κ(ξ ′ ) rather than κ(ξ ′ , µ). In order to capture this feature on the level of operator-valued symbols we shall introduce three further symbol spaces, by modifying the above defining estimate (5.1): i) Substituting κ 0 (ξ, µ) by κ 0 (ξ) yields the space S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ . ii) Substituting κ −1 1 (ξ, µ) by κ −1 1 (ξ) yields the space S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ . iii) Substituting both κ 0 (ξ, µ) and κ −1 1 (ξ, µ) by κ 0 (ξ) and κ −1 1 (ξ), respectively, yields the space S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ . Note the difference between κ (written in "boldface"), indicating the use of κ j (ξ, µ) for j = 0 or j = 1, and κ, indicating the use of κ j (ξ) for j = 0 or j = 1. Modifying in the analogous way Definition 2.7 of the space of κ-homogeneous functions S (d,ν) (R n × R + ; E 0 , E 1 ) κ,κ , leads to the three spaces S (d,ν) (R n × R + ; E 0 , E 1 ) κ,κ , S (d,ν) (R n × R + ; E 0 , E 1 ) κ,κ , S (d,ν) (R n × R + ; E 0 , E 1 ) κ,κ . Correspondingly, there are three classes of classical symbols, denoted by S d,ν (R n × R + ; E 0 , E 1 ) κ,κ , S d,ν (R n × R + ; E 0 , E 1 ) κ,κ , S d,ν (R n × R + ; E 0 , E 1 ) κ,κ . In any of the three cases we have the notion of homogeneous principal symbol, defined by a (d,ν) (x, ξ; µ) = lim λ→+∞ λ −d κ −1 1,λ a(x, λξ; λµ)κ 0,λ . The calculus for such symbols is pretty much the same as described in Section 2.3 but, additionally, one has to keep track of the indices κ and κ, respectively. Concerning the Leibniz product of symbols (respectively the composition of pseudodifferential operators), one has to pay attention that the κ-subscripts "fit together"; for example, if a 0 ∈ S d0,ν0 (R n ×R + ; E 0 , E 1 ) κ,κ and a 1 ∈ S d1,ν1 (R n ×R + ; E 1 , E 2 ) κ,κ , then a 1 #a 0 ∈ S d0+d1,ν0+ν1 (R n × R + ; E 0 , E 2 ) κ,κ . Taking the formal adjoint leads to an interchange of the two κ-indices; for example, if a ∈ S d,ν (R n × R + ; E 0 , E 1 ) κ,κ then a ( * ) ∈ S d,ν (R n × R + ; E 1 , E 0 ) κ,κ . Also asymptotic summations can be performed in any of these classes. Singular Green symbols of type 0. Below we shall work with various Hilbert spaces of the form E(R + ) ⊕ C N with E(R + ) ⊂ D ′ (R + ). On such a space we shall consider the standard group-action, extended trivially to C N , i.e., κ λ (u ⊕ z) = (κ λ u) ⊕ z, u ∈ E(R + ), z ∈ C N . First we focus on symbols of type (also called "class" in the literature) 0. Strongly parameter-dependent symbols coincide with the parameter-dependent Greensymbols of Schulze's version of the parameter-dependent Boutet de Monvel algebra, cf., for instance, [10, Section 2]. The following definition uses the formalism of [9, Section 3.2]. Definition 5.1. B d;0 G (R n−1 × R + ; (L, M ), (L ′ , M ′ )) denotes the space ∩ s,s ′ ,δ,δ ′ ∈R S d (R n−1 × R + ; H s,δ 0 (R + , C L ) ⊕ C M , H s ′ ,δ ′ (R + , C L ′ ) ⊕ C M ′ ). For convenience, we shall use the short-hand notation B d;0 G ((L, M ), (L ′ , M ′ )). The space B (d);0 G (R n−1 × R + ; (L, M ), (L ′ , M ′ )) = B (d);0 G ((L, M ), (L ′ , M ′ )) of strongly κ-homogeneous symbols is definied analogously by taking the intersection over the corresponding spaces of homogeneous symbols. The weakly parameter-dependent class is defined as follows: Definition 5.2. Denote by B d,ν;0 G (R n−1 × R + ; (L, M ), (L ′ , M ′ ) ) the space of all symbols that, for every choice of s, s ′ , δ, δ ′ , belong to each of the following four spaces: S d,ν (R n−1 × R + ; H s 0 (R + , C L ) ⊕ C M , H s ′ (R + , C L ′ ) ⊕ C M ′ ) κ,κ , S d,ν (R n−1 × R + ; H s 0 (R + , C L ) ⊕ C M , K s ′ ,0 (R + , C L ′ ) δ ′ ⊕ C M ′ ) κ,κ , S d,ν (R n−1 × R + ; K s,0 (R + , C L ) δ ⊕ C M , H s ′ (R + , C L ′ ) ⊕ C M ′ ) κ,κ , S d,ν (R n−1 × R + ; K s,0 (R + , C L ) δ ⊕ C M , K s ′ ,0 (R + , C L ′ ) δ ′ ⊕ C M ′ ) κ,κ . For convenience, we shall use the short-hand notation B d,ν;0 G ((L, M ), (L ′ , M ′ )). Again, in the obvious way, we can define the corresponding space B (d,ν);0 G (R n−1 × R + ; (L, M ), (L ′ , M ′ )) = B (d,ν);0 G (R n−1 × R + ; (L, M ), (L ′ , M ′ )) of weakly κ-homogeneous symbols. Given an operator A : E 0 ⊕ E 1 → F 0 ⊕ F 1 acting in between direct sums of ceratin spaces, we may write A in block-matrix form, i.e., A = A 11 A 12 A 21 A 22 : E 0 ⊕ E 1 −→ F 0 ⊕ F 1 . In this sense, we may represent any symbol g ∈ B d,ν;0 G ((L, M ), (L ′ , M ′ )) in blockmatrix form, i.e., g(x ′ , ξ ′ ; µ) = g(x ′ , ξ ′ ; µ) k(x ′ , ξ ′ ; µ) t(x ′ , ξ ′ ; µ) q(x ′ , ξ ′ ; µ) ; then g is called a singular Green symbol, k is a Poisson symbol, and t is a trace symbol. Clearly, q is a usual (matrix-valued) pseudodifferential symbol. The following observation is crucial for the calculus. Proof. For simplicity of notation let L = L ′ = 1 and M = M ′ = 0; the general case is treated in the same way and involves only more lengthy notation. Let g ∈ ∩ s,s ′ ,δ,δ ′ ∈R S d (H s,δ 0 (R + ), H s ′ ,δ ′ (R + )). For k ∈ N 0 , norms on K k,0 (R + ) k and H k,k (R + ) are given by u = 0≤ℓ ′ ≤k ℓ ′ ≤ℓ≤k t ℓ D ℓ ′ t u L 2 (R+) , v = 0≤ℓ,ℓ ′ ≤k t ℓ D ℓ ′ t v L 2 (R+) , respectively; in particular, H k,k (R + ) ֒→ K k,0 (R + ) k . Using these norms, it is easy to see that κ λ (k) := κ λ L (H k,k (R+),K k,0 (R+) k ) ≤ max(1, λ −k ). By duality it then follows that K −k,0 (R + ) −k ֒→ H −k,−k 0 (R + ) and κ λ (−k) := κ λ L (K −k,0 (R+) −k ,H −k,−k 0 (R+)) ≤ max(1, λ k ). By assumption, g ∈ S d,0 (H s,δ 0 (R + ), H k,k (R + )) κ,κ . Therefore κ −1 (ξ ′ ){D α ξ ′ D β x ′ D j µ g(x ′ , ξ ′ ; µ)}κ(ξ ′ , µ) L (H s,δ 0 (R+),K k,0 (R + ) k ) κ ξ ′ ,µ / ξ ′ (k) · · κ −1 (ξ ′ , µ){D α ξ ′ D β x ′ D j µ g(x ′ , ξ ′ ; µ)}κ(ξ ′ , µ) L (H s,δ 0 (R+),H k,k (R + )) ξ ′ −|α| ξ ′ , µ d−j , since ξ ′ , µ / ξ ′ ≥ 1. Choosing k ≥ max(s ′ , δ ′ ) we conclude that g belongs to S d,0 1,0 (H s 0 (R + ), K s ′ ,0 (R + ) δ ′ ) κ,κ . Using ξ ′ / ξ ′ , µ ≤ 1, κ −1 (ξ ′ , µ){D α ξ ′ D β x ′ D j µ g(x ′ , ξ ′ ; µ)}κ(ξ ′ ) L (K −k,0 (R + ) −k ,H s ′ (R+)) κ −1 (ξ ′ , µ){D α ξ ′ D β x ′ D j µ g(x ′ , ξ ′ ; µ)}κ(ξ ′ , µ) L (H −k,−k 0 (R + ),H s ′ (R+)) · · κ ξ ′ / ξ ′ ,µ (−k) ξ ′ −|α| ξ ′ , µ d−j . Choosing k large enough, it follows that g ∈ S d,0 1,0 (K s,0 (R + ) δ , H s ′ (R + )) κ,κ . Analogously one verifies that g ∈ S d,0 Given a function u on R + , denote by e + u its extension by 0 to R. It is known that e + extends to a mapping on the Sobolev spaces H s ( r z + ) provided s > − 1 2 . More precisely, e + induces maps 1,0 K s,0 (R + ) δ , K s ′ ,0 (R + ) δ ′ ) κ,e + : H s (R + ) → H s 0 (R + ), − 1 2 < s < 1 2 , e + : H s (R + ) → H 1 2 −ε 0 (R + ), s ≥ 1 2 , where ε > 0 can be chosen arbitrarily small. By r + we shall denote the operator of restriction D ′ (R) → D ′ (R + ). Obviously, the same is valid in a C L -valued setting. An operator or an operator-valued symbol a defined on the intersection over all s of all spaces H s 0 (R + , C L ) canonically induces an operator or operator-valued symbol defined on H s (R + , C L ), s > −1/2, defined as ae + . In particular, this applies to the singular Green symbols, i.e., B d,ν;0 G ((L, M ), (L ′ , M ′ )) can be considered as a subspace of (5.2) S d,ν (H s (R + , C L ) ⊕ C M , H s ′ (R + , C L ′ ) ⊕ C M ′ ), s > − 1 2 , s ′ ∈ R, in the sense of Definition 3.1. Actually, this will be the standard way of viewing singular Green symbols. 5.3. Leibniz product and formal adjoint symbol. Given a symbol g ∈ B d,ν;0 G ((L, M ), (L ′ , M ′ )) and some fixed µ ≥ 0, g := g(·, ·; µ) ∈ S d (R n−1 ξ ′ ; H s,δ 0 (R + , C L ) ⊕ C M , H s ′ ,δ ′ (R + , C L ′ ) ⊕ C M ′ ). for every choice of s, s ′ , δ, δ ′ ∈ R. In other words, g is a singular Green symbol of type 0 in Boutet de Monvel's algebra without parameter. This can be seen by writing g(x ′ , ξ ′ ) = 2 i,j=1 ω i (ξ ′ ) 0 0 1 g(x ′ , ξ ′ ) ω j (ξ ′ ) 0 0 1 , where ω ∈ C ∞ comp ([0, +∞)) is a cut-off function with ω ≡ 1 near the origin and ω 1 (ξ ′ ) denotes the operator of multiplication by ω(· ξ ′ ), while ω 2 (ξ ′ ) is the operator of multiplication by (1 − ω)(· ξ ′ ). In particular, it follows that op(g)(µ) : S (R n + , C L ) ⊕ S (R n−1 , C M ) −→ S (R n + , C L ′ ) ⊕ S (R n−1 , C M ′ ) ∀ µ ≥ 0. Using Theorems 3.3 and 3.4, the observations stated in the end of Section 5.1, and Proposition 5.5, the behaviour of singular Green symbols under composition (Leibniz product) and formal adjoint is as follows: Theorem 5.7. If g j ∈ B dj,νj ;0 G ((L j , M j ), (L j+1 , M j+1 )), j = 0, 1, then g 1 #g 0 ∈ B d,ν;0 G ((L 0 , M 0 ), (L 2 , M 2 )) with d = d 0 + d 1 , ν = min{ν 0 , ν 1 , ν 0 + ν 1 }. Moreover, for every N ∈ N, g 1 #g 0 − N −1 |α|=0 1 α! ∂ α ξ ′ g 1 D α x ′ g 0 ∈ B d−N,ν−N ;0 G ((L 0 , M 0 ), (L 2 , M 2 )). If g j ∈ B dj,νj ;0 G ((L j , M j ), (L j+1 , M j+1 )) for at least one of the values j = 0 or j = 1, then g 1 #g 0 ∈ B d,ν;0 G ((L 0 , M 0 ), (L 2 , M 2 )). In this sense, the weakly parameterdependent Green symbols form a two-sided ideal in the class of Green symbols. Moreover, the composition of two strongly parameter-dependent Green symbols is also strongly parameter-dependent. 6. Ellipticity and parametrix in the class 1 + B 0,ν;0 G With every g = g 0 + g ∈ B d,ν;0 G ((L, M ), (L ′ , M ′ )) we associate the homogeneous principal symbol ν) . Note that, in particular, g ( * ) − N −1 |α|=0 1 α! ∂ α ξ ′ D α x ′ g * ∈ B d(6.1) σ d,ν (g) := g (d) 0 + g (d,σ d,ν (g) ∈ S (d,ν) (H s 0 (R + , C L ) ⊕ C M , H s ′ (R + , C L ′ ) ⊕ C M ′ ), s, s ′ ∈ R, and σ d,ν (g) ∈ S (d,ν) (H s (R + , C L ) ⊕ C M , H s ′ (R + , C L ′ ) ⊕ C M ′ ), s > − 1 2 , s ′ ∈ R, where the spaces of κ-homogeneous symbols on the respective right-hand sides are understood in the sense of Definition 3.1. Combining the proof of Proposition 3.5 with the proof of Proposition 5.5 one finds: We shall now discuss ellipticity and parametrix construction for operators of the form identity plus zero order singular Green operator. Again, it is necessary to require positive regularity ν > 0. It is convenient to introduce the space Proof. For convenience of notation we shall assume L 0 = L 1 = L 2 = L 3 = 1; the general case is proved in the same way. Write a = a 0 + a with a 0 ∈ S (0) (H s0 (R + ), H s0 (R + )), a ∈ S (0,ν) (H s0 (R + ), H s0 (R + )) κ,κ . Similarly decompose g and h with g 0 , h 0 ∈ B (0);0 G (1, 1) and g, h ∈ B (0,ν);0 G (1, 1). Recall that also g 0 , h 0 ∈ B (0,0);0 G (1, 1). Now write hag = h 0 a 0 g 0 + h 0 ag 0 + h 0 a g + hag. Since h 0 ∈ S (0) (H s0 (R + ), H s ′ ,δ ′ (R + )) for every s ′ and δ ′ , it follows that h 0 a 0 g 0 ∈ B (0);0 G (1, 1). All the other terms in the above expression for hag belong to B (0,ν);0 G (1, 1). For example let us consider h 0 a g = h 0 a 0 g + h 0 a g. 1), and a 0 , a ∈ S (0,0) (H s0 (R + ), H s0 (R + )) κ,κ it is easy to see that both h 0 a 0 g and h 0 a g have all the mapping properties listed in Definition 5.2. The reasoning for all remaining terms in the expansion of hag is analogous. Since h 0 ∈ B (0,0);0 G (1, 1), g ∈ B (0,ν);0 G (1, Theorem 6.3. Let ν > 0 and g ∈ B 0,ν;0 G (L, L). Suppose that, for some s 0 > −1/2, 1 + σ (0,ν) (g)(x ′ , ξ ′ ; µ) : H s0 (R + , C L ) −→ H s0 (R + , C L ) is invertible for all x ′ and (ξ ′ , µ) = 0 with (1 + σ (0,ν) (g)(x ′ , ξ ′ ; µ)) −1 L (H s 0 (R+,C L )) 1 ∀ x ′ ∀ |ξ ′ , µ| = 1. Then there exists an h ∈ B 0,ν;0 G (L, L) such that (1 + g)#(1 + h) − 1, (1 + h)#(1 + g) − 1 ∈ B −∞,ν−∞;0 G (L, L). In particular, (1 + σ (0,ν) (g)(x ′ , ξ ′ ; µ)) −1 = 1 + σ (0,ν) (h)(x ′ , ξ ′ ; µ). Proof. For simplicity of notation write σ(g) = σ (0,ν) (g). Proposition 3.8 applied to a := 1 + σ(g) ∈ S (0,ν) (H s0 (R + , C L ), H s0 (R + , C L )) yields the existence of a b ∈ S (0,ν) (H s0 (R + , C L ), H s0 (R + , C L )) with (1 + σ(g)) −1 = 1 + b. Since 1 + b = 1 − σ(g) + σ(g)(1 + σ(g)) −1 σ(g) = 1 − σ(g) + σ(g) 2 + σ(g)bσ(g), it follows from Lemma 6.2 that b ∈ B (0,ν);0 G (L, L). Choose any symbol h ′ ∈ B 0,ν;0 G (L, L) with σ (0,ν) (h ′ ) = b. Then (1 + g)#(1 + h ′ ) = 1 − r ′ , r ′ ∈ B −1,ν−1;0 G (L, L), since r ′ has vanishing principal symbol. Choose any r ∈ B −1,ν−1;0 G (L, L) with r ∼ j≥1 (r ′ ) #j . By the standard von Neumann argument, (1 + h ′ )#(1 + r) = 1 + h ′ + r + h ′ #r =: 1 + h R is a right-parametrix of 1 + g. In the analogous way we construct a left-parametrix 1 + h L . Then 1 + h L and 1 + h R coincide up to a regularizing error. Hence the claim of the theorem follows by choosing h = h R or h = h L . The situation of general block-matrices can be reduced to the previous situation: 1 + σ (0,ν) (g)(x ′ , ξ ′ ; µ) : L 2 (R + , C L ) ⊕ C M −→ L 2 (R + , C L ) ⊕ C M is an isomorphism for all x ′ and (ξ ′ , µ) = 0 with (1 + σ (0,ν) (g)(x ′ , ξ ′ ; µ)) −1 1 ∀ x ′ ∀ |ξ ′ , µ| = 1.(1 + σ (0,ν) (g)(x ′ , ξ ′ ; µ)) −1 = 1 + σ (0,ν) (h)(x ′ , ξ ′ ; µ). Proof. Let us set a := 1 + σ (0,ν) (g). By assumption, a is pointwise invertible and there exist positive constants c ≤ C such that c u 2 ≤ a(x ′ , ξ ′ ; µ)u 2 ≤ C u 2 uniformly in u ∈ L 2 (R + , C L ) ⊕ C M and x ′ and |ξ ′ , µ| = 1. Obviously, a −1 = [a * a] −1 a * . We write Note that s(x ′ , ξ ′ ; µ) = 1 − C −1 (a * a)(x ′ , ξ ′ ; µ) ≤ δ with δ := C−c C < 1. Then 0 ≤ (s 11 u, u) L 2 (R+,C L ) a * a = 1 + g = C(1 − s), s = 1 − C −1 a * a;= [1 − C −1 a * a] u 0 , u 0 L 2 (R+,C L )⊕C M ≤ δ(u, u) L 2 (R+,C L ) shows that s 11 (x ′ , ξ ′ ; µ) L (L 2 (R+,C L )) ≤ δ ∀ x ′ ∀ |ξ ′ , µ| = 1. Therefore 1 + g is pointwise invertible with Similarly one sees that (1 + g) −1 = C −1 (1 − s 11 ) −1 ≤ C −1 1 1 − δ = 1 c ∀ x ′ ∀ |ξ ′ , µ| = 1.s 22 (x ′ , ξ ′ ; µ) L (C M ) ≤ 1 c ∀ x ′ ∀ |ξ ′ , µ| = 1. It follows that (1 + q) −1 = 1 + p, p ∈ S (0,ν) (R n−1 × R + ; C M , C M ). Therefore a * a = 1 + g k t 1 + q is invertible with (a * a) −1 = 1 0 −(1 + p) t 1 + p 1 + h 0 0 1 1 − k(1 + p) 0 1 = 1 + c, where c = h −(1 + h) k(1 + p) −(1 + p) t(1 + h) p + (1 + p) t(1 + h) k(1 + p) ∈ B (0,ν);0 G ((L, M ), (L, M )). We conclude the existence of an h ′ ∈ B (1 + σ (0,ν) (g)) −1 = 1 + σ (0,ν) (h ′ ). Now one proceeds as in the proof of Theorem 6.3 to construct the parametrix. 6.1. Invertibility for large parameter. In the previous section we have constructed the parametrix, i.e., an inverse modulo regularizing operators. In the present section we shall show that elliptic elements are invertible for sufficiently large values of µ and that one can find a parametrix that furnishes the exact inverse for these values of µ. Lemma 6.5. Let E be a Hilbert space equipped with trivial group-action κ ≡ 1. Let a ∈ S −∞,ν−∞ (R + × R n ; E, E) with some ν > 0. Then there exists a symbol b of the same form and a constant R ≥ 0 such that (1 − a)#(1 − b) = (1 − a)#(1 − b) = 1 ∀ µ ≥ R. Proof. Note that we have the identification S −∞,ν−∞ (R + × R n ; E, E) = S −ν 1,0 (R + ; S −∞ (R n ; E, E)). As shown in Corollary 9.5 of the appendix, A := {1 + op(r) | r ∈ S −∞ (R n ; E, E)} ⊂ L (L 2 (R n , E)) is a spectrally invariant Fréchet algebra (with multiplication #, i.e., the Leibniz product). In particular, A has an open group of invertible elements and inversion is a continuous operation in A due to a classical result of Waelbroeck [18]. It follows that (1 − a(µ)) −# (i.e., the inverse with respect to the product #) exists for µ ≥ R for some suitable R and that b 0 (µ) := χ(µ)(1 − a(µ)) −# ∈ S 0 1,0 (R + ; S −∞ (R n ; E, E)) if χ is a 0-excision function vanishing on [0, R]. But then the identity (1−T ) −1 = 1+T (1−T ) −1 shows that, for large µ, the inverse of 1−a(µ) coincides with 1−b(µ), where b(µ) = −a(µ)#b 0 (µ) ∈ S −ν 1,0 (R + ; S −∞ (R n ; E, E)). (6.3) (1 − g)#(1 − h) = (1 − h)#(1 − g) = 1 ∀ µ ≥ R. Proof. Note that g ∈ S −∞,ν−∞ (R + × R n ; L 2 (R, C L ) ⊕ C M , L 2 (R, C L ) ⊕ C M )) and recall that κ is a group of unitary operators on L 2 (R + , C L ) ⊕ C M , hence can be replaced here by the trivial group-action. According to Lemma 6.5 we find an h 0 from the same space which inverts 1 − g in the sense of (6.3). Thus, for large µ, 1 − h 0 = (1 − g) −# = 1 + g − g#(1 − g) −# #g = 1 + g − g#g + Singular Green symbols of non trivial type Recall that the maps u → d j u dt j (0) : S (R + , C L ) −→ C L , j ∈ N 0 , extend by continuity to maps γ j : H s,δ (R + , C L ) −→ C L , s > j + 1 2 , δ ∈ R, (we shall use the same notation, independent on the value of L). We can consider γ j as an operator-valued symbol, constant in (x ′ , ξ ′ ; µ); the following result is wellknown, cf. [9, Example 1.5], for instance. Proposition 7.1. For every j ∈ N 0 , γ j is κ-homogeneous of degree j + 1 2 . In particular, γ j ∈ S j+ 1 2 (H s,δ (R + , C L ), C L ), s > j + 1 2 , δ ∈ R. It is convenient to introduce the notation γ j := 0 0 γ j 0 . denotes the space of all symbols of the form g = g 0 + r−1 j=0 0 k j 0 q j γ j = g 0 + r−1 j=0 k j γ j 0 q j γ j 0 , where g 0 ∈ B d,ν;0 G ((L, M ), (L ′ , M ′ )) and 0 k j 0 q j ∈ B d−j− 1 2 ,ν;0 G ((L, L), (L ′ , M ′ )). The number r is called the type of the symbol g. For a symbol g as in the previous definition, the principal symbol is defined as σ (d,ν) (g) = σ (d,ν) (g 0 ) + r−1 j=0 0 σ (d,ν) (k j ) 0 σ (d,ν) (q j ) γ j , There is another, equivalent definition of symbols of type r > 0: If ∂ + is the operator of differentiation defined on D ′ (R + , C L ), it can be checked easily that ∂ + induces constant operator-valued symbols ∂ + := ∂ + 0 0 1 ∈ S 1 (H s,δ (R + , C L ) ⊕ C M , H s−1,δ (R + , C L ) ⊕ C M ) for every s, δ ∈ R. Then Using integration by parts, symbols from Definition 7.2 can be represented in this form, cf. [9, Example 3.10], for instance. For the reverse direction one needs to use a characterization of singular Green, Poisson, and trace symbols in terms of parameter-dependent integral kernels (as we have seen in Theorem 4.5 for Poisson symbols). However, since we shall not make use of this alternative representation, we shall not proof this equivalence here. Lemma 7.3. Let g 0 ∈ B d0,ν0;0 G ((L 0 , M 0 ), (L 1 , M 1 )) and g 1 ∈ B d1,ν1;0 G ((L 1 , L 1 ), (L 2 , M 2 )). Then (g 1 γ k )#g 0 ∈ B d0+d1+k+ 1 2 ,ν;0 G ((L 0 , M 0 ), (L 2 , M 2 )), ν = min{ν 0 , ν 1 , ν 0 + ν 1 }. Proof. Let us first remark that the Leibniz product of γ k with another operator-valued symbol (acting in appropriate spaces) coincides with the pointwise product of both, since γ k is constant. Therefore, (g 1 γ k )#g 0 = g 1 #γ k #g 0 = g 1 #(γ k g 0 ). It is enough to study the two cases where either both g 0 and g 1 are strongly parameter-dependent or both are weakly parameter-dependent; the mixed cases then follow from Proposition 5.5. For simplicity of notation let us now assume that M 0 = M 1 = M 2 = 0. Consider first the case of strongly parameter-dependent symbols. By Proposition 7.1, γ k g 0 ∈ S d0+k+ 1 2 (R n−1 × R + ; H s,δ 0 (R + , C L0 ), H s ′ ,δ ′ (R + , C L1 ) ⊕ C L1 ) for every s, s ′ , δ, δ ′ ∈ R. Recall that g 1 induces a symbol g 1 ∈ S d1 (R n−1 × R + ; H s,δ (R + , C L1 ) ⊕ C L1 , H s ′ ,δ ′ (R + , C L2 )) for every s > −1/2 and all s ′ , δ, δ ′ ∈ R. Then the result follows simply by using the Leibniz product of strongly parameter-dependent operator-valued symbols, cf. Section 2.3 and Example 2.10. In case of weak parameter-dependence we have to check that the Leibniz-product belongs to all four spaces listed in Definition 5.2. For example, due to Proposition 7.1, γ k g 0 ∈ S d0+k+ 1 2 ,ν0 (K s,0 (R + , C L0 ) δ , H s ′ ,δ ′ (R + , C L1 ) ⊕ C L1 ) κ,κ . Moreover, cf. the paragraph before (5.2), g 1 ∈ S d1,ν1 (H s,δ (R + , C L1 ) ⊕ C L1 , K s ′ ,0 (R + , C L2 ) δ ′ ) κ,κ for every s > −1/2 and all s ′ , δ, δ ′ ∈ R. By using the Leibniz product of weakly parameter-dependent operator-valued symbols as described in the end of Section 5.1 it follows that g 1 #∂ k + g 0 ∈ S d0+d1+k,ν0+ν1 (K s,0 (R + , C L0 ) δ , K s ′ ,0 (R + , C L2 ) δ ′ ) κ,κ . The argument for the other three spaces listed in Definition 5.2 is analogous. It is now clear that Theorem 5.7 on the Leibniz product of Green symbols with type 0 extends to general types as follows: Theorem 7.4. If g j ∈ B dj,νj ;rj G ((L j , M j ), (L j+1 , M j+1 )), j = 0, 1, then g 1 #g 0 ∈ B d,ν;r G ((L 0 , M 0 ), (L 2 , M 2 )) with d = d 0 + d 1 , ν = min{ν 0 , ν 1 , ν 0 + ν 1 }, r = r 0 . Moreover, for every N ∈ N, g 1 #g 0 − N −1 |α|=0 1 α! ∂ α ξ ′ g 1 D α x ′ g 0 ∈ B d−N,ν−N ;r G ((L 0 , M 0 ), (L 2 , M 2 )). Action in Sobolev spaces In this section we discuss the mapping properties of parameter-dependent singular Green operators in Sobolev spaces. 8.1. The abstract framework. Given a Hilbert space E with group-action κ, let W s (R n , E), s ∈ R, denote the space of all distributions u ∈ S ′ (R n , E) such that its Fourier transform u is a regular distribution and u W s (R n ,E) = ξ 2s κ −1 (ξ)û(ξ) 2 E dξ 1/2 < +∞. (note that the integral is defined, since together with u also κ −1 (·) u is measurable due to the strong continuity the group-action). These spaces are called abstract wedge Sobolev spaces and have been introduced by Schulze in [11]; cf. also [6]. Now consider two Hilbert spaces E 0 and E 1 with respective group-actions κ 0 and κ 1 . Let S d (R n ; E 0 , E 1 ) be the space of all symbols a(x, ξ) satisfying κ −1 1 (ξ){D α ξ D β x a(x, ξ)}κ 0 (ξ) L (E0,E1) ξ d−|α| . The following theorem follows easily from [17, Theorem 3.14] together with the calculus for such operator-valued symbols. Theorem 8.1. Every a ∈ S d (R n ; E 0 , E 1 ) induces continuous operators op(a) : W s (R n , E 0 ) −→ W s−d (R n , E 1 ), s ∈ R. Now let us introduce spaces with parameter-dependent norms. Definition 8.2. We denote by W (s,t),µ (R n , E) and W (s,t),µ (R n , E) the Sobolev space W s+t (R n , E) equipped with the norms u W (s,t),µ (R n ,E) = ξ, µ 2s ξ 2t κ −1 (ξ)û(ξ) 2 E dξ 1/2 and u W (s,t),µ (R n ,E) = ξ, µ 2s ξ 2t κ −1 (ξ, µ)û(ξ) 2 E dξ 1/2 , respectively. Assume now that on a Hilbert space E there exists a family of norms · µ , parametrized by a parameter µ, that are all equivalent to a fixed norm on E; denote by E µ the space E equipped with the norm · µ . Moreover, suppose we have a family A(µ) ∈ L (E). We shall write A(µ) ∈ L (E µ ) if A(µ) L (E µ ) = sup e∈E A(µ)e µ e µ 1. This concepts extends in the obvious way to two different spaces E 0 and E 1 equipped with µ-dependent norms. Theorem 8.3. Let s, t ∈ R. The following statements are valid: a) a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ implies op(a)(µ) ∈ L W (s,t),µ (R n , E 0 ), W (s+ν−d,t−ν),µ (R n , E 1 ) . b) a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ implies op(a)(µ) ∈ L W (s,t),µ (R n , E 0 ), W (s+ν−d,t−ν),µ (R n , E 1 ) . c) a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ implies op(a)(µ) ∈ L W (s,t),µ (R n , E 0 ), W (s+ν−d,t−ν),µ (R n , E 1 ) . d) a ∈ S d,ν 1,0 (R n × R + ; E 0 , E 1 ) κ,κ implies op(a)(µ) ∈ L W (s,t),µ (R n , E 0 ), W (s+ν−d,t−ν),µ (R n , E 1 ) . e) a ∈ S d 1,0 (R n × R + ; E 0 , E 1 ) implies op(a)(µ) ∈ L W (s,t),µ (R n , E 0 ), W (s−d,t),µ (R n , E 1 ) . Proof. a)-d) Let λ d,ν j (ξ; µ) = ξ ν ξ, µ d−ν id Ej and a = λ s−d+t,t−ν 1 #a#λ −s−t,−t 0 . Then a is of order and regularity 0 and the statements are equivalent to the continuity of op( a)(µ) between the corresponding parameter-dependent Sobolev spaces of order (0, 0). However, this can be shown following the proof of [17, Theorem 3.14], making only simple modifications that regard the use of the semi-group function κ j (ξ, µ) in place of κ j (ξ). e) is verified similarly. There are various estimates relating different parameter-dependent norms. For purposes below we mention that (8.1) u W (s,t),µ (R n ,E) ≤ µ r+ u W (s−r,t+r),µ (R n ,E) , u ∈ W s+t (R n , E), for every r, s, t, ∈ R. This simply follows from the fact that ξ, µ s ξ t = ξ, µ s−r ξ, µ r ξ t ≤ ξ, µ s−r ξ t+r µ r+ . 8.2. Example: Sobolev spaces on the half-space. We shall show how the previously described concept of edge Sobolev spaces allows to recover certain (anisotropic) Sobolev spaces with parameter-dependent norm. Let us first look at the full Euclidean space. Define H (s,t) (R n ) = {u ∈ S ′ (R n ) | ξ s ξ ′ t u(ξ) ∈ L 2 (R n )} with obvious definition of the norm. The parameter-dependent version, denoted by H (s,t),µ (R n ), carries the norm(s) u H (s,t),µ (R n ) = ξ, µ s ξ ′ , µ t u(ξ) L 2 (R n ) . Proposition 8.4. With equality of norms, H (s,t) (R n ) = W s+t (R n−1 , H s (R)) and H (s,t),µ (R n ) = W (s+t,0),µ (R n−1 , H s (R)). For an operator B : S (R n , E) → S ′ (R n , E) let L k B = iBx k − ix k B, M k B = D x k B − BD x k (1 ≤ k ≤ n), where x k means the operator of multiplikation by the function x → x k . Moreover, set B (α) (β) = L α1 1 · · · L αn n M α1 1 · · · M αn n B, α, β ∈ N 0 . Let us denote by S m ρ,δ (R n ; E, E) the operator-valued Hörmander class of all smooth functions b : R n × R n → L (E) satisfying, for every order of derivatives, D α ξ D β x b(x, ξ) ξ m−ρ|α|+δ|β| . Theorem 9.3. For a linear operator B : S (R n , E) → S ′ (R n , E) the following are equivalent: (1) There exists a symbol b(x, ξ) ∈ S 0 0,0 (R n ; E, E) with B = op(b). (2) All iterated commutators B (α) (β) extend to operators in L (L 2 (R n , E)). Proof. In case E = C this is a special case of [1,Theorem 1.4]. Let us summarize the proof given there: Let h be a smooth cut-off function with h ≡ 1 in a neighborhood of the origin. Define B ε = op(q ε )Bop(p ε ), q ε (x) = h(εx), p ε (ξ) = h(εξ). Let b 0,ε (x, y, ξ) = e −ξ (x)B(e ξ g(· − y)), where e ξ (x) = e ixξ and g ∈ S (R n ) is even with Fourier transform supported in the unit-ball centered at the origin. One shows that b 0,ε is a smooth function with |D β x D γ y D α ξ b 0,ε (x, y, ξ)| ≤ C αβγ uniformly in (x, y, ξ) and 0 < ε ≤ 1, with constants that are (respectively, can be chosen to be) finite linear combinations in the L (L 2 (R n ))-operator norms of the iterated commutators of B. Then b ε (x, ξ) = e iyη b 0,ε (x, x + y, ξ + η) dydη (oscillatory integral) is a smooth function satisfying analogous estimates. Then b = lim ε→0 b ε exists in S 0 0,0 (R n ) and B = op(b). Again, the constants in the symbol estimates are as described before. Now let us turn to the general case of a Hilbert space E. For an operator T : S (R n , E) → S ′ (R n , E) and e, f ∈ E define T e,f : S (R n ) → S ′ (R n ) by Then b ∈ S 0 0,0 (R n ; E, E) due to (9.1) and Corollary 9.2. Moreover, op(b)(φ ⊗ e), ψ ⊗ f = e −ixξ b(x, ξ) φ(ξ)e dξ, ψ(x)f E dx = e −ixξ (b(x, ξ)e, f ) E φ(ξ)ψ(x) dxdξ = op(b e,f )φ, ψ = B(φ ⊗ e), ψ ⊗ f . Since finite linear combinations of functions of the form u ⊗ e with u ∈ S (R n ) and e ∈ E are a dense subspace of S (R n , E) = S (R n ) ⊗ π E, we conclude that B = op(b). An algebra A ⊂ L (E), E an arbitrary Hilbert space, is called spectrally invariant if an element a ∈ A is invertible in A if and only if it is invertible in L (E), i.e., A −1 = A ∩ L (E) −1 , where X −1 denotes the group of invertible elements of the algebra X . Corollary 9.4. Let A = op(a) | a ∈ S 0 0,0 (R n ; E, E) . Then A ⊂ L (L 2 (R n , E)) is spectrally invariant. Proof. Let A = op(a) : L 2 (R n , E) → L 2 (R n , E) be an isomorphism with inverse A −1 . Then L k A −1 = iA −1 x k − ix k A −1 = A −1 ix k A − iAx k A −1 = −A −1 (L k A)A −1 and, analogously, M k A −1 = −A −1 (M k A)A −1 . It follows that an iterated commutator of A −1 is a finite linear combination of products, where each factor is either A −1 or an iterated commutator of A. Hence every such iterated commutator extends to a bounded map in L 2 (R n , E). By Theorem 9.3, A −1 ∈ A. Corollary 9.5. Let A = 1+op(a) | a ∈ S −∞ (R n ; E, E)}. Then A ⊂ L (L 2 (R n , E)) is spectrally invariant. Proof. Let A = 1 + op(a) : L 2 (R n , E) → L 2 (R n , E) be an isomorphism. By the preevious corollary, A −1 = op(b 0 ) for some b 0 ∈ S 0 0,0 (R n ; E, E). shows that A −1 = 1 + op(b) with b = −a + a#b 0 #a ∈ S −∞ (R n ; E, E). defines the Hilbert spaces K s,γ (R + ), s, γ ∈ R. Up to equivalence of norms, this construction is independent on the coice of ω. Finally, we consider spaces with power-weight at infinity, Theorem 4.3. A smooth function k(x ′ , x n , ξ ′ ; µ) is a weakly parameter-dependent Poisson symbol-kernel of order d + 1 2 and regularity ν if, and only if, there exist Remark 5 . 3 . 53Recall that the L 2 -inner product allows to identify H −s,−δ 0 (R + ) with the dual space of H s,δ (R + ) and K −s,0 (R + ) −δ with the dual space of K s,0 (R + ) δ for every choice of s, δ ∈ R. Hence it is clear from the definition of the symbol spaces, that taking pointwise the adjoint induces mapsB d;0 G ((L, M ), (L ′ , M ′ )) → B d;0 G ((L ′ , M ′ ), (L, M )) B d,ν;0 G ((L, M ), (L ′ , M ′ )) → B d,ν;0 G ((L ′ , M ′ ), (L, M )). Next we shall introduce the class of singular Green symbols of finite regularity: Definition 5.4. Denote by B d,ν;0 G (R n−1 × R + ; (L, M ), (L ′ , M ′ )) = B d,ν;0 G ((L, M ), (L ′ , M ′ )) the (non-direct) sum B d;0 G ((L, M ), (L ′ , M ′ )) + B d,ν;0 G ((L, M ), (L ′ , M ′ )). Proposition 5. 5 . 5We have the inclusionB d;0 G ((L, M ), (L ′ , M ′ )) ⊂ B d,0;0 G ((L, M ), (L ′ , M ′ ))and the analogous inclusion for the spaces of κ-homogeneous symbols. κ . Homogeneous components and corresponding expansions are treated similarly and yield the result for classical symbols. L, M ), (L ′ , M ′ )) = B d,ν;0 G ((L, M ), (L ′ , M ′ )), ν ≤ 0. Theorem 5. 8 . 8If g ∈ B d,ν;0 G ((L, M ), (L ′ , M ′ )) then g ( * ) ∈ B d,ν;0 G ((L ′ , M ′ ), (L, M )); for every N ∈ N, −N,ν−N ;0 G ((L ′ , M ′ ), (L, M )). Theorem 6. 1 . 1The principal symbol induces a surjective map σ (d,ν) : B d,ν;0 G ((L, M ), (L ′ , M ′ )) −→ B (d,ν);0 G ((L, M ), (L ′ , M ′ )) with kernel equal to B d−1,ν−1;0 G ((L, M ), (L ′ , M ′ )). , L ′ ) is the space of the entries g in the block matrices L, M ), (L ′ , M ′ )) with arbitrary M, M ′ .Lemma 6.2. Let ν > 0 and a ∈ S (0,ν) (H s0 (R + , C L1 ), H s0 (R + , C L2 )) for some fixed s 0 > −1/2 be a κ-homogeneous symbol. Further let g ∈ B L, M ), (L, M )) such that both (1 + g)#(1 + h) − 1 and (1 + h)#(1 + g) − 1 belong to B −∞,ν−∞;0 G ((L, M ), (L, M )). In particular, ( (L, M ), (L, M )). ( L, L) with (1 + g) −1 = 1 + h. ( (L, M ′ ), (L, M )) such that Proposition 6. 6 . 6Let g ∈ B −∞,ν−∞;0 G ((L, M ), (L, M )) with ν > 0. Then there exists a h ∈ B −∞,ν−∞;0 G ((L, M ), (L, M )) and a constant R ≥ 0 such that g#h 0 #g. Using Definition 5.2 and the calculus of operator-valued symbols, it is clear that h := −g + g#g − g#h 0 #g belongs to B −∞,ν−∞;0 G ((L, M ), (L, M )). The result follows, since 1 − h = 1 − h 0 for large µ. Theorem 6 . 7 . 67Using the notation of Theorem 6.4, there exists a parametrix 1 + h 0 that coincides, for sufficiently large values of µ, with the inverse (with respect to the Leibniz product) of 1 + g. Proof. Take h from Theorem 6.4. Then (1 + g)#(1 + h) = 1 − r 0 with r ∈ B −∞,ν−∞;0 G ((L, M ), (L, M )) and similarly (1 + h)#(1 + g) = 1 + r 1 . In view of Proposition 6.6 it suffices to take h 0 such that 1 + h 0 = (1 + h)#(1 + r 0 ) −# . Definition 7. 2 . 2Let r ∈ N be a positive integer. Then B d,ν;r G (R n−1 × R + ; (L, M ), (L ′ , M ′ )) = B d,ν;r G ((L, M ), (L ′ , M ′ )) L, M ), (L ′ , M ′ )) consists of all symbols of the form L, M ), (L ′ , M ′ )). T e,f φ, ψ = T (φ ⊗ e), ψ ⊗ f , φ, ψ ∈ S (R n ),where ·, · is the pairing of distributions and test functions and (φ ⊗ e)(x) = φ(x)e. Then (B e,f )(α) (β) = (B (α) (β) ) e,f and for the L 2 -extension holds (B e,f ) (α) (β) L (L 2 (R n )) ≤ B (α) (β) L (L 2 (R n ,E)) e f . By the scalar-valued result, there exists a symbol b e,f e,f (x, ξ)| ≤ C αβ e f . Now define b(x, ξ) : H → H by b(x, ξ)e, f = b e,f (x, ξ), e, f ∈ H. But then op(b 0 ) = (1 + op(a)) −1 = 1 − op(a) + op(a)(1 + op(a)) −1 op(a) H s,δ 0 0(R + ) := t −δ H s 0 (R + ),H s,δ (R + ) := t −δ H s (R + ), as well asK s,γ (R + ) δ := t −δ K s,γ (R + )with obvious definition of the corresponding norms.10.3. Duality. The standard L 2 (R + )-inner product induces non-degenerate sesqui-linear pairingsH s,δ (R + ) × H −s,−δ 0 (R + ) −→ C, K s,γ (R + ) δ × K −s,−γ (R + ) −δ → Cwhich allows for the following identification of dual spaces:H s,δ (R + ) ′ = H −s,−δ 0 (R + ), (K s,γ (R + ) δ ) ′ = K −s,−γ (R + ) −δ .10.4. Rapidly decreasing functions. Let I = R or I = R + and E a Hilbert space. We denote by S (I, E) the space of rapidly decreasing functions, i.e., of all smooth functions u : I → E with sup t∈I t N D k t u(t) < +∞ ∀ k, N ∈ N 0 . Definition 2.11. We call (E, H, E) a Hilbert-triple if the inner product of H induces a non-degenerate sesquilinear pairing E × E → C that permits to identify the dual spaces of E and E with E and E, respectively, and(κ λ e, e) H = (e, κ λ e) H ∀ e, e, λ > 0. Example 2.12. For every choice of s, δ, γ ∈ R, both in the proof of [10, Theorem 2.1.19], for instance.Theorem 4.2. A smooth function k(x ′ , x n , ξ ′ ; µ) is a strongly parameter-dependent Poisson symbol-kernel of order d + 1 2 and infinite regularity if, and only if, it has the form Proof. By definition,H (s,t),µ (R n ) = ξ ′ , ξ n , µ 2s ξ ′ , µ 2t | u(ξ ′ , ξ n )| 2 dξ n dξ ′ .By the change of variables ξ n = ξ ′ , µ τ n and noting that ξ ′ , ξ ′ , µ τ n , µ = ξ ′ , µ τ n we find u 2 H (s,t),µ (R n ) = ξ ′ , µ 2(s+t) τ n 2s | ξ ′ , µ 1/2 u(ξ ′ , ξ ′ , µ τ n )| 2 dξ n dξ ′ .Noting that Fourier transform and group-action satisfy κ λ F = F κ −1 λ , we findThis is exactly what has been claimed.Similarly, for the half-space R nwhere the first space consists of all elements of H (s,t),µ (R n ) having support contained in R n + , while the second denotes the space of restrictions to R n + , cf. the discussion in Section 10.1.Remark 8.5. Recall that C is equipped with the trivial group action κ ≡ 1. It is then rather obvious that All constructions from above extend trivially to C N -valued Sobolev spaces.for every choice of of s > r− 1 2 and s ′ ∈ R. Representing a ∈ B d,ν;r G ((L, M ), (L ′ , M ′ )) as a = a 0 + a in this sense, we obtain immediately the mapping properties stated in the parts a) and e) of Theorem 8.3. In particular, employing the identifications (8.3) and (8.4), we find thatNow, using the estimate (8.1) (with s subsituted by s + t + d − ν, t substituted by 0, and r substituted by −ν), we conclude the following:Appendix: A result on spectral invarianceWe shall proof a characterization of pseudodifferential operators with operatorvalued symbols in terms of mapping properties of certain commutators. In the scalar-valued setting, this result is a simple special case of a well-known theorem of Beals [1, Theorem 1.4]. We derive spectral invariance of two psudodifferential algebras.Let A ⊂ L (X, Y ) be a set of bounded operators between Banach spaces X and Y . By uniform boundedness principle, the following three statements are equivalent: i) A is a bounded set. ii) A x := {Ax | A ∈ A} is a bounded subset of Y for every x ∈ X. iii) y ′ (A x ) is a bounded set for every x ∈ X and y ′ ∈ Y ′ .If E is a Hilbert space we thus obtain the following: Proof. We claim that if all (u(·)e, f ) are (N + 1)-times continuously differentiable, then u is N -times continuously differentiable. By Induction it suffices to show this for N = 0. Given x 0 ∈ R, defineBy assumption on u, all sets {(Ae, f ) | A ∈ A} are bounded, hence A is bounded. Clearly this implies continuity of u in x 0 .Appendix: Function spaces for the half-lineWe recall here the definitions and some properties of all spaces of functions or distributions we shall need throughout this paper.10.1. Bessel potential spaces. We denote by H s (R) = H s 2 (R), s ∈ R, the standard L 2 -Sobolev paces, consisting of those tempered distributions u ∈ S ′ (R) whose Fourier transform is a measurable function withThe subspace of those distributions whose support is a subset of R + := [0, +∞) is denoted by H s 0 (R + ),Obviously, H s 0 (R + ) is a closed subspace of H s (R). Similarly, one definess H s (R − ) with R − := (−∞, 0]. Moreover, Then we defineMultiplication with powers t γ−1/2 yields the spaces .Note that for for k ∈ N 0 we have u ∈ H s,γ (R + ) ⇐⇒ t 1 2 −γ (t∂ t ) j u ∈ L 2 (R + , dt/t) ∀ 1 ≤ j ≤ k.Now let ω ∈ C ∞ comp (R + ) be a cut-off function with ω ≡ 1 near t = 0. Then u ∈ K s,γ (R + ) : ⇐⇒ ωu ∈ H s,γ (R + ) and (1 − ω)u ∈ H s (R + ) with norm u K s,γ (R+) = ωu H s,γ (R+) + (1 − ω)u H s (R+) Characterization of pseudodifferential operators and applications. R Beals, Duke Math. J. 444557R. Beals. Characterization of pseudodifferential operators and applications. Duke Math. J. 44 (1977), 4557. Correction to: "Characterization of pseudodifferential operators and applications. R Beals, Duke Math. J. 46215R. Beals. Correction to: "Characterization of pseudodifferential operators and applications". Duke Math. J. 46 (1979), 215. Boundary problems for pseudo-differential operators. L Boutet De Monvel, Acta Math. 126L. Boutet de Monvel. Boundary problems for pseudo-differential operators. Acta Math. 126 (1971), 11-51. Cone pseudodifferential operators in the edge symbolic calculus. J B Gil, B.-W Schulze, J Seiler, Osaka J. Math. 37J.B. Gil, B.-W. Schulze, J. Seiler. Cone pseudodifferential operators in the edge symbolic calculus. Osaka J. Math. 37 (2000), 221-260. Functional Calculus of Pseudo-differential Boundary Problems. G Grubb, 2nd ed.G. Grubb. Functional Calculus of Pseudo-differential Boundary Problems (2nd ed.). . Birkhäuser, Birkhäuser, 1996. Functional analysis in cone and edge Sobolev spaces. T Hirschmann, Ann. Global Anal. Geom. 8T. Hirschmann. Functional analysis in cone and edge Sobolev spaces. Ann. Global Anal. Geom., 8: 167-192, 1990. Pseudo-Differential Operators. Translated from the Japanese by the author, Rémi Vaillancourt and Michihiro Nagase. H Kumano-Go, MIT PressH. Kumano-go. Pseudo-Differential Operators. Translated from the Japanese by the author, Rémi Vaillancourt and Michihiro Nagase. MIT Press, 1981. Index Theory of Elliptic Boundary Problems. S Rempel, B.-W Schulze, Akademie VerlagBerlinS. Rempel, B.-W. Schulze. Index Theory of Elliptic Boundary Problems. Akademie Verlag, Berlin, 1982. An introduction to Boutet de Monvel's calculus. E Schrohe, Approaches to Singular Analysis. J.B. Gil et al.125Advances in PDE, BirkhäuserE. Schrohe. An introduction to Boutet de Monvel's calculus. In: J.B. Gil et al. (eds.), Ap- proaches to Singular Analysis, Operator Theory; Vol. 125: Advances in PDE, Birkhäuser, 2001. Boundary value problems in Boutet de Monvel's calculus for manifolds with conical singularities I. E Schrohe, B.-W Schulze, Pseudo-Differential Operators and Mathematical Physics. M. Demuth, E. Schrohe, B.-W. SchulzeAkademie Verlag5Math. TopicsE. Schrohe, B.-W. Schulze. Boundary value problems in Boutet de Monvel's calculus for manifolds with conical singularities I. In M. Demuth, E. Schrohe, B.-W. Schulze, editors, Pseudo-Differential Operators and Mathematical Physics, Math. Topics, Vol. 5: Advances in Part. Diff. Equ., Akademie Verlag, 1994. Pseudo-differential operators on manifolds with edges. B.-W Schulze, H. Triebel, B.-WB.-W. Schulze. Pseudo-differential operators on manifolds with edges. In H. Triebel, B.-W. Schulze, Symposium 'Partial Differential Equations. Teubner-Verlag112Teubner-Texte MathSchulze, editors, Symposium 'Partial Differential Equations', Holzhau 1988, Teubner-Texte Math. 112, Teubner-Verlag, 1989. Pseudo-differential Operators on Manifolds with Singularities. B.-W Schulze, Studies in Math. and its Appl. 24North-Holland Publishing CoB.-W. Schulze. Pseudo-differential Operators on Manifolds with Singularities. Studies in Math. and its Appl. 24, North-Holland Publishing Co., 1991. Boundary value problems with the transmission property. B.-W Schulze, Pseudodifferential operators: complex analysis and partial differential equations. Birkhäuser Verlag205B.-W. Schulze. Boundary value problems with the transmission property. In: Pseudo- differential operators: complex analysis and partial differential equations, 1-50, Oper. Theory Adv. Appl. 205, Birkhäuser Verlag, 2010. Complex powers of an elliptic operator. R Seeley, Singular Integrals (Proc. Sympos. Pure Math. Amer. Math. Soc288307R. Seeley. Complex powers of an elliptic operator. In: Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), 288307. Amer. Math. Soc., 1967. The resolvent of an elliptic boundary problem. R Seeley, Amer. J. Math. 91889920R. Seeley. The resolvent of an elliptic boundary problem. Amer. J. Math. 91 (1969), 889920. Pseudo-differential calculus on manifolds with non-compact edges. J Seiler, Institut für Mathematik, Universität PotsdamPhD-thesisJ. Seiler. Pseudo-differential calculus on manifolds with non-compact edges. PhD-thesis, In- stitut für Mathematik, Universität Potsdam, 1997. Continuity of edge and corner pseudo-differential operators. J Seiler, Mathematische Nachrichten. 205J. Seiler. Continuity of edge and corner pseudo-differential operators. Mathematische Nachrichten 205 (1999), 163-182. Topological Vector Spaces and Algebras. L Waelbroeck, Lect. Notes Math. 230Springer VerlagL.Waelbroeck. Topological Vector Spaces and Algebras. Springer Lect. Notes Math. 230, Springer Verlag, 1971. . Matematica Dipartimento Di, Università di TorinoItaly E-mail address: [email protected] di Matematica, Università di Torino, Italy E-mail address: [email protected]

In this short review, we discuss recent advances in exact solutions of models based on a onedimensional (1D) Coulomb gas by means of field-theoretic functional integral methods. The exact solutions can be used to assess the accuracy of various approximations such as the weak coupling Poisson-Boltzmann theory as well as the strong coupling theory of Coulomb gases. We consider three different 1D models: the Coulomb fluid configuration in the case of the soap film model consisting of positively and negatively charged particles between adsorbing boundaries, counterions between two charged surfaces, and an ionic liquid lattice capacitor with positively and negatively charged particles on a lattice between one positive and one negative bounding surface. arXiv:1209.3514v1 [cond-mat.stat-mech]

10.1201/b15597-7

1209.3514

da2f3b54e80cbf0a1d4ffda2e024197b9221d81a

Aspects of One-Dimensional Coulomb Gases Ronald R Horgan Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge CB3 0WACambridgeUnited Kingdom David S Dean Laboratoire Ondes et Matière d'Aquitaine (LOMA) UMR 5798 Université de Bordeaux CNRS F-33400TalenceFrance Vincent Démery UMR 7190 Institut Jean Le Rond d'Alembert CNRS UPMC Univ Paris 6F-75005ParisFrance Thomas C Hammant Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge CB3 0WACambridgeUnited Kingdom Ali Naji Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences University of Cambridge CB3 0WACambridgeUnited Kingdom School of Physics Institute for Research in Fundamental Sciences (IPM) 19395-5531TehranIran Rudolf Podgornik Department of Physics Faculty of Mathematics and Physics University of Ljubljana SI-1000LjubljanaSlovenia Aspects of One-Dimensional Coulomb Gases In this short review, we discuss recent advances in exact solutions of models based on a onedimensional (1D) Coulomb gas by means of field-theoretic functional integral methods. The exact solutions can be used to assess the accuracy of various approximations such as the weak coupling Poisson-Boltzmann theory as well as the strong coupling theory of Coulomb gases. We consider three different 1D models: the Coulomb fluid configuration in the case of the soap film model consisting of positively and negatively charged particles between adsorbing boundaries, counterions between two charged surfaces, and an ionic liquid lattice capacitor with positively and negatively charged particles on a lattice between one positive and one negative bounding surface. arXiv:1209.3514v1 [cond-mat.stat-mech] I. INTRODUCTION Field-theoretic functional integral methods can be used to study exact solutions of models based on a onedimensional (1D) Coulomb gas with charged boundaries. In 1D, exactly solvable Coulomb gas models can be then used as a testbed for assessing the accuracy of various approximations: the weak coupling expansion, Poisson-Boltzmann/mean-field equations, and the strong coupling expansion [1]. We review these approximations in the context of three 1D Coulomb gas systems and remark on whether or not they fail to predict important effects present in the exact solution. Some physical properties of the 1D system can be applicable at least qualitatively for dimensions d > 1 and can help us to understand whether pertaining approximation methods are reliable or not. In particular, our analysis gives insight into systems such as an array of charged smectic layers or lipid multilayers, and ionic liquids near charged interfaces, treated as effectively 1D systems. An important aspect of these endeavours is that we can test and develop the analysis and especially numerical methods that can then be tentatively applied also for d > 1. II. THEORETICAL METHODS The method of functional integrals applied to Coulomb gas systems has been developed over many years [2][3][4][5]. In any dimension this approach allows for both strong and weak coupling to be studied explicitly, but specifically in 1D the functional integral representation can be applied using a variety of methods to obtain exact solutions to a number of models which are generally characterized by a Coulomb gas of ions of possibly non-zero size confined between boundaries with properties that allow their potential or charge to be determined either dynamically or as an external field condition. Three varieties of a 1D Coulomb gas model discussed below are presented in Fig. 1. The functional integral representation of the Coulomb gas partition function allows us to formulate two effective solution techniques. The Schrödinger kernel technique is applicable in all dimensions and has been used to analyze a number of models [6]. In 1D it corresponds to solving the Schrödinger equation [2] which is in principle exact. In d > 1 the Schrödinger kernel field theoretic representation of the partition function is derived, often using a Hubbard-Stratonovich transformation, and is analyzed by perturbative and graphical methods. For d > 1 this approach does require that a preferred co-ordinate can be designated as the Euclidean time and so the approach is limited to symmetrically layered systems [6]. The transfer matrix and Fourier methods technique is an alternative to the Schrödinger kernel approach. Though it is more general, it is only practical in 1D. Its implementation exploits periodicity in the (imaginary) electrostatic potential φ which also restricts its general applicability. Examples of this technique in 1D are the counterion gas and lattice ionic liquids. The actual formulation of the functional integral method relies on the action for the full QED of a general system that is then reduced to the electrostatic action proper. The relevant electrostatic Lagrangian is then L(ψ) = 1 2 ε dx(∇ψ(x)) 2 − e i q i ψ(x i ) − dxρ e (x)ψ(x). Here q i is the charge of the i-th ion at position x i and ρ e (x) is the external charge distribution. The partition function is obtained by tracing the Boltzmann weight of the above Lagrangian over the electrostatic field [ψ]. Tracing furthermore over ion positions, changing the axis of functional integration ψ = iφ and introducing fugacity µ − = µ + ≡ µ by the Gibbs technique, the partition function for monovalent ions (with q i = ±1) assumes the form Z = d[φ] exp(S(φ)) ,(1) with the "field action" S(φ) = − β 2 ε dx(∇φ(x)) 2 + 2µ dx cos(eβφ(x)) ,(2) where β = 1/(k B T ). The charge density operator is then given by ρ = µ d dµ log Z(µ) ⇒ ρ = 2µ cos(eβφ) ,(3) where · · · stands for the φ average. III. BILAYER SOAP FILM IN IONIC SOLUTION Because hydrophobic heads of the surfactant molecules preferentially migrate to the surfaces charging them up dynamically, the configuration of the bilayer soap film consists of two planar (surfactant) surfaces separated by a distance L confining a solution of a symmetric electrolyte. We calculate the surface charge, the density profile of electrolyte near the interfaces, and the disjoining pressure P as a function of the thickness L of the soap film, defined as P = P film − P bulk = − 1 β ∂J film ∂L − ∂J bulk ∂L . i.e. the difference between the film and bulk pressures. Here J is the grand-canonical partition function/unit area. An important phenomenon to predict is the first-order collapse transition of the film to a Newton black film expected as the electrostatic coupling in the film is increased. We model this system by a Coulomb gas confined to z ∈ [0, L], schematically presented in Fig. 1 (top), with potentials on the boundaries that account for the hydrophillic nature of the head group of the surfactant molecule. The Debye length is given by l D = εk B T /2ρe 2 , and the Bjerrum length in 1D by l B = 2k B T ε/e 2 . Perturbation theory is an expansion in the coupling parameter g = l D /l B . We use the partition function described earlier but now includes surface free energy f (φ) to model the surface potentials, which are attractive for the negatively charged hydrophillic surfactant head groups whose surface density is denoted by ρ − (φ): f (φ) = e λρ − (φ) , where λ controls the potential strength. To simplify the notation we scale the variables: φ → eβφ, x → x l B . The charge density operators for ± charges are then given by the Boltzmann weights ρ ± (φ) = e ±iφ . The 1D partition function then becomes Z = 1 2π 2π 0 dφ 0 dφ L f (φ 0 )K(φ 0 , φ L ; L)f (φ L ) , where K(φ 0 , φ x ; x) = Dφ(x) exp x 0 dx L(φ(x )) is the Schrödinger kernel for evolution in the "Euclidean time" x: Ψ(φ, x) = dφ K(φ, φ ; x)f (φ ). It satisfies the Schrödinger (Feynman-Kac) equation HΨ = 2ε βe 2 ∂ ∂x Ψ , H = ∂ 2 ∂φ 2 + Z(g) 2g 2 cos(φ), with Z(g) = 1/ cos(φ) . The above equation is also known as the Mathieu equation, and the harmonic term gives the Debye length in units of l B . Z(g) = 2µ/ρ is the renormalization that relates the fugacity to the observable charge density and is given by Eq. (3). We now consider the solution in various limiting regimes. A. Large L: bulk pressure Strong coupling (SC) g → ∞: The Mathieu ground state dominates in this regime and so we can use the Schrödinger perturbation theory for the ground-state energy of H. The result, derived originally in [2], is P bulk = 1 2 ρk B T 1 + 7 32 1 g 2 − 23 4608 1 g 4 − 4897 7826432 1 g 6 + . . . . The leading term is the free gas term but for density ρ/2, which therefore signals the onset of the dimerization process, i.e., the Bjerrum pair formation of positive and negative mobile charges. Weak coupling (WC) g → 0: Feynman perturbation theory is applicable in this case and so we use the Feynman diagram expansion to find P bulk = ρk B T 1 − 1 2 g + 1 128 g 3 + . . . . The leading term is the free gas term and the second-order term is the familiar Debye-Hückel result in its 1D variant. Note that there is no O(g 2 ) term; this is cancelled by the counter term in Z(g). The strong and weak coupling dependencies of the bulk pressure P bulk on g compare well with the exact solution of the problem. Both approximations are accurate across a wide range of g in their regime of validity. More details can be found in [4]. B. Finite L: exact methods For finite L we expand the kernel K(φ 0 , φ L ; L) over periodic eigenfunctions of the Mathieu equation. We can then use a numerical approach for eigenfunctions/eigenenergies which will give an exact solution for all L. This method is described fully in [4] and we do not delve into details here. It gives the same answers as the Fourier approach that we describe below. The Fourier method for obtaining an exact solution to problems in 1D is more general than the Schrödinger approach since it works also when the Hamiltonian is not hermitian, which is the case for the counterion gas considered in the next section. It also forms the basis for the transfer matrix method. The theory is periodic under φ → φ + 2π and we can define Ψ(φ, x) = K(φ, φ ; x)f (φ ) dφ = e x/2g 2 n=∞ n=−∞ b n (x)e inφ , where the coefficients b n (x) obey the evolution equation db n dx = −n 2 b n + 1 4g 2 (b n+1 + b n−1 − 2b n ). This is the Fourier version of the Schrödinger equation but can be derived generally from the convolution property of the Schrödinger kernel. The partition function can then be obtained from Z(T, L) = 2π 0 f (φ)Ψ(φ, L) dφ = e L/2g 2 n=∞ n=0 λ n n! b n (L), The exact solution for the disjoining pressure as a function of the separation L for different values of the surface potential strength parameter λ clearly predicts a collapse transition to a Newton black film that can not be accounted for by the mean-field theory, which we address next. C. Classical or mean-field (MF) theory Standard variational methods applied to the expression for the partition function gives the classical MF equation: the Poisson-Boltzmann (PB) equation for φ cl (x) as the saddle point equation of the corresponding field theory. In this case the disjoining pressure P is given by the value of ion density at the midpoint x = L/2 between the bounding surfaces. The MF theory predicts that universally P > 0, contrary to our exact result and also to experiment; it does not predict any collapse transition, which is thus obviously a consequence of the non-MF correlation effects and is intrinsically a fluctuation phenomenon. IV. COUNTERIONS BETWEEN CHARGED SURFACES The 1D model here is a Coulomb gas of counterions confined between two oppositely charged surfaces; the system is overall neutral. We compare exact results with strong and weak coupling calculations, which are the same as in a 3D system. More details can be found in [7]. The system is shown in Fig. 1 (middle) and consists of N counterions, each of valency q, with surface charges σ 1 and σ 2 , respectively. We define ζ = σ 2 /σ 1 , with −1 < ζ < 1, and define α = 1/(1 + ζ). The 1D Bjerrum length is l B = 2k B T ε/e 2 , and the Gouy-Chapman length is µ ≡ µ 1 = l B e/q|σ 1 |, where we have chosen σ 1 to be non-zero and have µ 2 = µ/|ζ|. The electrostatic coupling constant, g, is then given by g ≡ q 2 µ l B = 1 + ζ N , where N → ∞ corresponds to the MF/PB theory and N → 1 to the SC theory. The partition function is derived as Z N = 1 2π 2π 0 dφ(0) ∞ −∞ dφ(L) e i σ1 qe φ(0) φ(L) φ(0) dφ e −S(φ) e i σ2 qe φ(L) , with S(φ) = L 0 dx 1 2q 2 e 2 β dφ(x) dx 2 − e iφ(x) . A. Exact results In this system H is not hermitian because the counterions are, by definition, of one charge only. We therefore analyze the model using the Fourier method. We exploit periodicity in H of φ → φ + 2π in order to write f (φ; x) = e −xH f (φ; 0) with f (φ; x) = b(n, x)e inφ .P (L) = − (M 1 + 1 − η 2 ) 2 2 q 2 e 2 + b(M 1 , M 2 , L) Z(σ 1 , ζ, N, L) can be evaluated exactly. Since the second term in the above equation can be seen to be just the counterion density at the boundary of the system, the above form of the pressure is thus a clear example of the contact value theorem; it connects the pressure with the value of the particle density at the confining wall of the system. B. Weak coupling We consider the WC expansion g → 0 which is equivalent in the lowest order to the MF/PB theory. In the d > 1 case, the MF theory treats the potential field φ(x) as constant in the directions transverse to the normal to the bounding interfaces, and so the results are independent of the dimensionality. The leading contribution arises from the saddle-point configuration φ 0 (x) = iψ 0 (x) with ψ 0 real. The PB equation and the boundary conditions have the form d 2 ψ 0 (x) dx 2 = −q 2 e 2 β e −ψ 0 (x) , with dψ 0 dx 0 = −σ 1 βqe, dψ 0 dx L = σ 2 βqe. The leading PB contribution to the disjoining pressure, P , is then expressed as βP = − 1 2q 2 e 2 β dψ 0 dx 2 + ρ 0 (x), where ρ 0 (x) is the density of counterions between the boundaries, given by the standard Boltzmann form ρ 0 (x) = Ce −ψ 0 (x) , where C is a normalization constant. This furthermore implies that the MF/PB disjoining pressure P is obtained as follows: When the pressure is repulsive (P > 0), we have P = µ 2 σ 2 1 Γ 2 /2, where Γ satisfies tan(ΓL) = Γ(1 + ζ)µ Γ 2 µ 2 − ζ , and when the pressure is attractive (P < 0), which may be the case within the MF/PB theory only for ζ < 0, we have P = −µ 2 σ 2 1 Γ 2 /2, where Γ is now given as a solution of coth(ΓL) = − ζ + µ 2 Γ 2 µΓ(1 + ζ) . C. Strong coupling The strong coupling limit is formally identical to the one-particle limit [1]. In the present case it is easily evaluated from the partition function in the case of a single counterion in the system. The partition function in an explicit one-particle form leads to the disjoining pressure P (L) = σ 2 1 2 − 1 2 (1 + ζ 2 ) + 1 2 (1 − ζ 2 ) coth (1 − ζ) L 2µ . The range of validity of this limiting expression is of course defined by the number of counterions in the system. As this number decreases towards one, N → 1, the above expression for the disjoining pressure becomes exact. D. Comparison Both the weak and strong coupling approximations are independent of dimension d and the comparison with the exact results can test their validity. For symmetric surface charges (ζ = 1) the PB/MF pressure is positive (repulsive) for all intersurface separations, whereas the SC expansion and the exact result for N = 1 predict attraction at large separations; this distinction holds for 0 < ζ ≤ 1. For the asymmetric configuration with ζ < 0, there is little difference between the different approaches; on trivial grounds there is attraction for large separations but there is repulsion for sufficiently small separations, see Fig. 2, where a comparison is made with Monte-Carlo (MC) simulations at different numbers of counterions N [7]. V. IONIC LIQUID LATTICE CAPACITOR In the models above the ions have been chosen to be point-like. Here we address the question of changes wrought by their finite size. In this case the system consists of a 1D lattice of M sites with spacing a, with the i−th site, 0 ≤ i < M , occupied by ion with charge qS i with S i ∈ [−q, 0, q], see Fig. 1 (bottom). Within this model the finite ion size is ∼ a, which is crucial to the phenomena observed in experiments on confined ionic liquids. The configuration described is one of the 1D ionic liquid capacitor. The external fields are imposed either by fixing the charges of the boundaries at i = −1 and i = M to be ±qQ, respectively, or by imposing a fixed voltage/potential difference, ∆v, across the capacitor. More details can be found in [8]. The electrostatic Hamiltonian in this case is expressed through a spin-like variable S i = 0, ±1 βH = − γ 4 M −1 i.j=0 |i − j|S i S j γ = βq 2 a ε . After a Hubbard-Stratonovich transformation this yields the action S(φ) = M −2 j=0 (φ j+1 − φ j ) 2 2γ − M −1 j=0 ln[1 + 2µ cos(φ)] + iQ(φ −1 − φ M ). The system includes boundary charges ±qQ at sites −1, M . The electrostatic potential is defined as V = −iφ/βq. In limit a → 0, q/a fixed, the MF equations obtained from the saddle-point of the above field action reduce to those of Kornyshev [9] and Borukhov et al. [10]. For non-zero a the action is not positive definite for µ ≥ 0.5 and so we seem to have a sign problem and certainly cannot use the Schrödinger approach a priori. Nevertheless, in the case of 1D the partition function can be computed exactly by using the transfer matrix approach, with the Fourier method described earlier. This can be seen as follows: write y i = φ i and define p 1/2 (y, y ) = 1 √ πγ e −(y−y ) 2 /γ , with K(y, y ) = dz p 1/2 (y, z)[1 + µ cos(z)]p 1/2 (z, y )(5) with Kf (y) = ∞ −∞ dy K(y, y )f (y ). The free energy for the fixed Q ensemble, Ω Q , then follows as e −βΩ Q = π −π dx e iQx p 1/2 K M p 1/2 e −iQx ≡ ψ Q |K M |ψ Q with ψ Q |y = e iQy , y|K|y = K(y, y ) The conjugate free energy for the fixed ∆v ensemble, Ω ∆v , then follows from a Legendre transform, e −βΩ ∆v = dQ e −∆vQ−βΩ Q , while the capacitance C ∆v is obtained from the first derivative of ∂ Q ∆v w.r.t. ∆v. and can thus be calculated directly from the partition function. A. Results The transfer matrix and Fourier approach can be formulated in order to evaluate the free energy explicitly. Details of this procedure can be found in Ref. [8]. Enthalpy G M = Ω M + M P bulk , the disjoining pressure P = G M − G M +1 , and the capacitance C ∆v , can all be calculated as a function of µ, Q, ∆v. We show explicitly only the capacitance results, C ∆v , as a function of ∆v in Fig. 3, both for large µ and small µ. For large µ the curve shows shows the typical "bell" shape in contrast to the curve for smaller µ, which shows the non-monotonic "camel" shape and so C ∆v has a minimum at the point of zero charge confirming the Fermi MF results of Kornyshev [9]. For smaller γ (increasing T ) the periodic non-monotonicity both for large µ and small µ disappears and the solution approaches the Fermi MF result of Kornyshev [9]. It is interesting that the exact solution dances around the Kornyshev solution with an ever increasing amplitude but the system nevertheless always remains thermodynamically stable, as can be straightforwardly ascertained. VI. LESSONS We have demonstrated that in 1D one can use the Schrödinger approach for continuum models of Coulomb fluids, but that for discrete models a more general approach is needed which exploits the transfer matrix and the periodicity of the field to use Fourier methods. We tested the PB/MF and the strong coupling limiting expressions and demonstrated that they need correcting although the exact analytic result clearly supports the two limiting analyses. We also confirmed that the MF theory does not capture the important effects which are due to correlations, either the attractive intersurface forces in the case of a counterion-only system or non-monotonic periodic variation of the capacitance in the confined ionic liquid case. FIG. 1 : 1Top: The 1D Coulomb fluid configuration in the case of the soap film model (positively and negatively charged particles and adsorbing surfaces), middle: counterions between charged surfaces (positively charged particles and oppositely charged bounding surfaces) and bottom: an ionic liquid lattice capacitor (positively and negatively charged particles on a lattice with one positive and one negative bounding surface). Positively (negatively) charged particles are shown schematically as light (dark) gray spheres. The integral over dφ(0) ensures charge neutrality: N qe+σ 1 +σ 2 = 0. The corresponding Hamiltonian (Feynman-Kac) and the partition function are thenH = − q 2 e 2 β 2 2 d 2 dφ 2 − e iφ , Z N = 2π 0 dφ e i σ1qe φ e −LH e i σ2 qe φ . By introducing σ 1 = 1−M 1 qe, σ 2 = −M 2 qe, M 1 = Int(αN ), M 2 = N − M 1 − 1, η 1 = αN − M 1 , and η 2 = 1 − η 1 and α = 1/(1 + ζ), we can derive the Fourier evolution equation from surface 2 to surface 1 in the form db(n, M 2 , L) dL = − (n − η 2 ) 2 2 βq 2 e 2 b(n, M 2 , L) + b(n − 1, M 2 , L), (4) with b(n, M 2 , 0) = δ n,−M2 . This Fourier evolution equation can be integrated numerically and the corresponding partition function Z(σ 1 , ζ, N, L) = b(M 1 + 1, M 2 , L), and disjoining pressure FIG. 2 : 2Rescaled disjoining pressure, 2P/σ 2 1 , for a 1D counterion gas between charged surfaces as function of the rescaled intersurface separation L/µ. Thick (red) solid lines represent the PB result (Section IV B), dashed lines are the SC results (Section IV C) and thin solid lines are the exact results (Section IV A) compared with MC simulations data (symbols) at different numbers of counterions N and for ζ = 0.5 (left pane) and ζ = −0.5 (right pane)[7]. ) Bell shaped C∆v as a function of ∆v for large µ = 0.5. (Right) Camel shaped C∆v as a function of ∆v for small µ = 0.03. In both cases, we have γ = 1. The dashed line is the MF result[8]. Statics and dynamics of strongly charged soft matter. H Boroudjerdi, Y.-W Kim, A Naji, R R Netz, X Schlagberger, A Serr, Phys. Rep. 416129H. Boroudjerdi, Y.-W. Kim, A. Naji, R. R. Netz, X. Schlagberger, and A. Serr. Statics and dynamics of strongly charged soft matter. Phys. Rep., 416:129, 2005. Exact Statistical Mechanics of a 1 Dimensional System with Coulomb Forces. 2. Method of Functional Integrals. S F Edwards, J Lenard, J. Math. Phys. 3778S. F. Edwards and J. Lenard. 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The field theory of symmetrical layered electrolytic systems and the thermal casimir effect. J. Phys. C, 17:3473, 2005. One-dimensional counterion gas between charged surfaces: Exact results compared with weak-and strong-coupling analysis. D S Dean, A Naji, R Podgornik, J. Chem. Phys. 13094504D. S. Dean, A. Naji, and R. Podgornik. One-dimensional counterion gas between charged surfaces: Exact results compared with weak-and strong-coupling analysis. J. Chem. Phys., 130:094504, 2009. The one-dimensional coulomb lattice fluid capacitor. V Demery, D S Dean, T C Hammant, R R Horgan, R Podgornik, J. Chem. Phys. 13764901V. Demery, D. S. Dean, T. C. Hammant, R. R. Horgan, and R. Podgornik. The one-dimensional coulomb lattice fluid capacitor. J. Chem. Phys., 137:064901, 2012. Double-Layer in Ionic Liquids: Paradigm Change?. A A Kornyshev, J. Phys. Chem. B. 1115545A. A. Kornyshev. Double-Layer in Ionic Liquids: Paradigm Change? J. Phys. Chem. B, 111:5545, 2007. 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Material attributes have been shown to provide a discriminative intermediate representation for materials, as they help tame the otherwise challenging intra-class appearance variation that materials exhibit. In the past, however, the space of material attributes has been manuallydefined and treated as merely an intermediate feature for other recognition tasks. Recent neuroscience studies on material perception, as well as computer vision research on object and place recognition, show that attributes may in fact arise naturally during the course of higher-level recognition. These results suggest that, analogous to scenes and objects, perceptual material attributes may arise during material recognition. Part of the challenge in investigating this conjecture is that existing attribute discovery methods cannot leverage large amounts of data likely required to cause the attributes to arise. In this paper, we design a novel network architecture and database in order to show that perceptual attributes can in fact arise from large-scale end-to-end material recognition. We focus on local material recognition, from small patches, in order to separate the materials from surrounding object and scene context. We extract the inherent material attributes by adding auxiliary loss functions to a material recognition CNN, enabling perceptual attributes to be produced as a side product. Our results show that the discovered attributes correspond well with semantically-meaningful visual material traits, and enable recognition of previously unseen material categories given only a few examples.

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Perceptual Material Attributes Arise in Local Material Recognition Gabriel Schwartz Department of Computer Science Drexel University Ko Nishino Department of Computer Science Drexel University Perceptual Material Attributes Arise in Local Material Recognition Material attributes have been shown to provide a discriminative intermediate representation for materials, as they help tame the otherwise challenging intra-class appearance variation that materials exhibit. In the past, however, the space of material attributes has been manuallydefined and treated as merely an intermediate feature for other recognition tasks. Recent neuroscience studies on material perception, as well as computer vision research on object and place recognition, show that attributes may in fact arise naturally during the course of higher-level recognition. These results suggest that, analogous to scenes and objects, perceptual material attributes may arise during material recognition. Part of the challenge in investigating this conjecture is that existing attribute discovery methods cannot leverage large amounts of data likely required to cause the attributes to arise. In this paper, we design a novel network architecture and database in order to show that perceptual attributes can in fact arise from large-scale end-to-end material recognition. We focus on local material recognition, from small patches, in order to separate the materials from surrounding object and scene context. We extract the inherent material attributes by adding auxiliary loss functions to a material recognition CNN, enabling perceptual attributes to be produced as a side product. Our results show that the discovered attributes correspond well with semantically-meaningful visual material traits, and enable recognition of previously unseen material categories given only a few examples. Introduction Attributes have proven to be a valuable intermediate representation for higher-level image understanding tasks. Material attributes, attributes used for material recognition, are particularly useful as they provide a discriminative representation for materials whose appearance otherwise exhibits large intra-class appearance variation [15]. Beyond just suggesting the presence of various materials, material attributes can inform us as to the potential physical proper- Local Materials Material Attributes MAC-CNN Figure 1. Existing attempts to leverage attributes for material recognition treat them simply as an intermediate representation. Following recently-discovered parallels between human material perception and material recognition algorithms, we show that we can automatically discover discriminative and semantically meaningful material attributes inside a local material recognition network trained end-to-end for category recognition. ties, such as "rough" or "soft", a material might exhibit. These cues can, for instance, guide autonomous interaction with real-world surfaces made of various materials. Attributes also have the desirable property that they can form a compact representation for novel categories from few examples (N-shot learning). Existing material recognition and material attribute discovery methods consider attributes separately from category recognition. Attributes are used either solely as an intermediate representation [15], or as an automatically discovered perceptual representation for the same purpose [16,22]. Similarly for conventional object and scene recognition, attributes like "sunset" or "natural," have also been extracted for use as independent features. Shankar et al. [17] generate pseudo-labels to improve the attribute prediction accuracy of a Convolutional Neural Network, and Zhou et al. [24] discover concepts from weakly-supervised image data. In both cases, the attributes are considered on their own, not within the context of higher-level categories. In object and scene recognition, however, recent work shows that semantic attributes seem to arise in networks that are trained end-to-end for category recognition [25]. We would like to take advantage of the benefits of endto-end learning to incorporate automatically-discovered attributes with material recognition in one seamless process. Material attribute recognition, however, is not easily scalable. Past approaches rely on semantic attributes, such as "shiny" or "fuzzy", that need careful annotation by a consistent annotator as their appearance may not be readily agreed upon. We are also specifically interested in scaling up local material recognition: recognizing materials using only information from small patches inside object boundaries so as to separate materials from the surrounding objects. Schwartz and Nishino [15,16] argue the importance of recognizing materials locally: methods which rely on context like object shape to recognize a material fundamentally confuse objects and materials, precluding the use of materials as a cue for object recognition (since the object was required to recognize the material). Along with results in object and scene recognition, recent neuroscience studies reveal that human material perception, in fact, relies on internal representations that correspond to semantic material attributes. Hiramatsu et al. and Goda et al. [7,8] have investigated how visual information is transformed in the brain during the human and animal recognition of materials. They find that the material representation in our visual system shifts from raw image features at lower levels (V1/V2) to perceptual properties (such as matte, colorful, fuzzy, shiny, etc.) in higher-level brain regions dedicated to recognition (FG/CoS). In this paper, we realize large-scale end-to-end learning for local material recognition and show that perceptual material attributes like those discovered by [16] are indeed present in and may be extracted from the network. We introduce a novel material attribute-category CNN architecture (MAC-CNN, Figure 1) to show that perceptual material attributes recognizable at the local level can be discovered during material recognition. By introducing additional auxiliary attribute layers (layers connected to the network but not participating in the final classification loss) and constraints derived from human material perception, we find that we may extract the perceptual material attributes present in the network. Unlike methods that rely on images and text (along with material annotations), we require only a perceptual distance matrix as weak supervision for the attributes. As part of our work, we also introduce a novel local material image database. We agree with their argument, but find that existing material databases are lacking in a few areas key to local material recognition. The Flickr Materials Database of Sharan et al. [19] provides sufficient information for local recognition, but uses only Flickr images which biases the dataset towards more artistic or professional images. Recent datasets, such as the Materials in Context (MINC) dataset [3], take steps to address this, but have inconsistencies in the definition of what makes a material category. The patches they extract are also large enough to include entire objects, further confusing the recognition of objects and materials. To support the experiments performed in this paper, we introduce a database explicitly targeted at local material recognition. We derive a systematically organized hierarchy for material categories, and we collect annotations for images from a wide variety of sources while carefully ensuring that object information, such as shape, is not present. Our results show that our MAC-CNN produces a generalizable internal material representation following the same principles proposed to be integral to human material perception. We show that the attributes we extract exhibit the same properties, such as spatial consistency, as existing automatically-discovered perceptual material attributes. By visualizing the arrangement of material categories in the space of attribute probabilities, we show that attributes separate materials into distinct clusters. We perform true local material recognition, predicting categories for single small image patches with no aggregation, a significantly more challenging task than previous approaches. We may also recognize manually-identified semantic material traits, such as "fuzzy" or "smooth", based on our attributes. Most important, we demonstrate that the extracted material attributes add significant information to recognize previously unseen material categories from a small number of training examples (i.e., N-shot learning with material attributes). These results show that our method successfully extracts effective and semantically meaningful internal representations of complex material appearance from a local material recognition network. Related Work In this paper, we investigate Convolutional Neural Networks (CNNs) as the framework within which we should find perceptual attributes. Shankar et al. [17] have recently proposed a modified CNN training procedure to improve attribute recognition. Their "deep carving" algorithm provides the CNN with attribute pseudo-label targets, updated periodically during training. This causes the resulting network to be better-suited for attribute prediction. Escorcia et al. [4] show that known semantic attributes can be extracted from a CNN. Most important for our work, they show that attributes depend on features in all layers of the CNN. Con-ceptLearner, proposed by Zhou et al. [24] uses weak supervision, in the form of images with associated text content, to discover semantic attributes. These attributes correspond to terms within the text that appear in the images. All of these frameworks predict a single set of attributes for an entire image, as opposed to the per-pixel attributes discussed in our work. Furthermore, our extracted attributes do not require semantic information (which may be challenging to collect in a consistent manner), and are defined based on human perceptual information. At the intersection of neuroscience and computer vision, Yamins et al. [23] find that feature responses from highperforming CNNs can accurately model the neural response of the human visual system in the inferior temporal (IT) cortex (an area of the human brain that responds to complex visual stimuli). They perform a linear regression from CNN feature outputs to IT neural response measurements and find that the CNN features are good predictors of neural responses despite the fact that the CNN was not explicitly trained to match the neural responses. Their work focuses on object recognition CNNs, not materials. Hiramatsu et al. [8] take functional magnetic resonance imaging (fMRI) measurements and investigate their correlation with both direct visual information and perceptual material properties (similar to the material traits of [15]) at various areas of the human visual system. They find that pairwise material dissimilarities derived from fMRI data correlate best with direct visual information (analogous to pixels) at the lowerorder areas and with perceptual attributes at higher-order areas. Goda et al. [7] obtain similar findings in non-human primates. Of particular importance is the fact that their work inherently considers materials independently from objects. Material samples are shown as cylinders of the material, thus avoiding any distracting cues from the surrounding object. These studies suggest the existence of perceptual attributes in human material recognition, but do not actually derive a process to extract them from novel images. Our work is closely related to the non-semantic perceptual material attributes discovered by Schwartz and Nishino [16]. In their work, they collect measurements of human perceptual distances between material categories and use those distances to discover perceptual material attributes that reproduce these distances. These attributes are subsequently used to recognize material categories. We use the constraints derived in their work as a basis for our auxiliary attribute layers. This approach can be considered similar to the work of Lee et al. [10], which introduced "deep supervision" via auxiliary loss functions to betterpropagate gradient information during CNN training. They do so by adding additional SVM-like loss functions that encourage classification at lower levels of the network. Rather than simply replicating the final classification loss, we impose new constraints to explicitly output additional information about the input, in our case the perceptual material attributes. Perceptual Material Attributes from Local Material Recognition In this paper, we show that perceptual material attributes arise in a material recognition framework. This agrees with the findings of Hiramatsu et al. [8] which indicate that perceptual attributes form an integral component of the human material recognition process. Based on correlations between Convolutional Neural Network (CNN) feature maps and human visual system neural output discovered by Yamins et al. [23], a CNN architecture appears to be a very suitable framework in which to discover attributes analogous to those in human material perception. We must derive our own method to realize material attributes, however, as their work focuses on object recognition and does not extract any attributes. We find the human-perceptionbased attributes of Schwartz and Nishino [16] to be particularly relevant. In this section we derive a novel framework to discover perceptual attributes similar to those in [16] inside a material recognition CNN framework. Finding Material Attributes in a Material Recognition CNN A simple experiment to verify the presence of perceptual attributes in a CNN trained to recognize materials would be to add an attribute prediction layer at the top of the network, immediately before the final material category probability softmax layer. If we could predict attributes from this layer without affecting the material recognition accuracy, it would suggest that the attributes were indeed present in the network. We implemented this approach with the goal of predicting the perceptual attributes derived from [16] and found that, while the material accuracy was unaffected, the attribute predictions were less accurate than those of their relatively simple attribute-only model (mean average error of 0.2 vs 0.08). The key issue with the straightforward approach is that it is not an entirely faithful model to the process described in [8]. They note that the human neural representation of material categories transitions from visual (raw image features) to perceptual (visual properties like "shiny") in an hierarchical fashion. This implies, in agreement with findings of Escorcia et al. [4], that attributes require information from multiple levels of the material recognition network. We show that this is indeed the case by successfully discovering the attributes using input from multiple layers of the material recognition network. Material Attribute-Category CNN We need a means of extracting attribute information at multiple levels of the network. Simply combining all feature maps from all network layers and using them to predict attributes would be computationally impractical. Rather than directly using all features at once, we augment an initial CNN designed for material classification with a set of auxiliary fully-connected layers attached to the spatial pooling layers. This allows the attribute layers to use information from multiple levels of the network without needing direct access to every feature map. We treat the additional layers as a set of weak learners, each auxiliary layer discovering the attributes available at the corresponding level of the network. This concept is similar to deep supervision by Lee et al. [10]. Their goal, however, is to inject the category recognition loss function into intermediate layers for better end recognition (in their case, object recognition) by simply propagating the same classification targets (via SVM-like loss functions) to the lower layers. Our goal is to discover and extract perceptual material attributes through this internal supervision using loss functions different from that for material category recognition. For the auxiliary layer loss functions, we extend the perceptual attribute loss functions of [16] and apply them to the outputs of each auxiliary fully-connected layer. Schwartz and Nishino's proposed method begins with a set of pairwise perceptual distances between material categories measured via human yes/no binary similarity annotations on material image patches. From these distances, they learn a mapping matrix A between categories and unknown, nonsemantic attributes. The mapping preserves the pairwise human perceptual distances while causing the resulting attributes to exhibit the behaviors, such as spatial consistency, of semantic attributes. We derive our attribute layer loss functions from these learning constraints. Specifically, assuming the output of a given pooling layer i in the network for image j is h ij , and given categories C, |C| = K and a set of sample points P ∈ (0, 1) for density estimation, we add these auxiliary loss functions: u i = 1 K k∈C a k − 1 N k j|cj =k f W T i h ij + b i 1 (1) d i = p∈P β (p; a, b) ln β (p; a, b) q p; f W T i h ij + b i ,(2) where f (x) = min (max (x, 0) , 1) clamps the outputs within (0, 1) to conform to attribute probabilities, and weights W i , b i represent the auxiliary fully-connected layers we add to the network. a k represents a row in the category-attribute mapping matrix we derived from our data by collecting the yes/no similarity answers used in [16] for patches in our database (see Sec. 4). Equation 1 causes the attribute layer to discover attributes which match the perceptual distances measured from human annotations. As certain attributes are expected to appear at different levels of the network, some layers will be unable to extract them. This implies that their error should be sparse, either predicting an attribute well or not at all. For this reason we use an L1 error norm. Equation 2, applied only to the final attribute layer, encourages the distribution of the attributes to match those of known semantic material traits. It takes the form of a KL-divergence between a Beta distribution (empirically observed by [16] to match the distribution of semantic attribute probabilities), and a Kernel Density Estimate q (·) of the extracted attribute probability sampled at points p ∈ P . The reference network we build on is based on the high-performing VGG-16 network of Simonyan and Zisserman [20]. We use their trained convolutional weights as initialization where applicable, and add new fully-connected layers for material classification. Figure 2 shows our architecture for material attribute discovery and category recognition. We refer to this network as the Material Attribute-Category CNN (MAC-CNN). Local Material Database In order to train the category recognition portion of the MAC-CNN, we need a proper local material recognition dataset. We find existing material databases lacking in a few key areas necessary to properly perform local material recognition. Previous material recognition datasets [2,3,18] have relied on ad-hoc choices regarding the selection and granularity of material categories (e.g., carpet and wall are considered materials). When patches are involved, as in [3], the patches can be as large as 24% of the image size surrounding a single pixel identified as corresponding to a material. These patches are large enough to include entire objects. These issues make it difficult to separate challenges inherent to material recognition from those related to general recognition tasks and inevitably lead to material recognition based on object and scene information, which would not be beneficial for scene understanding tasks. We also find that image diversity is still lacking in modern datasets. For these reasons, we introduce a new local material recognition dataset to support the experiments in this paper. Material Category Hierarchy Material categories in existing datasets have not been carefully selected. Examples of this issue include the proposed material categories "mirror" (actually an object), and "brick" (an object or group of objects). Existing categories also confuse materials and their properties (e.g., surface finish), for example, separating "stone" from "polished stone". To address the issue of material category definition, we propose a more carefully-selected set of material categories for local material recognition. We derive a taxonomy of materials based on their properties from materials science [1] and create a hierarchy based on the generality of each material family. Figure 3 shows an example of one tree of the hierarchy. Please see our supplemental material for a complete diagram of the hierarchy including all categories at all levels. Our hierarchy consists of a set of three-level material trees. The highest level corresponds to major structural differences between materials in the category. Metals are conductive, polymers are composed of long chain molecules, ceramics have a crystalline structure, and composites are fusions of materials either bonded together or in a matrix. We define the mid-level (also referred to as entry-level [12]) categories as groups that separate materials based primarily on their visual properties. Rubber and paper are flexible, for Figure 4. Local material patches extracted as the final step in our database creation process. These patches are used to compute human perceptual distances, and also form the training input for our combined material attribute-category CNN. example, but paper is generally matte and rubber exhibits little color variation. The lowest level, fine-grained categories, can often only be distinguished via a combination of physical and visual properties. Silver and steel, for example, may be challenging to distinguish based solely on visual information. Such a hierarchy is sufficient to cover most natural and manmade materials. In creating our hierarchy, however, we found that certain categories that are in fact materials did not fit within the strict definitions described above. For the sake of completeness, we make the conscious decision to add these mid-level categories to our data collection process. These categories are: food, water, and non-water liquids. While food is both a material and an object, we rely on our annotation process (Sec. 4.2) to ensure we obtain examples of the former and not the latter. Data Collection and Annotation The mid-level set of categories forms the basis for a crowdsourced annotation pipeline to obtain material regions from which we may extract local material patches (Figure 4). We employ a multi-stage process to efficiently extract both material presence and segmentation information for a set of images. The first stage asks annotators to identify materials present in the image. Given a set of images with materials identified in each image, the second stage presents annotators with a user interface that allows them to draw multiple regions in an image. Each annotator is given a single imagematerial pair and asked to mark regions where that material is present. While not required, our interface allows users to create and modify multiple disjoint regions in a single image. Images undergo a final validation step to ensure no poorly drawn or incorrect regions are included. Each image in the first stage is shown to multiple annotators and a consensus is taken to filter out unclear or incorrect identifications. While sentinels and validation were not used to collect segmentations in other datasets, ours is intended for local material recognition. This implies that identified regions should contain only the material of interest. During collection, annotators are given instructions Figure 5. Annotators did not hesitate to take advantage of the ability to draw multiple regions, and most understood the guidelines concerning regions crossing object boundaries. As a result, we have a rich database of segmented local material regions. to keep regions within object boundaries, and we validate the final image regions to insure this. Image diversity is an issue present to varying degrees in current material image datasets. The Flickr Materials Database (FMD) [19] contains images from Flickr which, due to the nature of the website, are generally more artistic in nature. The OpenSurfaces and Materials in Context datasets [2,3] attempt to address this, but still draw from a limited variety of sources (e.g., real estate photographs). We source our images from multiple existing image datasets spanning the space of indoor, outdoor, professional, and amateur photographs. We use images from the PASCAL VOC database [5], the Microsoft COCO database [11], the FMD [19], and the imagenet database [14]. Examples in Figure 5 show that our annotation pipeline successfully provides properly-segmented material regions within many images. Many images also contain multiple regions. While the level of detail for provided regions varies from simple polygons to detailed material boundaries, the regions all contain single materials. Perceptual Material Attributes Discovered in the MAC-CNN To verify that the perceptual attributes we seek are indeed present in and can be extracted with our MAC-CNN, we augment our dataset with annotations to compute the necessary perceptual distances described in [16]. Using our dataset and these distances, we derive a category-attribute matrix A and train an implementation of the MAC-CNN described in Sec. 3.2. We train the network on~200,000 48×48 image patches Figure 6. Attribute Space Embedding via t-SNE [21]: Many categories, such as water, food, foliage, soil, and wood, are very well-separated in the attribute space. We find that this separation corresponds roughly with per-category accuracy. extracted from segmented material regions. Optimization is performed using mini-batch stochastic gradient descent with momentum. The learning rate is decreased by a factor of 10 whenever the validation error increases, until the learning rate falls below 1 × 10 −8 . Properties of the Perceptual Material Attributes We examine the properties of our perceptual material attributes by visualizing how they separate materials, computing per-pixel attribute maps to verify that the attributes are being recognized consistently, and linking the non-semantic attributes with known semantic material traits ("fuzzy", "smooth", etc...) to visualize semantic content. Figs. 6, 7, and 8 are generated using a test set of held-out images. A 2D embedding of material image patches shows that the perceptual attributes ( Figure 6) separate material categories. A number of materials are almost completely distinct in the attribute space, while a few form overlapping but still distinguishable regions. Foliage, food, and water form particularly clear clusters. The quality of the clusters matches the per-category recognition rates, with accuratelyrecognized categories forming more separate clusters. Visualizations of per-pixel attribute probabilities in Figure 7 show that the attributes are spatially consistent. While overfitting is difficult to measure for weakly-supervised attributes, we use spatial consistency as a proxy. Spatial consistency is an indicator that the attributes are not overlysensitive to minute changes in local appearance, something that would appear if overfitting were present. The attributes exhibit correlation with the materials that induced them: attributes with a strong presence in a material region in one image often appear similarly in others. The visualizations also clearly show that the attributes are representing more than trivial properties such as "flat color" or "textured". Figure 8. By performing logic regression from our MAC-CNN extracted attributes to material traits, we are able to extract semantic information from our non-semantic attributes. Doing so in a sliding window gives per-pixel semantic material trait information. The predictions show crisp regions that correspond well with their associated semantic traits. Traits are independent, and thus the maps contain mixed colors. Fuzzy and organic in the lower right image, for example, creates a yellow tint. These semantic material traits computed from discovered material attributes provide rich information about the underlying surface properties that can be leveraged to determine how to interact with them. Logic regression [13] is a method for building trees that convert a set of boolean variables into a probability value via logical operations (AND, OR, NOT). It is well-suited for collections of binary attributes such as ours. Results of performing logic regression (Figure 8) from extracted attribute predictions to known semantic material traits (such as fuzzy, shiny, smooth etc...) show that our MAC-CNN attributes encode material traits with the same average ac-curacy (75%) as the attributes of [16]. For per-trait accuracy comparisons, please see our supplemental materials. We may also predict per-pixel trait probabilities in a sliding window fashion, showing that the attributes are encoding both perceptual and semantic material properties. The material attributes provide rich information regarding the surface properties that may benefit, for instance, action planning for autonomous agents. Local Material Recognition If perceptual material attributes are naturally present in the material classification network, we must be able to extract them without compromising the network's ability to recognize materials. Our results in Section 5 show that we can extract the perceptual attributes in the combined material-attribute network. We compare local material recognition accuracy with and without the auxiliary attribute loss functions to verify the second requirement. The average accuracy is 60.2% across all categories. Foliage is the most accurately recognized, consistent with past material recognition results in which foliage is the most visually-distinct category. Paper is the least wellrecognized category. Unlike the artistic closeup images of the FMD, many of the images in our database come from ordinary images of scenes. Paper, in these situations, shares its appearance with a number of other materials such as fabric. It is important to note that we are recognizing materials directly from single small image patches, with none of the region-based aggregation or large patches used in [3,15,16]. This is a much more challenging task as the available information is restricted. For a breakdown of percategory accuracy, please see our supplemental material. We find that the average material category accuracy does not change when the attribute layers are removed. While the attribute layers are auxiliary, they are connected to spatial pooling layers at every level and thus the attribute constraints affect the entire network. If the attributes were not in fact encoding visual material properties, constraining the network to extract them would negatively affect the material recognition performance. A full semantic segmentation framework is beyond the scope of this paper. We are, however, able to use the same attribute/material CNN to produce per-pixel material probability predictions. Results in Figure 10 show that we may still generate reasonable material probability maps even from purely local information. Novel Material Category Recognition One prominent application of attributes is in novel category recognition tasks. Examples of these tasks include one-shot [6] or zero-shot learning [9]. Zero-shot learning allows recognition of a novel category from a human-supplied list of applicable semantic attributes. Since our attributes Figure 9. Graphs of novel category recognition accuracy vs. training set size for various held-out categories. The rapid plateau shows that we need only a small number of examples to define a previously-unseen category. The accuracy difference between feature sets shows that the attributes are contributing novel information. Even when the attributes do not outperform material probabilities on their own, the combination is still superior demonstrating the rich discriminative information carried by the extracted material attributes. Figure 10. These material maps, obtained by applying the MAC-CNN in a sliding window, show that we may obtain coherent regions using only small local patches as input. The foliage predictions on the couch are reasonable, as the local appearance pattern is indeed a flower. In the baseball image, the local appearance of the fence resembles lace (a fabric). are non-semantic, zero-shot learning is not applicable here. We may, however, investigate the generalization of our attributes through a form of one-shot learning in which we use image patches extracted from a small number of images to learn a novel category. To evaluate the use of perceptual material attributes for novel category recognition, we train a set of MAC-CNNs on modified datasets each containing a single held-out category. No examples of the held-out category are present during training. The corresponding row of the categoryattribute matrix is also removed. The same number of attributes are defined based on the remaining categories. For the novel category training, we use a balanced dataset consisting of unseen examples of training categories and a matching number of images from the held-out category. We also separate a number of images of the heldout category as final testing samples. We train a simple binary classifier (a linear SVM) to distinguish between the training categories and the held-out category based on either their attribute probabilities, material probabilities, or both, computed on patches extracted from each input image. We measure the effectiveness of novel category recognition by the fraction of final held-out category samples properly identified as belonging to that category. Figure 9 shows plots of novel category recognition effectiveness as the number of training examples for the heldout category varies. We can see that the accuracy plateaus quickly, indicating that the attributes provide a compact and accurate representation for novel material categories. The number of images we are required to extract patches from to obtain reasonable accuracy is generally quite small (on the order of 10) compared to full material category recognition frameworks which require hundreds of examples. Furthermore, we include accuracy for the same predictions based on only material probabilities instead of attribute probabilities, as well as using a concatenation of both. Attributes alone offer better recognition for some novel categories. Even when they do not, the addition of attributes still increases performance. This clearly shows that the extracted attributes can expose novel discriminative information in the MAC-CNN that would not ordinarily be available. Conclusion We proposed that, analogous to human and animal findings in neuroscience, perceptual attributes inherently arise in the material recognition process. To show this, we derived a CNN architecture, the MAC-CNN, for discovering perceptual material attributes within a local material recognition network, collected a new image database on which to evaluate our method, and showed that the extracted enable accurate recognition of novel materials. The accuracy of novel category recognition based solely on the extracted attributes of a few sample images shows that the attributes form a compact representation for novel materials. We find the parallels between our own human visual perception of materials and the material attributes discovered in the MAC-CNN architecture particularly interesting. Our integration of attribute and category recognition with a single network likely has implications in other tasks such as object and scene recognition, and we may find similar parallels there as well. Figure 2 . 2Material Attribute-Category CNN (MAC-CNN) Architecture: We introduce auxiliary fully-connected attribute layers to each spatial pooling layer and combine the per-layer predictions into a final attribute output via an additional set of weights. The loss functions attached to the attribute layers encourage the extraction of attributes that match the human material representation encoded in perceptual distances. The first set of attribute layers acts as a set of weak learners to extract attributes wherever they are present. The final layer combines them to form a single prediction. Figure 3 . 3One tree in our material category hierarchy. Categories at the top level separate materials with notable differences in physical properties. Mid-level categories are visually distinct. The lowest level of categories are fine-grained and may require both physical and visual properties and expert knowledge to distinguish them. The full set of trees may be found in our supplemental material. Figure 7 . 7Each column after the first (the input image) shows per-pixel probabilities for an extracted perceptual attribute. The attributes form clearly delineated regions, similar to semantic attributes, and their distributions match as well. . Matbase: Chemical, Mechanical, Physical and Environmental Properties of Materials. 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We construct a universal action of a countable locally finite group (the Hall's group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces as subactions. The main ingredient is the amalgamation of actions by isometries. We show that an equivalence class of this universal action is generic.We show that the restriction to locally finite groups in our results is necessary as analogous results do not hold for infinite non-locally finite groups.We discuss the problem also for actions by linear isometries on Banach spaces.

10.1007/s11856-019-1856-8

1612.09448

01af75b7e26b01fcb740bd08621171d0438b51e2

UNIVERSAL ACTIONS AND REPRESENTATIONS OF LOCALLY FINITE GROUPS ON METRIC SPACES 2 Aug 2018 Michal Doucha UNIVERSAL ACTIONS AND REPRESENTATIONS OF LOCALLY FINITE GROUPS ON METRIC SPACES 2 Aug 2018 We construct a universal action of a countable locally finite group (the Hall's group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces as subactions. The main ingredient is the amalgamation of actions by isometries. We show that an equivalence class of this universal action is generic.We show that the restriction to locally finite groups in our results is necessary as analogous results do not hold for infinite non-locally finite groups.We discuss the problem also for actions by linear isometries on Banach spaces. Introduction Groups acting by isometries on metric and Banach spaces are one of the active areas of research in geometry, group theory and functional analysis. In this paper, we are interested in amalgamation of group actions and constructing universal actions on metric spaces and universal representations in Banach spaces. It is well known from the beginnings of combinatorial group theory that one can construct an amalgam of two groups over some common subgroup. At least as old is the amalgamation of metric spaces, or amalgamation of normed vector spaces. However, to the best of our knowledge, nobody has considered yet amalgamation of actions of groups on metric or Banach spaces by isometries. In metric geometry or functional analysis, amalgamation techniques are often used to construct various universal metric or Banach spaces (consider for instance the Urysohn universal metric space [25], or the Gurarij universal Banach space [10]). The well-known Hall's universal locally finite group ( [11]) is essentially made by amalgamating finite groups. Here by amalgamating actions of finite groups on finite metric spaces by isometries we obtain the following result. Theorem 0.1. There exists a universal action of the Hall's locally finite group G on the Urysohn space U by isometries. That is, for any action of a countable locally finite group H on a separable metric space X by isometries, there exists a subgroup H ′ ≤ G isomorphic to H such that, after identifying H and H ′ , there is an H-equivariant isometric embedding of X into U. The meaning of the theorem is that there is a single action of a countable locally finite group on a separable metric space by isometries that contains all actions of all countable locally finite groups on all separable metric spaces as subactions. One of the main ingredients is the amalgamation of actions and we have the following general theorem. Theorem 0.2. Let G 1 , G 2 be two groups (countable or not) with a common subgroup G 0 . Suppose that G 1 acts on a metric space X 1 and G 2 acts on X 2 , by isometries in both cases. Let X 0 be a common G 0 -invariant subspace of X 1 and X 2 , i.e. the restrictions of the two actions on G 0 and X 0 coincide. Then there is an amalgam of the action, which is an action of G 1 * G 0 G 2 on a metric space of density character max{|G 1 * G 0 G 2 |, dens(X 1 ), dens(X 2 )}. Following the research of Rosendal in [24] and of Glasner, Kitroser and Melleray in [8] we investigate the genericity of the universal action from Theorem 0.1. Theorem 0.3. The universal action from Theorem 0.1 is weakly generic in some sense. That is, the set of those actions in the Polish space Hom(G, Iso(U)) that are naturally equivalent to the universal one is dense G δ . As a consequence, we derive the following result which was originally asked by Melleray and Tsankov (in [17]) for abelian groups. Theorem 0.4. There exists a Polish group H such that for comeager many actions α ∈ Hom(G, Iso(U)) we have that the closure α[G] is topologically isomorphic to H. We show that the restriction to locally finite groups in our results is essential. Theorem 0.5. There are no analogously universal actions of infinite groups that are not locally finite. Moreover, we show that the amalgamation of the actions does not work in the abelian category. Theorem 0.6. The class of actions of finite abelian groups on finite metric spaces does not have the amalgamation property. Finally, we discuss universal actions on Banach spaces. General actions by isometries are by affine isometries. Unfortunately, we show that no universal action by affine isometries can exist, even of finite groups. Thus we are forced to restrict to actions by linear isometries, i.e. representations in Banach spaces. We propose a class of actions of finite groups on finite-dimensional Banach spaces such that, provided this class has the amalgamation property, its Fraïssé limit would be a universal action of the Hall's group on the Gurarij space by linear isometries. Preliminaries Let us start with our notational convention. All the group actions in this paper are by isometries. We usually denote actions by the symbol 'α : G X', where G is a group and X is a metric space. However, as it is common, we usualy write g · x instead of α(g, x). Regarding groups, we are mostly concerned with locally finite ones, where a group is locally finite if every finitely generated subgroup is finite. Since we shall work solely with countable groups, it is the same as saying that the group is a direct limit of a sequence of finite groups. Our constructions of universal objects are based on techniques commonly referred as "Fraïssé theory". We refer to Chapter 7 in [13] for more information about this subject. For a reader unfamiliar with this method we briefly and informally describe the basics of Fraïssé theory that we use in the paper. Let K be some countable class of mathematical objects of some type with some notion of embedding between these objects. Suppose that direct limits of objects from K exist. Think of the class of finite groups for instance. We say it is a Fraïssé class if any two objects from K can be embedded into a single object from K, such a property is called joint embedding property, and if whenever we have objects A, B, C ∈ K such that A embeds into both B and C, witnessed by embeddings ι B , resp. ι C , then there exists an object D ∈ K and embeddings ρ B , resp. ρ C of B into D, resp. C into D such that ρ C • ι C = ρ B • ι B ; i.e we can do amalgamation with object from K. The latter property is called amalgamation property. The Fraïssé theorem (see Chapter 7 in [13]) then asserts that there exists a unique object K, called the Fraïssé limit of K, which is a direct limit of a sequence of objects from K satisfying • every object A ∈ K embeds into K; • whenever we have objects A, B ∈ K such that A embeds via ρ A into K and via ι A into B, then there exists an embedding ρ B of B into K such that ρ A = ρ B • ι A . The second property is called the extension property and will be used in our proofs of universality of certain actions. Note that whenever X is some direct limit of a sequence of objects from K, then successive application of the extension property gives an embedding of X into K. We note that the Fraïssé theorem stated above is the only tool which we shall use and its proof is actually shorter that the discussion on Fraïssé theory above and may be left as an exercise. Since we are going to work with Fraïssé classes which are 'metric' we note that recently a general theory for metric Fraïssé classes was developed independently in [2] and [15]. However, we shall not directly use their results in our paper. Example 1 Consider the countable class of all finite graphs. It is easy to show it has the joint and amalgamation properties, thus by the Fraïssé theorem there exists a Fraïssé limit, a certain direct limit of a sequence of finite graphs, which is a countable graph commonly known as the random graph, or the Rado graph. The extension property allows to show that it contains as a subgraph a copy of every countable graph. Example 2 Consider now the countable class of all finite abelian groups. It is again easy to show the joint and amalgamation properties and one can even show that the Fraïssé limit is nothing else than n∈N Q/Z. Example 3 Consider now the countable class of all finite groups, not necessarily abelian. This is the most important example for us regarding the topic of our paper. It is less straightforward, nevertheless possible to show (see [19]), that this class has the amalgamation property, and thus also the joint embedding property. The Fraïssé limit is what is commonly known as the Hall's universal locally finite group ( [11]). Example 4 Consider the countable class of all finitely presented groups. It is again easy to show the amalgamation property. We are not aware anyone has considered the Fraïssé limit of this class yet. Example 5 Consider the countable class of all finite metric spaces with rational distances. The amalgamation and joint embedding is again straightforward. The Fraïssé limit is what is known as the rational Urysohn space. Its completion is the Urysohn universal space (see [25]). Example 6 As the last example, we present another 'metric Fraïssé class' recently discovered by the author in [5]. It is the class of all finitely generated free abelian groups with a 'finitely presented rational metric'. The completion of its limit gives the metrically universal abelian separable group. See the paper for details. Universal actions Let us start with the discussion on the notion of universality, which can be naturally done in the category-theoretical language. Given a category, consisting of objects and morphism (or embeddings) between them, an object is universal if for every object from the category there is a morphism (embedding) into the universal one. Our objects are groups acting on metric spaces by isometries. If the acting group, say G, is fixed, the natural notion of embedding is that of 'G-equivariant isometric embedding'. If the acting group is allowed to vary, then the embedding should consist of both group monomorphism and equivariant isometric embedding. We propose two notions of universality based on these two choices of embeddings. Definition 2.1. Let C be a class of actions of countable groups on separable metric spaces by isometries. Say that α : G X ∈ C is a universal action from C if for any action β : H Y ∈ C there is a subgroup H ′ ≤ G isomorphic to H and an isometric embedding of Y into X which is, after identifying H and H ′ , H-equivariant. The previous notion of universality corresponds to the universality from Theorem 0.1 if one takes as C the class of all actions of all countable locally finite groups on all separable metric spaces. We however state another notion of universality which is natural. As it will turn out, it is too strong. Definition 2.2. Let G be a fixed countable group. Let C be a class of actions of G on separable metric spaces. Say that α : G X ∈ C is a universal action from C if for any action β : G Y ∈ C there is an G-equivariant isometric embdedding of Y into X. 2.1. Proof of the main theorem. In this subsection, we define a natural Fraïssé class of actions of finite groups on finite metric spaces. Using that, we will prove Theorem 0.1. Definition 2.3. Let G be a group and X a metric space. A pointed free action of G on X by isometries is a tuple (G X, (x i ) i∈I ), where G X is a free action of G on X by isometries and I is some index set for the orbits of the action and (x i ) i∈I is a selector on the orbits, i.e. X = i∈I G · x i and for i = j, x i and x j lie in different orbits. There is also a natural notion of an embedding between two pointed free actions. Suppose we are given two such actions (H Y, (y i ) i∈I ) and (G X, (x j ) j∈J ). An embedding of (H Y, (y i ) i∈I ) into (G X, (x j ) j∈J ) is a pair (φ, ψ), where φ : H ֒→ G is a group embedding and ψ : Y ֒→ X is an isometric embedding that sends the distinguished points (y i ) i∈I into the set of distinguished points (x j ) j∈J and such that for any i, j ∈ I and f, h ∈ H we have d Y (f · y i , h · y j ) = d X (φ(f ) · ψ(y i ), φ(h) · ψ(y j )). Notice that in the case when the group monomorphism φ is just an inclusion, the previous definition says that ψ is an H-equivariant isometric embedding that sends the set of distinguished points into the set of distinguished points. Theorem 2.4. The pointed free actions can be amalgamated. Remark 2.5. It means that for any embeddings ψ i : (G 0 X 0 , (x j ) j∈I 0 ) ֒→ (G i X i , (x j ) j∈I i ), for j ∈ {1, 2}, where we assume that G 0 ≤ G 1 and G 0 ≤ G 2 , there are a group G 1 , G 2 ≤ G 3 , pointed action (G 3 X 3 , (x j ) j∈I 3 ) and embeddings ρ j : (G i X i , (x j ) j∈I i ) ֒→ (G 3 X 3 , (x j ) j∈I 3 ), for j ∈ {1, 2}, such that ρ 2 • ψ 2 = ρ 1 • ψ 1 . Proof. Consider such actions from the remark above, i.e. (G i X i , (x j ) j∈I i ), for i ∈ {0, 1, 2}. We may also suppose that I 0 ⊆ I i , for i = 1, 2, and that I 0 = I 1 ∩ I 2 . Let G 3 be G 1 * G 0 G 2 , i.e. the free product of G 1 and G 2 amalgamated over G 0 (we refer to [16] for constructions of amalgamated free products of groups). Let I 3 = I 1 ∪ I 2 and set X 3 = j∈I 3 G 3 · j. Clearly, X i ⊆ X 3 , for i = 1, 2. We shall define a metric on X 3 so that the canonical action of G 3 on X 3 is by isometries and that the inclusion of X i into X 3 is isometric (it is obviously G i -equivariant), for i = 1, 2. We define a structure of a weighted graph on X 3 that will help us define a metric there. That is, we define edges on X 3 and then associate a certain weight function w giving positive real numbers to these edges. For g, h ∈ G 3 and i, j ∈ I 3 , the elements g · x i and h · x j are connected by an edge if and only if • either g −1 h ∈ G 1 and i, j ∈ I 1 , then its weight is w(g · x i , h · x j ) = d X 1 (x i , g −1 h · x j ); • or g −1 h ∈ G 2 and i, j ∈ I 2 , then analogously its weight is w(g · x i , h · x j ) = d X 2 (x i , g −1 h · x j ). In case that g −1 h ∈ G 0 and i, j ∈ I 0 there is no ambiguity in the definition. Indeed, by assumption, in such a case we have d X 0 (x i , g −1 h · x j ) = d X 1 (x i , g −1 h · x j ) = d X 2 (x i , g −1 h · x j ). It is clear that this graph is connected, so we define the graph metric d on X 3 as follows: for x, y ∈ X 3 we set d(x, y) = inf{ n i=1 w(e i ) : e 1 . . . e n is a path from x to y}. In case the groups and the index sets are finite we may replace the infimum above by minimum. It follows immediately from the definition that the natural action of G 3 on X 3 is a weighted graph automorphism, i.e. it preserves the edges including their weight. It follows that G 3 acts by isometries on X 3 . We shall check that the canonical embeddings (inclusions) of X 1 and X 2 into X 3 are isometric. We shall check it for both X 1 and X 2 . Thus fix some g, h ∈ G and i, j ∈ I 3 such that either both g, h ∈ G 1 and both i, j ∈ I 1 , or both g, h ∈ G 2 and both i, j ∈ I 2 . We need to check that d X l (g·x i h·x j ) = d(g·x i , h·x j ), where l ∈ {1, 2} depending on whether g, h ∈ G 1 , i, j ∈ I 1 , or g, h ∈ G 2 , i, j ∈ I 2 . It is clear that d X l (g · x i , h · x j ) ≥ d(g · x i , h · x j ) , so suppose there is a strict inequality and we shall reach a contradiction. There is then an edge-path e 1 . . . e n from x = g · x i to y = h · x j . By induction on n, the length of the path, we shall show that d X l (g · x i , h · x j ) ≤ w(e 1 ) + . . . + w(e n ). The case n = 1 is clear, so we suppose that n ≥ 2 and we have proved it for all paths of length strictly less than n between all pairs of elements from X 1 and all pairs of elements from X 2 . Now without loss of generality we suppose that g, h ∈ G 1 , i, j ∈ I 1 , the other case is analogous. For 1 ≤ l ≤ n, let z l = g l · x i l be the start vertex of e l and z l+1 = g l+1 · x i l +1 the end vertex. Set h l = g −1 l g l+1 , for 1 ≤ l ≤ n. It follows that gh 1 h 2 . . . h n = h and each h l belongs to either G 1 or G 2 . If all the h l 's belong to G 1 then also all the i l 's belong to I 1 and the path goes within X 1 and we can use the triangle inequalities there. So we suppose that some h l , 1 ≤ l ≤ n, is from G 2 ; equivalently, that the path leaves X 1 at some point. Let 1 ≤ l < n be the least index where the path leaves X 1 , i.e. z l ∈ X 1 , while z l+1 / ∈ X 1 . It follows that i l ∈ I 0 . Indeed, by assumption for all k ≤ l we have i k ∈ I 1 , however since h l ∈ G 2 , by the definition of the edges of the graph we must have also that i l ∈ I 2 ; thus i l ∈ I 1 ∩ I 2 = I 0 . Now let l < l ′ ≤ n be the least index such that the path returns back to X 1 , i.e. the least index l < l ′ such that z l ′ ∈ X 1 . Again necessarily i l ′ ∈ I 0 . If 1 < l or l ′ < n, then the subpath e l . . . e l ′ −1 between two elements of X 1 is strictly shorter than n and thus by the inductive hypothesis we have d X 1 (z l , z l ′ ) ≤ w(e l ) + . . . + w(e l ′ −1 ). So we may replace this subpath by a single edge going from z l to z l ′ , hereby again shortening the path, so by the inductive hypothesis we get d X 1 (g · x i , h · x j ) ≤ w(e 1 ) + . . . + w(e n ). Thus we are left with the case l = 1 and l ′ = n. In such a case we have h 1 ∈ G 2 and h n ∈ G 2 , i 1 , i 2 , i n−1 , i n ∈ I 0 , and also, since n is the least number l such that h 1 . . . h l ∈ G 1 , we must actually have h 1 . . . h n ∈ G 0 . It follows that g and h lie in the same left-coset of G 0 in G 1 , i.e. g −1 h ∈ G 0 . It follows that d(g · x i , h · x j ) = d(x i , g −1 h · x j ). Thus it suffices to show that d(x i , g −1 h · x j ) = d X 0 (x i , g −1 h · x j ) = d X 1 (x i , g −1 h · x j ), where the latter equality is known and we need to show the former. In other words, we shall thus now, without loss of generality, assume that g = 1, so h = h 1 . . . h n ∈ G 0 and x i , h · x j ∈ X 0 . We have two cases: (1) If n = 2, i.e. h = h 1 h 2 , then the path e 1 e 2 is within X 2 between two elements from X 0 . Therefore, by the triangle inequality in X 2 , its length is greater or equal to the path consisting of a single edge from x i to h · x j , that means we have w(e 1 ) + w(e 2 ) = d X 2 (x i , h 1 · x i 2 ) + d X 2 (h 1 · x i 2 , h · x j ) ≥ d X 2 (x i , h · x j ) = d X 0 (x i , h · x j ) = d X 1 (x i , h · x j ), and we are done. (2) If n > 2, then the non-trivial subpath e 2 . . . e n−1 is a path of length strictly less than n between two elements from X 2 (note that z 2 = h 1 · x i 2 ∈ X 2 and also z n = hh −1 n · x in ∈ X 2 ), thus by the inductive hypothesis we get that w(e 2 ) + . . . + w(e n−1 ) ≥ d X 2 (z 2 , z n ). It follows that n l=1 w(e l ) ≥ d X 2 (x i , z 2 ) + d X 2 (z 2 , z n ) + d X 2 (z n , h · x j ) ≥ d X 2 (x i , h · x j ) = d X 0 (x i , h · x j ) = d X 1 (x i , h · x j ), and we are again done. Remark 2.6. The previous theorem was stated and proved for free actions. However, the proof can be modified to work for non-free actions as follows: Replace the metric by a pseudometric so that the action becomes free. Then proceed completely analogously working with pseudometrics instead of metrics and at the end make a metric quotient. Remark 2.7. We were informed by the referee that the proof of the previous theorem is related to the constructions from [4] where the authors show (among other things) that countable discrete groups that are uniformly embeddable into a Hilbert space are closed under taking amalgamated free products. In the next theorem we shall restrict our attention to actions of finite groups on finite metric spaces. In the proof, it will turn out that the theorem is a corollary of Theorem 2.4. That means that even if we are interested only in the theorem that follows it is natural to first prove Theorem 2.4 and use the existence of the general amalgam to show the existence of the finite amalgam. Theorem 2.8. The class of pointed free actions of finite groups on finite metric spaces has the amalgamation property Proof. Let us start as in the previous theorem with three pointed actions (G i X i , (x j ) j∈I i ), for i ∈ {0, 1, 2} such that G 0 ≤ G i , I 0 ⊆ I i , for i = 1, 2, I 0 = I 1 ∩ I 2 . Now the difference is that all the sets are finite. Let G be now any amalgam group of G 1 and G 2 over G 0 , e.g. the free product with amalgamation G 1 * G 0 G 2 . Set I 3 = I 1 ∪ I 2 and X G = j∈I 3 G · x j . As in the proof of Theorem 2.4 we define a weighted graph structure on X G . That is, for g, h ∈ G and i, j ∈ I 3 , the elements g · x i and h · x j are connected by an edge if and only if • either g −1 h ∈ G 1 and i, j ∈ I 1 , then its weight is w(g · x i , h · x j ) = d X 1 (x i , g −1 h · x j ); • or g −1 h ∈ G 2 and i, j ∈ I 2 , then analogously its weight is w(g · x i , h · x j ) = d X 2 (x i , g −1 h · x j ). There is no ambiguity when g −1 h ∈ G 0 and i, j ∈ I 0 . We again define the graph metric as follows: for x, y ∈ X G we set d G (x, y) = min{ n i=1 w(e i ) : e 1 . . . e n is a path from x to y}. Notice that now w assumes only finitely many values, so we may indeed use the minimum. Again, G acts on X G by graph automorphisms preserving the weight function, thus also by isometries. In the proof of Theorem 2.4 we showed that d G extends d X 1 and d X 2 in case G = G 1 * G 0 G 2 . We shall now find a finite amalgam G with the same property. First set G ′ = G 1 * G 0 G 2 . Set M = max{w(e) : e is an edge in X G ′ } and m = min{w(e) : w(e) = 0 and e is an edge in X G ′ }. Set K = ⌈ M m ⌉. Consider the finite set G 1 ∪ G 2 as the set of generators of G ′ and let λ : G ′ → [0, ∞) be the corresponding length function, i.e. the distance from the unit in G ′ in the Cayley graph of G ′ with G 1 ∪ G 2 as the generating set. Now we use the fact that free amalgams of residually finite groups over finite groups are residually finite, see Theorems 2 and 3 in [1]. We note that when the common subgroup is not finite, there are counterexamples (see [12]). Thus let G 3 be a finite group such that there is an onto homomorphism φ : G ′ → G 3 which is injective on the ball {g ∈ G ′ : λ(g) ≤ K + 1}. Clearly, G 1 and G 2 are subgroups of G 3 with the identified common subgroup G 0 . Thus in particular, G 3 is a finite amalgamation of G 1 and G 2 over G 0 . Moreover, we may suppose that G 1 ∪ G 2 generates G 3 . Let ρ be the length function on G 3 with respect to these generators. We have that φ is isometric with respect to λ and ρ on the ball {g ∈ G : λ(g) ≤ K + 1} We now set X 3 to be the finite set X G 3 = j∈I 3 G 3 · x j . We again consider X 0 , X 1 , X 2 to be subsets of X 3 . We have a metric d X 3 = d G 3 defined using the weight function. What remains to check is that the canonical inclusions of X 1 , resp. X 2 into X 3 are isometric. We shall do it for X 1 , for X 2 it is analogous. So take some g, h ∈ G 1 and i, j ∈ I 1 . We must check that d X 3 (g·x i , h·x j ) = d X 1 (g·x i , h·x j ). Again, it is clear that d X 3 (g · x i , h · x j ) ≤ d X 1 (g · x i , h · x j ); suppose that there is a strict inequality. It follows that there is a path e 1 . . . e n from g · x i to h · x j such that n l=1 w(e l ) < d X 1 (g · x i , h · x j ) . We claim that the length of the path n is less or equal to K. Suppose that n > K. Then since for every 1 ≤ l ≤ n we have w(e l ) ≥ m, we get n l=1 w(e l ) ≥ n · m > K · n ≥ M. However, by assumption d X 1 (g · x i , h · x j ) ≤ M, a contradiction. Now, it follows that the path e 1 . . . e n lies within the finite set i∈I 3 {g ∈ G 3 : ρ(g) ≤ K + 1} · x i . Since φ is isometric with respect to λ and ρ on the ball {g ∈ G : λ(g) ≤ K + 1} it follows that the path e 1 . . . e n from G 3 also exists in X G ′ , and is, by definition, of the same length. However, we showed in the proof of Theorem 2.4 that in X G ′ its weight was greater or equal to d X 1 (g · x i , h · x j ). This finishes the proof. Remark 2.9. We note that in the previous theorem it was essential that the actions were free. In that case, the residual finiteness of the free product of finite groups with amalgamation was sufficient. A strictly stronger notion than residual finiteness is the Ribes-Zalesskiǐ property (see [22]). Note that a one way how to formulate residual finiteness of G is to say that the unit 1 G is closed in the profinite topology on G. In a similar spirit, one says that G has the Ribes-Zalesskiǐ property if for every tuple of finitely generated subgroups H 1 , . . . , H n ≤ G their product H 1 · . . . · H n = {h 1 · . . . · h n : ∀i ≤ n (h i ∈ H i )} is closed in the profinite topology. In [23], Rosendal used the Ribes-Zalesskiǐ property for finitary approximations of actions of groups on metric spaces by isometries. Similar ideas could be used to prove the amalgamation property for general, not necessarily free, actions if amalgamated free products had the Ribes-Zalesskiǐ property. After proving the preceding theorem, it was pointed out to us by Julien Melleray that indeed amalgamated free products of two finite groups do have the Ribes-Zalesskii property. The proof follows the lines of Theorem 2 in [1] and uses the fact that free groups have this property. Let (G n X n , (x i ) i∈In ) n∈N be an enumeration of all pointed free actions of finite groups on finite metric spaces with rational distances. It follows from the previous theorem that it is a Fraïssé class. Indeed, it is clear from the proof that when working with rational spaces the amalgam will be rational as well. Moreover, the joint-embedding property is just a special case of the amalgamation property (note that any two actions have a common subaction, namely the action of a trivial group on a one-point space). So it has some Fraïssé limit (α 0 : G X, (x i ) i∈I ), where G is some countably infinite locally finite group, X is a countably infinite rational metric space with countably infinite distinguished set of points (x i ) i∈I and α 0 : G X is a free action by isometries. It follows from the Fraïssé theorem that (α 0 : G X, (x i ) i∈I ) has the following extension property: Fact 2.10 (The extension property). Let F ≤ G be a finite subgroup, A ⊆ I a finite subset, and denote by X 0 the finite metric space i∈A F · x i . Consider the free pointed action (F X 0 , (x i ) i∈A ). Let (H Y, (y j ) j∈B ) be some free pointed action of a finite group on a finite rational metric space and let (ψ, φ) is an embedding from (F X 0 , (x i ) i∈A ) to (H Y, (y j ) j∈B ). Then there exists an embedding (ψ,φ) from (H Y, (y j ) j∈B ) to (G X, (x i ) i∈I ) such thatψ • ψ = id F andφ • φ = id X 0 . Now let X be the metric completion of X. The action α 0 : G X obviously extends to the action α : G X by isometries, which is no longer free though. The following is a restatement of Theorem 0.1 from Introduction using the just constructed action α : G X. Theorem 2.11. The action α : G X is a universal action in the class of all actions of countable locally finite groups on all separable metric spaces by isometries. Before we prove the theorem we shall need few notions and lemmas. Definition 2.12. Let X be a set equipped with two pseudometrics d and p. We define the distance D(d, p) between these two pseudometrics as their supremum distance, i.e. D(d, p) = sup x,y∈X |d(x, y) − p(x, y)|. Lemma 2.13. Let (H X, (x i ) i∈I ) be a free pointed action by isometries of some finite group H on a finite pseudometric space X = i∈I H · x i with pseudometric d. Then for any ε > 0 there exists a rational metric p on X such that the free action of H on (X, p) is still by isometries and D(d, p) < ε. Proof of Lemma 2.13. Enumerate by (d i ) i≤n the distances from (X, d) in an increasing order. Also, we may suppose that ε < min{|k − l| : k = l, k, l ∈ {d i : i ≤ n} ∪ {0}}. For i ≤ n, let p i be an arbitrary rational number from the open interval (d i + (n−i)ε n+1 , d i + (n+1−i)ε n+1 ). Now for a pair x, y ∈ X set p(x, y) = 0 if x = y and for x = y ∈ X set p(x, y) = p i iff d(x, y) = d i . Let us check that p is a rational metric. By definition it is rational. It is clear that p(x, y) = 0 iff x = y, and that it is symmetric, so we must just check the triangle inequality. Take a triple x, y, z ∈ X. We check that p(x, z) ≤ p(x, y) + p(y, z). If either d(x, y) or d(y, z) is bigger or equal to d(x, z), then the same is true for p(x, y), p(y, z), p(x, z) by definition. So we may suppose that d(x, z) > max{d(x, y), d(y, z)}. Then by setting d(x, z) = d i , d(x, y) = d j and d(y, z) = d k , we have that i > max{j, k}. We must check that p i ≤ p j + p k . However, we have p i ≤ d i + (n + 1 − i)ε n + 1 ≤ d j + (n − j)ε n + 1 + d k + (n − k)ε n + 1 ≤ p j + p k , and we are done. Lemma 2.14. Let H 1 ≤ H 2 be two finite groups and I ⊆ J two finite sets. Let d be a metric on X = i∈I H 1 · x i and p be a metric on Y = j∈J H 2 · x j ⊇ X. Suppose that the canonical actions of H 1 on (X, d), resp. of H 2 on (Y, p) are by isometries. Suppose further that D(d, p ↾ X) ≤ ε. Then there exists a metric ρ on Z, the disjoint union X ⊆ i∈I H 2 · x i j∈J H 2 · x j = Y which is equal to i∈I H 2 · x i ∪ j∈J H 2 · y j such that • ρ extends both d and p on the corresponding subspaces, • for every i ∈ I, ρ(x i , y i ) ≤ ε, • the canonical action of H 2 on Z is by isometries. Proof of Lemma 2.14. As before, we define a weighted graph structure on Z. A pair x, y is connected by an edge if and only if • either x, y ∈ X, resp. x, y ∈ Y , in such a case w(x, y) = d(x, y), resp. w(x, y) = p(x, y); • or there are i ∈ I ⊆ J and h ∈ H 2 such that x = h · x i and y = h · y i or vice versa, in such a case we set w(x, y) = ε; • or x = g · x i , y = h · x j such that i ∈ I and g −1 h ∈ H 1 ; in such a case we set w(x, y) = d(x i , g −1 hx j ). It is again immediate that the graph is connected, thus it determines a metric ρ on Z, and the canonical action of H 2 on Z is by isometries. We need to check that ρ extends d and p. We check both simultaneously. Fix x, y such that either x, y ∈ X or x, y ∈ Y . Suppose that ρ(x, y) < d(x, y) (it is again clear that ρ(x, y) ≤ d(x, y)), resp. ρ(x, y) < p(x, y) depending on where x, y lie. Then there is an edge path e 1 . . . e n such that w(e 1 ) + . . . + w(e n ) < d(x, y), resp. w(e 1 ) + . . . + w(e n ) < p(x, y). We shall again prove the claim by induction on the length of the edge path. The case n = 1 is clear. Suppose we have proved it for all l < n and all edge paths of length at most l between all pairs x, y ∈ X and all pairs x, y ∈ Y . We may suppose that there are not two neighboring edges e i and e i+1 such that both of them lie in X or both of them lie in Y , for otherwise we could contract them into a single edge using triangle inequality in X, resp. Y . Suppose first that x, y ∈ X and let x = g · x i and y = h · x j , for some g, h ∈ H 1 and i, j ∈ I. Denote byX the set i∈I H 2 · x i ⊇ X. Notice that there is no edge between an element z ∈ X and an element z ′ ∈X \ X. Thus we may suppose that e 1 is an edge between x = g · x i and g · y i and e n is an edge between h · y j and h · x j = y. Indeed, otherwise either e 1 is an edge within X, so we may use the inductive assumption for the subpath e 2 , . . . , e n , or e n is an edge within X and we may use the inductive assumption for the subpath e 1 , . . . , e n−1 . It follows that e 2 , . . . , e n−1 is an edge path of length strictly less than n between two elements of Y , thus by inductive assumption we may suppose that n = 3 and e 2 is an edge between g · y i and h · y j and we have w(g · y i , h · y j ) = p(g · y i , h · y j ) ≥ d(g · x i , h · x j ) − ε. However, since w(e 1 ) = w(e 3 ) = ε, we get that d(x, y) = d(g · x i , h · x j ) < w(e 1 ) + w(e 2 ) + w(e 3 ), a contradiction. Suppose now that x, y ∈ Y and again let x = g · y i and y = h · y j , for some g, h ∈ H 2 and i, j ∈ J. As in the paragraph above, we may without loss of generality assume that e 1 is an edge between x = g · y i and g · x i and e n is an edge between h · x j and h · y j = y; thus in particular i, j ∈ I. If both g, h ∈ H 1 then g · x i , h · x j ∈ X and we are done by the same argument as in the paragraph above. So suppose that at least one of g, h is in H 2 \ H 1 . Say g ∈ H 2 \ H 1 , i.e. g · x i ∈X \ X. Since there is no edge between an element from X and an element fromX \ X there exists a minimal l ≤ n such that e 2 , . . . , e l−1 is a path withinX \ X and e l is an edge between an element fromX \ X and an element from Y . If l < n then we use the inductive hypothesis, so suppose that l = n, i.e. the subpath e 2 , . . . , e n−1 is withinX \ X. Note also that there is an edge between elements f · x k and f ′ · x k ′ inX \ X if and only if f −1 f ′ ∈ H 1 . It follows that g −1 h ∈ H 1 . Translating the whole path e 1 , . . . , e n by g −1 we does not change the distance (g −1 acts as an isometry). Thus we may assume that g = 1 and h ∈ H 1 . However, then we are again done by an argument used above. Proof of Theorem 2.11. Let H Z be an action of an infinite locally finite group by isometries on a separable metric space. It is sufficient to prove the theorem in case Z is countable. Indeed, in the general case we would find a countable dense H-invariant subspace Z ′ . Then the action on the metric completion of Z ′ , thus in particular on Z, is uniquely determined by its behavior on Z ′ . Since the space X is complete, we are done. So assume that both H and Z are countable. Without loss of generality we assume that Z has infinitely many H-orbits and let (z n ) n∈N be a sequence which picks one single element from each orbit, i.e. we may write the metric space Z as n∈N H · z n with a pseudometric d. Also, without loss of generality we shall assume that H is infinite and write H as H 1 ≤ H 2 ≤ H 3 ≤ . . . which is an increasing chain of finite subgroups of H whose union is H. Moreover, for every n define Z n to be the finite pseudometric subspace i≤n H n · z i ⊆ Z. For every n consider the free pointed action (H n Z n , (z i ) i≤n ). By Lemma 2.13 there exists a rational metric p n on Z n such that D(d ↾ Z n , p n ) < 1/2 n+1 . In particular, we get a free action of H n on (Z n , p n ) by isometries. It follows that for every n we have D(p n , p n+1 ↾ Z n ) < 1/2 n . By Lemma 2.14, for every n we can define a rational metric ρ n on a disjoint union of Z n Z n+1 = i≤n H n · z i ∪ j≤n+1 H n+1 · z ′ j which is free and by isometries, and which extends the original metrics and for i ≤ n we have ρ n (z i , z ′ i ) = 1/2 n . Now by a successive application of Fact 2.10, the extension property of (G X, (x i∈I ), we obtain • an increasing chain of finite subgroups H ′ 1 ≤ H ′ 2 ≤ . . . ≤ G and isomorphisms ψ i : H ′ i → H i , for i ∈ N, such that ψ i ⊆ ψ i+1 for every i. Thus ψ = i ψ is an isomorphism between H ′ = n H ′ n and H; • isometric embeddings φ n : Z n Z n+1 ֒→ X such that φ n ↾ Z n+1 = φ n+1 ↾ Z n+1 , for every n; • for every n, we have that the free actions H n Z n and H ′ n φ n [Z n ] are isometric. For every i we have that the sequence (φ n (z i )) n≥i is Cauchy, since d X (φ n (z i ), φ n+1 (z i )) = 1/2 n . Let y i ∈ X be the limit of that sequence. Consider the subset Z ′ = i∈N H ′ · y i ⊆ X. It follows it is naturally isometric to Z. Indeed, take any x, y ∈ Z and write them as x = h · z i and y = g · z j for some h, g ∈ H and i, j ∈ N. Since H and H ′ ≤ G are isomorphic, let h ′ , g ′ be the corresponding elements of H ′ ≤ G and consider the elements h ′ · y i , g ′ · y j ∈ Z ′ ⊆ X. Then d(h ′ · y i , g ′ · y j ) = lim n d(h ′ · φ n (z i ), g ′ · φ n (z j )) = lim n d Z (h · z i , g · z j ) + o(n), where o(n) ∈ [0, 1/2 n ], so the claim is proved. Finally, consider the restriction of the action G X on H ′ i∈N H ′ · y i . It follows from the approximation above that it is isometric to the action H Z, and we are done. Finally, we show that the group G is isomorphic to the Hall's universal locally finite group and that the space X is isometric to the rational Urysohn space, so the completion X is isometric to the Urysohn universal space. It is just the use of the extension property of G X, (x i ) i∈I from Fact 2.10. These are standard arguments, so we omit some details. For the former it is necessary to show that G has the extension property. That is, whenever F ≤ G is some finite subgroup and H ≥ F is some abstract finite supergroup of F , i.e. a supergroup of F that does not in principle lie in G, then we can actually find a copy H ′ of H within G so that it is a supergroup of F there, i.e. F ≤ H ′ ≤ G. So pick some finite subgroup F ≤ G and some abstract supergroup H ≥ F . The Fraïssé limit G X, (x i ) i∈I is a direct limit of a sequence of some finite actions (G n X n , (x i ) i∈In ) n∈N . Take n so that F ≤ G n and consider the subaction F X ′ n , (x i ) i∈In , where X ′ n = i∈In F · x i . It is possible to use Lemma 2.14 to extend this action to an action of H on i∈In H · x i . Then we use the extension property of G X, (x i ) i∈I to find the action H i∈In H · x i within the universal one, thus in particular to find a copy of H within G that is a supergroup of F . Now for the latter, it is necessary to show that the countable rational metric space X has the extension property. That is, whenever A ⊆ X is some finite subspace and A ⊆ B is finite abstract extension, still a rational metric space, then we can actually find this extension within X. So take some finite A ⊆ X. As above, find some n so that A ⊆ X n . By extending the metric by metric amalgamation if necessary we may assume that A = X n . Set I ′ n = I ∪(B \X n ) and X ′ n = i∈I ′ n G n ·x i . Clearly, A = X n ⊆ B ⊆ X ′ n . By using the technique with defining a weighted graph structure on X ′ n we can extend the metric from B to X ′ n so that G n acts on X ′ n by isometries. Then we use the extension property of G X, (x i ) i∈I to get a copy of X ′ n , thus also of B, in X so that it is an extension of A there. 2.2. Amalgamation property for actions of abelian groups. A possible modification of the ideas above would be to consider the class of (pointed free) actions of abelian groups on finite metric spaces. One might expect that it has the amalgamation property as well and the (completion of the) limit will be a universal action of n∈N Q/Z on the Urysohn space U, where n∈N Q/Z is easily checked to be the Fraïssé limit of the class of all finite abelian groups. We show that it is not the case. More precisely, we prove the following proposition. Proof. In the proof we stick to the same notation we used for actions of general non-abelian groups. Let us consider actions (G i X i , (x 1 , x 2 , x 3 )), for i = 0, 1, 2, where G 0 is the trivial group and G 1 = G 2 = Z/2Z. We denote the single non-zero element of G 1 as g and the single non-zero element of G 2 as h. The embeddings of (G 0 X 0 , (x 1 , x 2 , x 3 )) into (G 1 X 1 , (x 1 , x 2 , x 3 )), resp. (G 2 X 2 , (x 1 , x 2 , x 3 )) are obvious. We set d(x 1 , x 2 ) = d(x 2 , x 3 ) = 10 and d(x 1 , x 3 ) = 16. On X 1 = i≤3 G 1 · x i , we then have d(g · x 1 , g · x 2 ) = d(g · x 2 , g · x 3 ) = 10 and d(g · x 1 , g · x 3 ) = 18. Additionally, we set d(x 1 , g · x 2 ) = d(x 2 , g · x 3 ) = d(g · x 1 , x 2 ) = d(g · x 2 , x 3 ) = 8. This gives X 1 a structure of a connected weighted graph on which G 1 acts by preserving the weighted graph structure. So we can define a metric on X 1 as the graph metric. It is easy to check that this metric extends d and G 1 acts on X 1 by isometries. On X 2 = i≤3 G 2 · x i , we also have d(h · x 1 , h · x 2 ) = d(h · x 2 , h · x 3 ) = 10 and d(h · x 1 , h · x 3 ) = 18. Additionally, we set d(x 2 , h · x 2 ) = 1, d(x 1 , h · x 3 ) = d(h · x 1 , x 3 ) = 18 . This again gives X 2 a structure of a connected weighted graph on which G 2 acts by preserving the weighted graph structure. So, as for X 1 , we define a metric on X 2 as the graph metric, which clearly extends d and G 2 acts on X 2 by isometries. We claim that these two actions cannot be amalgamated over G 0 X 0 . Suppose that there is an amalgam action of some G 3 on X 3 = i≤3 G 3 · x i . We denote the metric on X 3 again just by d. Moreover, we denote the image of g ∈ G 1 , resp. of h ∈ G 2 , in G 3 again by g, resp. by h. Then we have 18 = d(x 1 , h · x 3 ) ≤ d(x 1 , g · x 2 ) + d(g · x 2 , (h + g) · x 2 ) + d((h + g) · x 2 , h · x 3 ) = 8 + 1 + 8 = 17, a contradiction. However, we do not know if this class has the cofinal amalgamation property. That is, whether there exists a proper subclass C having the amalgamation property such that every pointed free action of an abelian group on a finite metric space embeds to an action from C. The existence of such a class would also imply the existence of a universal action of n∈N Q/Z on the Urysohn space by isometries. So in particular the following is left open. Proof. Suppose the contrary, first for the metric spaces. That is, suppose there exists a separable metric space X and an action α : G X by isometries such that for any action β : G Y by isometries on a separable metric space Y there is a G-equivariant isometric embedding of Y into X. Since G is countably infinite we can find a sequence (g n ) n that generates G and moreover, for any n = m we have g n / ∈ {g m , g −1 m }. For any x ∈ 2 N let λ ′ x : {g n , g −1 n : n ∈ N} → N be defined as follows: λ ′ x (g) = 1 g ∈ {g n , g −1 n } ∧ x(n) = 0, 2 g ∈ {g n , g −1 n } ∧ x(n) = 1. Finally, for any x ∈ 2 N define a length function λ x : G → N as follows: for any g ∈ G, set λ x (g) = min{ m i=1 λ ′ x (h i ) : g = h 1 . . . h m , (h i ) m i=1 ⊆ {g n , g −1 n : n ∈ N}}. We claim that λ x extends λ ′ x , i.e. for every g ∈ {g n , g −1 n : n ∈ N}, λ x (g) = λ ′ x (g). It suffices to show that for any n such that x(n) = 1 we have λ x (g n ) = 2. Suppose the contrary. Then necessarily λ x (g n ) = 1, so by definition g n = g m or g n = g −1 m for m such that x(m) = 0. However, that contradicts our assumption . Now for every x ∈ 2 N take the left-invariant metric d x on G induced by λ x . The action of G on itself by left translations is then an action of G on (G, d x ) by isometries. We claim that there is x ∈ 2 N such that there is no G-equivariant isometric embedding of (G, d x ) into X. Suppose otherwise that for every x ∈ 2 N there is a G-equivariant isometric embedding ι x of (G, d x ) into X. For every x ∈ 2 N denote ι x (1 G ) ∈ X by z x . Then for x = y ∈ 2 N we have d X (z x , z y ) ≥ 1/2, for if d X (z x , z y ) < 1/2 and n ∈ N is such that x(n) = y(n), say x(n) = 1, y(n) = 0, then 1/2 > d X (z x , z y ) = d X (g n · z x , g n · z y ) ≥ |d X (g n · z x , z x ) − d X (z x , z y ) − d X (z y , g n · z y )| > 1/2, a contradiction. Thus we get that {z x : x ∈ 2 N } ⊆ X is a 1/2-separated uncountable set in X which contradicts the separability of X. To prove the same for the category of Banach spaces, we can for example extend the action of G on (G, d x ), for every x ∈ 2 N , to an action of G on the Lipschitz-free Banach space F (G, d x ) over (G, d x ) (see [9] and [26] for information about Lipschitz-free Banach spaces). That is, consider a real vector space V G with G \ {1 G } as the free basis, and 1 G as a zero. Define a norm · x on V G as follows: for v = α 1 g 1 + . . . + α n g n set v x = min{ m i=1 |β i | · d x (h i , h ′ i ) : v = m i=1 β i (h i − h ′ i )}. Then it is easy to check (and it is a standard fact about Lipschitz-free spaces) that for any g, h ∈ G, g − h x = d x (g, h). G acts by (affine) isometries on (V G , · x ) in the following way: for h ∈ G and α 1 g 1 + . . . + α n g n ∈ V G we set h · (α 1 g 1 + . . . + α n g n ) = (α 1 hg 1 + . . . + α n hg n ) − (α 1 + . . . + α n − 1)h. It is easy to check that this gives an action of G on (V G , · x ) by isometries which extends the action of G on itself by translation. It also extends to an action of G on the completion W x . Then arguing exactly the same as with the metric space one can show that it is not possible to embed in a G-equivariant way all the spaces W x ,x ∈ 2 N , into a single separable Banach space with an action of G. Second, we explain that the universality cannot be naturally extended beyond the class of locally finite groups -at least in the case when we do not restrict the class of separable metric spaces. Proposition 2.18. Let G be a countably infinite non-locally finite group. Let C be the class of all actions of groups isomorphic to subgroups of G on all separable metric spaces. Then C does not admit a universal action. Proof. Suppose there is such an action α : F X, where F is some subgroup of G and X is some separable metric space. Since G is not locally finite it contains a finitely generated infinite subgroup H ≤ G. By the proof of Theorem 2.17 there are continuum many somewhat different left-invariant metrics (d x ) x∈2 N on H. Note that F contains at most countably many subgroups isomorphic to H. By the pigeonhole principle there is one fixed subgroup H ′ ≤ F isomorphic to H and an uncountable subset I ⊆ 2 N such that for each x ∈ I there is an H ′ -equivariant isometric embedding of (H ′ , d x ) into X. We reach a contradiction with separability of X by the same argument as in the proof of Theorem 2.17. We have not considered any classes of actions where the class of separable metric spaces is restricted. Problem 2.19. Find a natural class of separable metric spaces for which there are universal actions of non-locally finite groups. Genericity of the action The objects constructed using the Fraïssé theory enjoy two interesting properties, the universality and the homogeneity. In the previous section we focused solely on the universality of the constructed action as that was the more interesting part in our point of view. We shall not explore the homogeneity here, however we want to focus on a property which often accompanies homogeneity, in fact is a homogeneity in disguise in a sense. That is the 'genericity'. Let us start with a general discussion. Fix some countable group G. Let X be some Polish metric space, i.e. a complete separable metric space. We want to define a space of all actions of G on X by isometries. We have that Iso(X), the group of all isometries on X with the pointwise-convergence, or equivalently compact-open, topology is a Polish group, i.e. a completely metrizable second-countable topological group. Fixing a countable dense subset {x i : i ∈ N} ⊆ X we may define a compatible complete metric ρ on Iso(X) as follows: for φ, ψ ∈ Iso(X) we set ρ(φ, ψ) = ∞ i=1 min{d X (φ(x i ), ψ(x i )), 1} 2 i . Since every action α : G X by isometries is in unique correspondence with some homomorphism f : G → Iso(X), we may define the space Act G (X) of all actions of G on X by isometries as the space of all homomorphisms of G into Iso(X). Act G (X) is a closed subspace of the product space Iso(X) G , thus a Polish space. There has been a recent research on investigating which countable groups admit generic homomorphisms into certain Polish groups. That means, fix a countable group G and a Polish group H. Denote by Hom(G, H) the Polish space of all homomorphisms of G into H. Note that there is a natural equivalence relation on the space Hom(G, H), that is of conjugation, where two homomorphisms f, g : G → H are conjugate if there exists an element x ∈ H such that f = x −1 gx. Say that a homorphism f is generic if it has a comeager conjugacy class. In [7], the author with Malicki and Valette prove that a countable group G with property (T) and such that finite-dimensional representations are dense in the unitary dualĜ has a generic unitary representation. That is, a generic homomorphism into the unitary group of a separable infinite-dimensional Hilbert space equipped with the strong operator topology. This was implicitly present already in [14] (see Theorem 2.5), from which one can also derive the converse. The existence of such a countably infinite group seems to be open though. In [24] Rosendal proved that every finitely generated group with the Ribes-Zalesskiǐ property ( [22]), i.e. products of finitely generated subgroups are closed in the profinite topology of the group, has a generic action on the rational Urysohn space. More recently, Glasner, Kitroser and Melleray ([8]) characterized those countable groups that have generic permutation representations, i.e. generic homomorphisms into S ∞ , the full permutation group of the natural numbers. See also the results about generic representations in other metric structures [6]. We shall prove a genericity result for the universal action from Theorem 2.11. However, it turns out that the standard equivalence relation on the space of actions is too strong. Indeed, by Melleray in [18] (also independently proved in [6]), the conjugacy class of every action of a countably infinite group on the Urysohn space is meager, so not generic. Thus we shall weaken the equivalence relation of being conjugate by also allowing group automorphisms. Let us state that precisely in the following definition. Definition 3.1. Let G be a countable group and X a Polish metric space. Say that two homomorphisms f, g : G → Iso(X) are weakly equivalent if there exist an autoisometry φ : X → X and an automorphism ψ : G → G such that for all x ∈ X and v ∈ G we have f (v)x = φ −1 g(ψ(v))φx. Moreover, we say that an element f ∈ Act G (X) is weakly generic if it has a comeager equivalence class in the weak equivalence. We shall prove the following. Theorem 3.2. Let G be the Hall's universal locally finite group. The universal action α : G U from Theorem 2.11 is weakly generic. Theorem 3.2 has an interesting corollary which we state after the proof the theorem. We need some notions. When F and F ′ are two isomorphic finite groups, I and I ′ two finite bijective sets and d, resp. d ′ a metric on i∈I F · x i , resp. on i∈I ′ F ′ · y i , we denote by D((F, {x i : i ∈ I}, d), (F ′ , {y i : i ∈ I ′ }, d ′ )), analogously as in the previous section, the supremum distance sup i,j∈I,g,h∈F |d(g · x i , h · x j ) − d ′ (g ′ · y i ′ , h ′ · y j ′ )|, where g ′ , h ′ ∈ F ′ are the images of g, h ∈ F under the given isomorphism between F and F ′ and i ′ , j ′ ∈ I ′ are the images of i, j ∈ I under the given bijection between I and I ′ . Such an isomorphism and a bijection will be never explicitly mentioned, it should be always clear from the context. Also, we shall often write D((F, {x i : i ∈ I}), (F ′ , {y i : i ∈ I ′ })), thus suppressing the metrics from the notation; they should also be clear from the context. The following fact follows from Lemma 2.14, however we will state it here since it will be used extensively. Fact 3.3. Suppose we are given two finite isomorphic groups F and F ′ , finite bijective sets I and I ′ , and metrics d and d ′ on i∈I F · x i , resp. on i∈I ′ F ′ · y i . Suppose moreover that D((F, {x i : i ∈ I}), (F ′ , {y i : i ∈ I ′ })) < ε for some ε > 0. Then there exists a metric ρ on i∈I∪I ′ F · x i such that • D((F, {x i : i ∈ I}, d), (F, {x i : i ∈ I}, ρ)) = 0, i.e. ρ extends d; • D((F, {x i : i ∈ I ′ }, ρ), (F ′ , {y i : i ∈ I ′ }, d ′ )) = 0; • for every i ∈ I we have ρ(x i , x i ′ ) ≤ ε. Remark 3.4. Conversely, suppose that a finite group F acts freely on some metric space Y and let {y i : i ∈ I} and {z i : i ∈ I} be two finite subsets of Y indexed by the same set such that for every i ∈ I, d Y (y i , z i ) < ε. Then D((F, {y i : i ∈ I}), (F, {z i : i ∈ I})) < 2ε. We shall now define a subset of Act G (U). By QU we denote the rational Urysohn space, a countable dense subset of U. We shall denote the pointed free rational actions of finite groups by (F, {x i :∈ I}) and we write (F, {x i : i ∈ I}) ≤ (H, {x i : i ∈ I ′ }) to denote that the former actions embeds into the latter. To simplify the notation, we always assume in such a case that F ≤ H and I ⊆ I ′ . Recall that the class K of all pointed free rational actions by finite groups is countable. By D we denote the subset of Act G (U) of all actions G U satisfying: for all ε > ε ′ > 0, for all (F, {x i : i ∈ I}) ≤ (H, {x i : i ∈ I ′ }) ∈ K and for every subgroup F ′ ≤ G isomorphic to F and all {u i : i ∈ I} ⊆ QU such that D((F, {x i : i ∈ I}), (F ′ , {u i : i ∈ I}) < ε ′ there exist a subgroup F ′ ≤ H ′ ≤ G isomorphic to H, and points {u i : i ∈ I ′ } ⊆ QU such that D((H, {x i : i ∈ I ′ }), (H ′ , {u i : i ∈ I ′ })) < ε. We shall refer to the property above as D-property. A simple computation shows that the D-property is a G δ condition, i.e. D is a G δ set. It is non-empty since the universal action α : G U from Theorem 2.11 clearly belongs to D. We claim that its weak equivalence class is dense. Indeed, fix an open neighborhood of some action β : G U which is given by finitely many group elements, finitely many elements from U and some ε > 0. Without loss of generality, we may assume that these group elements form a finite subgroup F ≤ G and these finitely many elements from U form a finite subset A ⊆ U invariant under the subaction of this finite group. We may view this subaction as a pointed free action on a finite pseudometric space (Y, p). Let τ : Y → A be the corresponding surjection (which is not injective unless the action of F on A is free). By Lemma 2.13, we may approximate this pseudometric by a rational metric r such that the action is still by isometries and D((F, Y, p), (F, Y, r)) < ε/2. This finite action F Y (or its equivalence class) on a rational metric space belongs to the Fraïssé class of all pointed free actions of finite groups on finite rational metric spaces. Therefore, there is a subaction F ′ X n of the universal action α that is equivalent to F Y . That is, F is isomorphic to F ′ , and between Y and X n there is an equivariant isometry. Moreover, by the density of QU in U and the extension property of QU we may realize Y as a subset of QU ⊆ U so that for every y ∈ Y we have d U (y, τ (y)) < ε/2, where this last inequality is possible because of our assumption D((F, Y, p), (F, Y, r)) < ε/2. Since Y and X n are isometric, by the homogeneity of U there exists an autoisometry ψ : U → U such that ψ(X n ) = Y . Conjugating the action α with ψ gives us an action α ′ where F ′ acts on Y in the same way as F acts on Y . Since F ′ and F are isomorphic, by the homogeneity of the Hall's group G there exists an automorphism φ : G → G with φ(F ) = F ′ . 'Shifting' the action α ′ by φ, gives us a weakly equivalent action α ′′ , i.e. α ′′ (g, x) = α ′ (φ(g), x). The action α ′′ is then easily checked to be in the given neighborhood of β. Now we need to show that any two actions from D are weakly equivalent. That is, for actions α, β ∈ D we need to find an automorphism φ : G → G and an autoisometry of ψ : U → U such that for all g ∈ G and x ∈ U we have α(g, x) = β(φ(g), ψ(x)). We now fix two actions α, β ∈ D and show that. Let (z n ) n∈N be some enumeration of QU such that for each i 0 ∈ N both sets {z i : i ≥ i 0 , i is odd} and {z i : i ≥ i 0 , i is even} are dense in U. Also, write G as an increasing union G 1 ≤ G 2 ≤ G 3 ≤ . . . of finite subgroups. By induction, we shall find for each n ∈ N: • an increasing sequence of finite groups H 1 ≤ . . . ≤ H n ≤ G and H ′ 1 ≤ . . . ≤ H ′ n such that for each i ≤ n, H i and H ′ i are isomorphic by some φ i and φ i ⊇ φ i−1 , and for every odd i ≤ n we have that G i ≤ H i , and for every even i ≤ n we have that G i ≤ H ′ i ; • for each i ≤ n, sequences (u j i ) n j=i ⊆ QU and (v j i ) n j=i ⊆ QU such that for each i ≤ n and i ≤ j < k ≤ n, d(u j i , u k i ) ≤ 1/2 j+1 and d(v j i , v k i ) ≤ 1/2 j+1 , -for every odd i ≤ n, u i i = z i , and for every even i ≤ n, v i i = z i ; • D((H n , {u n i : i ≤ n}), (H ′ n , {v n i : i ≤ n})) < 1/2 n+1 . Once the induction is finished, we have that G = n H n = n H ′ n , i.e φ = n φ n : G → G is an isomorphism. Also we have that for every i ≤ n the sequences (u n i ) n and (v n i ) n are Cauchy in QU, thus they have some limit u i ∈ U, resp. v i ∈ U. It follows from the inductive assumption that both {u i : i ∈ N} and {v i : i ∈ N} are dense in U and that the map sending u i to v i is an isometry which extends to an autoisometry ψ of U. By the limit argument we get that the actions α and β are weakly equivalent witnessed by φ and ψ. Thus we need to describe the inductive steps to finish the proof. The first and second step of the induction. Set H 1 = G 1 , u ′ 2 , {x 1 , x 2 }) ∈ K such that D((H ′ 2 , {v 2 1 , v 2 2 }), (H ′ 2 , {x 1 , x 2 }) < 1/4 . By the D-property of α, Fact 3.3 and also Remark 3.4 we can find H 1 ≤ H 2 ≤ G isomorphic to H ′ 2 (via some φ 2 extending φ 1 ) and u 2 1 , u 2 2 ∈ QU such that d(u 1 1 , u 2 1 ) < 1/2 and D((H 2 , {u 2 1 , u 2 2 }), (H ′ 2 , {v 2 1 , v 2 2 })) < 1/2. This finishes the second step of the induction. The general odd and even step of the induction. The general steps are treated analogously as the second step of the induction. So we only briefly show the general odd n-th step of the induction, i.e. n is now odd greater than 2. For i < n we set u n i = u n−1 i and we set u n n = z n . Let H n ≤ G be an arbitrary finite subgroup containing both H n−1 and G n . By Lemma 2.13 there exists (H n , {x 1 , . . . , x n }) ∈ K such that D((H n , {u n 1 , . . . , u n n }), (H n , {x 1 , . . . , x n }) < 1/2 n . By the D-property of β, Fact 3.3 and also Remark 3.4 we can find H ′ n−1 ≤ H ′ n ≤ G isomorphic to H n (via some φ n extending φ n−1 ) and v n 1 , . . . , v n n ∈ QU such that d(v n−1 i , v n i ) < 1/2 n−1 , for all i < n, and D((H n , {u n 1 , . . . , u n n }), (H ′ n , {v n 1 , . . . , v n n })) < 1/2 n−1 . That finishes the inductive construction and the whole proof of Theorem 3.2. In [17], Melleray and Tsankov ask whether there is a Polish group H and a countable abelian group G such that for a generic element α ∈ Hom(G, Iso(U)) the closure of α[G] in Iso(U) is topologically isomorphic to H. The most interesting case is when G = Z, however the results from [17] suggest that the choice of G is not important. That means, it seems likely that if the result holds for an unbounded countable abelian group G, then it holds for Z as well. We show that the Melleray-Tsankov's problem has an affirmative answer in the non-abelian case when G is the Hall's group. Proof. Let (φ, ψ) be a pair of a group automorphism and an autoisometry that witnesses that β and γ are weakly equivalent. Let N β , resp. N γ be the kernels of β, resp. γ. It is easy to see that φ[N γ ] = N β , therefore φ induces an automorphism φ ′ : H/N γ → H/N β . H/N γ , resp. H/N β are countable dense subgroups of γ[H], resp. of β[H]. We claim that φ ′ induces also a topological group isomorphism φ ′′ : γ[H] → β [H]. It suffices to show that φ ′ is a homeomorphism between H/N γ and H/N β with the topologies inherited from Iso(X). We show that it is continuous. Showing that the inverse is continuous is analogous. Fix some h ∈ H \ N γ and some neighborhood U of [φ(h)] N β which is given by some x 1 , . . . , x n ∈ X and ε > 0 (viewing the equivalence class [φ(h)] N β as an isometry of X). However, then the neighborhood V of h (or [h] Nγ ) given by ψ −1 (x 1 ), . . . , ψ −1 (x n ) and ε is such that φ ′ [V ] = U. That follows immediately from the definition of weak equivalence. The following is now an immediate consequence of Theorem 3.2 and Proposition 3.5. Corollary 3.6. Let G be the Hall's group. There exists a Polish group H such that for comeager many α ∈ Hom(G, Iso(U)) we have α[G] ∼ = H. Question 3.7. What is H from the corollary? Melleray and Tsankov pointed out to us that most likely H is going to be the full isometry group Iso(U). Open problems about universal actions on Banach spaces In the last section we discuss universal actions by isometries on Banach spaces. Recall that by the theorem of Mazur and Ulam (see e.g. Theorem 14.1 in [3]) every (onto) isometry on a Banach space is affine, that means it is a linear isometry plus translation. From that, one can derive that every action α : G X of some group G on a Banach space X by isometries is determined by an action α 0 : G X, which is by linear isometries, and by a cocycle map b : G → X which determines the corresponding translates. That is, for any g ∈ G and x ∈ X we have α(g)x = α 0 (g)x + b(g). Conversely, whenever we have an action α 0 : G X by linear isometries and a map b : G → X satisfying the so-called 'cocycle condition', i.e. for every g, h ∈ G, we have b(gh) = α 0 (g)b(h) + b(g), we can get an action of G on X by affine isometries. We refer the reader to Chapter 6 of [20] for more information. This implies that the natural notion of embedding between two actions of groups on Banach spaces involves a group monomorphism and an equivariant affine isometric embedding. That motivates the following question. Question 4.1. Does there exist an action of a countable locally finite group G on a separable Banach space X by affine isometries such that for every action of a locally finite countable group H on a separable Banach space Y by isometries there is a subgroup H ′ ≤ G, isomorphic to H, and an affine isometric embedding φ : Y ֒→ X that is, after identifying H and H ′ , H-equivariant? If one considers a less natural type of embeddings between actions of groups on Banach spaces by isometries, namely just equivariant isometric embeddings (thus treating Banach spaces just as metric spaces), the universality problem has a positive answer. Let F be the functor which sends a pointed metric space (X, 0) to its Lipschitz-free Banach space F (X). By functoriality, any autoisometry of X which preserves 0 extends to a linear autoisometry of F (X). However, even every autoisometry of X extends to an affine autoisometry of F (X). Indeed, let φ : X → X be some autoisometry. In what follows, we view X as a metric subspace of F (X), i.e. we view every point x ∈ X as a point in F (X) also. Then the map x → φ(x) − φ(0) from X to F (X) is an isometric embedding of X into F (X) which preserves 0, thus extends to a linear isometric embedding from F (X) into F (X). It is easy to check that it is actually onto. Composing it with the translation '+φ(0)' gives the affine autoisometry that extends φ. Moreover, it is easy to check that by the same method every action of a group G on a pointed metric space (X, 0) by isometries extends to the action of G on F (X) by affine isometries; i.e. the cocycle condition is satisfied. Thus from Theorem 2.11 we get the following corollary. We refer to Chapter 5 in [21] for information about the Holmes space. There exists an action of the Hall's group G on the Holmes space F (U), the Lipschitzfree Banach space over the Urysohn space, such that for any action of a countable locally finite group H on a separable Banach space Y by isometries there exists H ′ ≤ G isomorphic to H and an isometric embedding Y into F (U) which is H-equivariant, after identifying H and H ′ . Finally, one can consider actions of groups on Banach spaces by linear isometries; in other words, representations of groups in linear isometry groups of Banach spaces. Below we propose a possible way how to prove an analogue of Theorem 0.1 for such representations of locally finite groups by similar methods, again using Fraïssé theory. The proofs seem to be much more technical than in the case of plain metric spaces without algebraic structure. We suggest a class of actions of finite groups on finite-dimensional Banach spaces by isometries. We do not have a full proof of the amalgamation property for this class. However, provided it does exist it follows there is a universal representation of the Hall's group in the Gurarij space. Let F be a finite group and I a non-empty finite set. By F I we denote the finite set F × I = {x g,i : g ∈ F, i ∈ I}. Instead of x 0,i , where i ∈ I and 0 ∈ F is the group zero, we may just write x i . Consider now a finite-dimensional real vector space E F,I with F I as a basis. The canonical action of F on F I , where g · x f,i = x gf,i (or the permutation representation of F on F I ), extends to a linear action of F on E F,I (resp. the representation of F in GL(E F,I )). Now let W ⊆ E F,I be any finite subset satisfying: • 0 ∈ W ; if w ∈ W , then −w ∈ W ; • for every i = j ∈ I, x i − x j ∈ W ; • for every i ∈ I, g ∈ F , x g,i ∈ W ; • for any g ∈ F and w ∈ W , g · w ∈ W . A partial F -norm · ′ on W is a partial norm on the finite set W compatible with the action of F ; that is, a function satisfying • w ′ = 0 iff w = 0; (positivity); • αw ′ = |α| w ′ provided that w, αw ∈ W , for α ∈ R; (homogeneity) • w ′ ≤ n i=1 |α i | w i ′ , where w = n i=1 α i w i w, (w i ) n i=1 ⊆ W , (α i ) n i=1 ⊆ R; (triangle inequality) • w ′ = g · w ′ , for g ∈ F , w ∈ W . (compatibility with the action) Having · ′ we define a norm · on E F,I as the maximal extension of · ′ to the whole E F,I . That is, for any x ∈ E F,I we set x = min{ n j=1 |β j | w j ′ : x = n j=1 β j w j , (w j ) i≤n ⊆ W }. It follows by compactness (from the finite-dimensionality) that the minimum is indeed attained. Now it is straightforward to check that · is a norm that extends · ′ and that the action of F on E F,I with · is by linear isometries. Notice that there are several other equivalent ways how to define · using · ′ . For instance one can take the closed convex hull of the set {w/ w ′ : w ∈ W } and then consider the Minkowski functional of such a set. The resulting norm will be · . Another way is to consider the following set of functions F = {f : F I ∪ {0} → R : f (0) = 0, |f (w)| ≤ w ′ }, wherẽ f is the unique linear extension of f on E F,I . Then we have, for every x ∈ E F,I , x = sup f ∈F |f (x)|. We shall call such an action of F on such a finite-dimensional space finitely presented. If the partial norm · ′ is defined only on linear combinations of basis vectors with rational coefficients and it has a rational range, we shall call such a finitely presented action rational. Let us have finite-dimensional spaces E F,I and E H,J , where F, H are finite groups and I, J finite sets. Suppose there are embeddings φ : F ֒→ H and ψ : I ֒→ J. Then they naturally induce a linear embedding of E F,I into E H,I which is also, after identifying F and ψ[F ] ≤ H, F -equivariant. If E F,I , resp. E H,J are equipped with the finitely presented norm, invariant by the action, and the linear embedding given by φ and ψ is also isometric, we call such a pair (φ, ψ) an embedding between two finitely presented actions. If the answer is affirmative, the Fraïssé limit would be an action of the Hall's group G on a normed vector space E G,J with the corresponding extension property such that G, the completion of E G,J , is isometric to the Gurarij space. Similar arguments as in the proof of Theorem 2.11 show that it is a universal action within the class of actions of countable locally finite groups on separable Banach spaces by linear isometries. We conclude with few more open questions. Of special interest is whether one can amalgamate unitary representations. Question 4.5. Let G 1 and G 2 be two groups with a common subgroup G 0 . Let π 1 and π 2 be two unitary representations of G 1 , resp. G 2 , on Hilbert spaces H 1 , resp. H 2 with a common subspace H 0 , on which the restrictions π 1 ↾ G 0 and π 2 ↾ G 0 are equivalent. Does there exist an amalgam of these representations? Does this problem have a positive answer when the groups and spaces in question are finite, resp. finite-dimensional? Proposition 2 . 15 . 215The class of all pointed free actions of finite abelian groups on finite metric space does not have the amalgamation property. Question 2 . 16 . 216Does there exist an action of a countable torsion abelian group on a separable metric space by isometries which is universal for the class of all actions of torsion abelian groups on all separable metric spaces?2.3. Non-universality results. In this subsection we justify why we work with Definition 2.1 rather than with Definition 2.2, where a different notion of universality was defined.Theorem 2.17. Let G be a countably infinite group. Then there is no universal action of G in both the class of all actions of G on all separable metric spaces and the class of all actions of G on all separable Banach spaces. Proposition 3 . 5 . 35Let H be a countable group and X a Polish metric space. Suppose that β, γ ∈ Hom(H, Iso(X)) are weakly equivalent. Then the topological groups β[H] and γ[H] are topologically isomorphic. Question 4 . 3 . 43Does the class of all finitely presented rational actions have the amalgamation property? Question 4 . 4 . 44Does there exist a universal unitary representation of a universal locally finite group? On the residual finiteness of generalized free products of nilpotent groups. G Baumslag, Trans. Amer. Math. Soc. 106G. Baumslag, On the residual finiteness of generalized free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193-209 I , Ben Yaacov, arXiv:1203.4459Fraïssé limits of metric structures. math.LOI. Ben Yaacov, Fraïssé limits of metric structures, arXiv:1203.4459 [math.LO], 2012. Y Benyamini, J Lindenstrauss, Geometric nonlinear functional analysis. Providence, RIAmerican Mathematical Society1Y. Benyamini, J. Lindenstrauss, Geometric nonlinear functional analysis Vol. 1., American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000. M Dadarlat, E Guentner, Constructions Preserving Hilbert Space Uniform Embeddability of Discrete Group. 355M. Dadarlat, E. Guentner, Constructions Preserving Hilbert Space Uniform Embeddability of Discrete Group, Trans. Amer. Math. Soc. 355 (8), 2003, 3253-3275. Metrically universal abelian groups. M Doucha, Trans. Amer. Math. Soc. 3698M. Doucha, Metrically universal abelian groups, Trans. Amer. Math. Soc. 369 (8), 2017, 5981-5998. M Doucha, M Malicki, arXiv:1710.08170Generic representations of countable groups. math.GRM. Doucha, M. Malicki, Generic representations of countable groups, arXiv:1710.08170 [math.GR], 2017. M Doucha, M Malicki, A Valette, arXiv:1711.04584Property (T), finite-dimensional representations, and generic representations. math.GRM. Doucha, M. Malicki, A. Valette, Property (T), finite-dimensional representations, and generic representations, arXiv:1711.04584 [math.GR], 2017. From isolated subgroups to generic permutation representations. Y Glasner, D Kitroser, J Melleray, J. London Math. Soc. 943Y. Glasner, D. Kitroser, J. Melleray, From isolated subgroups to generic permutation representations, J. London Math. Soc. (2016) 94 (3): 688-708. Lipschitz-free Banach spaces. G Godefroy, N J Kalton, Studia Math. 159Dedicated to Professor Aleksander Pe lczyński on the occasion of his 70th birthdayG. Godefroy and N. J. Kalton, Lipschitz-free Banach spaces, Studia Math., 159 (2003), pp. 121-141. Dedicated to Professor Aleksander Pe lczyński on the occasion of his 70th birthday. Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces. V I Gurarij, Sibirsk. Mat. Zh. 7in RussianV.I. Gurarij, Spaces of universal placement, isotropic spaces and a problem of Mazur on rotations of Banach spaces, Sibirsk. Mat. Zh. 7 (1966) 1002-1013 (in Russian). Some constructions for locally finite groups. P Hall, J. London Math. Soc. 34P. Hall, Some constructions for locally finite groups, J. London Math. Soc. 34 1959, 305-319. A finitely generated group with an isomorphic proper factor group. G Higman, J. London Math. Soc. 26G. Higman, A finitely generated group with an isomorphic proper factor group, J. London Math. Soc. 26 (1951), 59-61. W Hodges, Model theory. Encyclopedia of Mathematics and its Applications. CambridgeCambridge University Press42W. Hodges, Model theory. Encyclopedia of Mathematics and its Applications, 42, Cambridge University Press, Cambridge, 1993. Turbulence, representations, and trace-preserving actions. D Kerr, H Li, M Pichot, Proc. Lond. Math. Soc. 3D. Kerr, H. Li, M. Pichot, Turbulence, representations, and trace-preserving actions, Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 459-484. Fraïssé sequences: category-theoretic approach to universal homogeneous structures. W Kubiś, Ann. Pure Appl. Logic. 16511W. Kubiś, Fraïssé sequences: category-theoretic approach to universal homogeneous structures, Ann. Pure Appl. Logic 165 (2014), no. 11, 1755-1811. . R C Lyndon, P E Schupp, Springer-VerlagBerlin-Heidleberg-New YorkCombinatorial Group TheoryR. C. Lyndon, P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin-Heidleberg-New York, 1977. Generic representations of abelian groups and extreme amenability. J Melleray, T Tsankov, Israel J. Math. 1981J. Melleray, T. Tsankov, Generic representations of abelian groups and extreme amenability, Israel J. Math. 198 (2013), no. 1, 129-167. Polish groups and Baire category methods. J Melleray, Confluentes Math. 81J. Melleray, Polish groups and Baire category methods, Confluentes Math. 8 (2016), no. 1, 89-164. Permutational products of groups. B H Neumann, J. Austral. Math. Soc. 1B. H. Neumann, Permutational products of groups, J. Austral. Math. Soc. 1 1959/1960, 299-310. Large scale geometry. P Nowak, G Yu, EMS Textbooks in Mathematics. European Mathematical Society (EMS). P. Nowak, G. Yu, Large scale geometry, EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2012. Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon. V Pestov, University Lecture Series. 40American Mathematical SocietyV. Pestov, Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon. University Lecture Series, 40. American Mathematical Society, Providence, RI, 2006. On the profinite topology on a free group. L Ribes, P A Zalesskiǐ, Bull. London Math. Soc. 25L. Ribes, P. A. Zalesskiǐ, On the profinite topology on a free group, Bull. London Math. Soc., vol. 25 (1993), 37-43. Finitely approximable groups and actions part I: The Ribes-Zalesskiǐ property. C , J. Symbolic Logic. 764C. Rosendal, Finitely approximable groups and actions part I: The Ribes-Zalesskiǐ property, J. Symbolic Logic 76 (2011), no. 4, 1297-1306. Finitely approximable groups and actions part II: Generic representations. C , J. Symbolic Logic. 764C. Rosendal, Finitely approximable groups and actions part II: Generic representations, J. Symbolic Logic 76 (2011), no. 4, 1307-1321. Sur un espace métrique universel. P S Urysohn, Bull. Sci. Math. 51P. S. Urysohn, Sur un espace métrique universel, Bull. Sci. Math. 51 (1927), 43-64, 74-96. N Weaver, Lipschitz algebras. River Edge, NJ; Prague, Czech republic E-mail addressWorld Scientific Publishing Co., IncN. Weaver, Lipschitz algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. Institute of Mathematics, Academy of Sciences, Prague, Czech republic E-mail address: [email protected]

Magnetotactic bacteria (MTB) are fascinating micro-organisms which possess embodied biomineralized nanomagnets providing them the ability to orient with the Earth's magnetic field. This property is presumably related to an evolutionary advantage in finding the oxic-anoxic interface along the up and down direction in aquatic environments. So far the magnetic field response by MTB, called magnetotaxis, has been well described by a paramagnetic model where bacteria orient passively along the field lines according to a purely physical mechanism where magnetic torque and orientational Brownian noise compete. Here we demonstrate using Magnetococcus marinus strain MC-1 as MTB model that magnetotaxis shows more complex behaviors, which are affected by environmental conditions of different types. Indeed while MC-1 swimmers are found to essentially obey the paramagnetic paradigm when swimming in their growth medium, they exhibit a run-and-tumble dynamics in a medium devoid of energy source. Tumbling events are found to provide isotropic reorientation capabilities causing the cells to escape from their prescribed field direction. This behavior has a major influence on the capabilities of the cells to explore their environment across field lines and represents an alternative search strategy to the back-and-forth motion along field-imposed tracks. Moreover, we show that aside chemical conditions, steric/geometrical constraints are also able to trigger tumbling events through obstacle encountering. Overall, physico-chemical environmental conditions appear to be important parameters involved in the swimming properties of MTB. Depending on environmental conditions, the run-and-tumble mobility may provide advantages in the search for nutrient or ecological niche, in complement to classical magnetotaxis.

1603.00490

023778ff3f32ad1611cdf7770ac86c1c6c27e03d

Environment-dependent swimming strategy of Magnetococcus marinus under magnetic field 1 Mar 2016 Nicolas Waisbord Institut Lumière Matière UMR 5306 CNRS Université Claude Bernard Lyon1 Université de Lyon France Christopher T Lefèvre Biosciences and Biotechnologies Institute CNRS CEA/Aix-Marseille University 13108Saint Paul lez DuranceFrance Lydéric Bocquet Laboratoire de Physique Statistique UMR CNRS 8535 Ecole Normale Suprieure 75005ParisFrance Christophe Ybert Institut Lumière Matière UMR 5306 CNRS Université Claude Bernard Lyon1 Université de Lyon France Cécile Cottin-Bizonne Institut Lumière Matière UMR 5306 CNRS Université Claude Bernard Lyon1 Université de Lyon France Environment-dependent swimming strategy of Magnetococcus marinus under magnetic field 1 Mar 2016 Magnetotactic bacteria (MTB) are fascinating micro-organisms which possess embodied biomineralized nanomagnets providing them the ability to orient with the Earth's magnetic field. This property is presumably related to an evolutionary advantage in finding the oxic-anoxic interface along the up and down direction in aquatic environments. So far the magnetic field response by MTB, called magnetotaxis, has been well described by a paramagnetic model where bacteria orient passively along the field lines according to a purely physical mechanism where magnetic torque and orientational Brownian noise compete. Here we demonstrate using Magnetococcus marinus strain MC-1 as MTB model that magnetotaxis shows more complex behaviors, which are affected by environmental conditions of different types. Indeed while MC-1 swimmers are found to essentially obey the paramagnetic paradigm when swimming in their growth medium, they exhibit a run-and-tumble dynamics in a medium devoid of energy source. Tumbling events are found to provide isotropic reorientation capabilities causing the cells to escape from their prescribed field direction. This behavior has a major influence on the capabilities of the cells to explore their environment across field lines and represents an alternative search strategy to the back-and-forth motion along field-imposed tracks. Moreover, we show that aside chemical conditions, steric/geometrical constraints are also able to trigger tumbling events through obstacle encountering. Overall, physico-chemical environmental conditions appear to be important parameters involved in the swimming properties of MTB. Depending on environmental conditions, the run-and-tumble mobility may provide advantages in the search for nutrient or ecological niche, in complement to classical magnetotaxis. Many micro-organisms have the ability to move in their environment in order to respond to their needs, looking for instance for nutrients or for an optimum of oxygen concentration [1]. Among these micro-organisms magneto-aerotactic bacteria (MTB) are aquatic prokaryotes that show the specificity of a magnetically-assisted aerotaxis: they passively orient with magnetic field lines along which they actively swim, the so-called magnetoaerotaxis [2,3]. Combined with the fact that -except at the equator-magnetic field lines are indicative of the top to bottom location, this constitutes the basis of the magneto-aerotaxis paradigm. Indeed, by switching their motions from 3-dimensional to 1-dimensional (1D), MTB find more efficiently their ecological niche, that is anaerobic or micro-aerobic conditions [4]. Thus, in the Northern Hemisphere, the local magnetic field is antiparallel to the oxygen gradient and MTB are north-seeking while in the Southern Hemisphere, the local magnetic field is parallel to the oxygen gradient and MTB are south-seeking. The 1D motility of the cells is regulated back and forth to remain in the vicinity of the oxic-anoxic interface. Magneto-aerotaxis is a common feature shared by a large number of bacterial species disseminated all around the globe in almost each and every freshwater and marine environment [5]. MTB have in common the biomineralization of magnetosomes, a prokaryotic organelle, generally aligned in the cell and responsible for their magnetic orientation [3]. Our study is focused on the model magnetotactic bacterium Magnetococcus marinus strain MC-1 [6]. This bacterium has a polar magneto-aerotaxis: in a homogenous chemical environment it swims towards the magnetic north (or south) for north-seekers in the Northern Hemisphere (respectively south-seekers) [2,7]. Since the early studies on the motility of MTB, the response to magnetic field alone has been well described by a paramagnetic model. In this framework, the bacterium direction results from a competition between (i) a passive orientation of the bacterium along the magnetic field lines associated to the torque exerted by the field on the magnetic moment of the bacterium and (ii) thermal energy that tends to disorient the bacterium [8]. Within this description, MTB have no active control on their swimming orientation apart from back and forth switches. Magneto-aerotaxis could be seen as a limitation in the ability of MTB for optimizing their chemical environment and thus one may wonder if MTB can override this paramagnetic swimming behavior to let emerge other swimming strategies [9]. This interrogation is legitimated by the fact that i) MTB are found around the equator [10] where the field lines no longer provide a relevant direction for exploring the oxygen gradient and ii) south-seeking MTB are sometimes found in the Northen Hemisphere [11] suggesting that alternate search mechanisms may be possible. In this contribution we explore the behavior of strain MC-1 under various environmental and magnetic field conditions. Starting with a canonical experiment of chemical waves and bacterial aerotactic band, we first show that the field-mediated 1D motion strategy encompassed in the classical paramagnetic model cannot always account for the observed response of MC-1. FIG. 1. A: Set up of the chemical wave experiment. Up: initial stage, the glass capillary aligned on the axis of a Helmoltz Coils set-up and MC-1 bacteria are concentrated in a central "droplet" using focusing magnetic fields from each Coil. After a few minute, a front grows from the droplet and a propagating chemical wave develops along the capillary. Down: 1D propagation stage, the magnetic field is set homogeneous by reversing one of the coils' current and the capillary is either oriented perpendicular or along (not shown) the field direction. Remarkably, bacteria band keep moving driven by the chemical gradient even when the magnetic field is perpendicular to the propagation direction. B: Speed of the chemical wave band with the magnetic field intensity for: perpendicular field (blue •), and parallel field towards propagation (green ♦) or against it (red ×). C: Example of tumbling tracks in the propagating front region, showing distinct tumbling events. chemical environments, we evidence the existence of a swimming strategy alternative to the paramagnetic response. This strategy takes the form of a run-and-tumble swimming pattern by analogy to the well-known behavior exemplified by Escherichia coli and more generally by a large variety of chemotactic bacteria. This strategy, which can override the magnetically-set orientation, is found to be promoted by the chemical composition of the surrounding environment but also by the geometry of the surrounding environment through collisions with obstacles. Overall, our results provide a clear evidence that some MTB do not rely on the sole magnetically-assisted aerotaxis while swimming toward the North or South but are also capable of supplementary strategies that deflect from their prescribed field direction. I. CHEMICAL WAVES UNDER MAGNETIC FIELD We performed the benchmark aerotactic band experiment commonly used to characterize the magnetoaerotactic behavior of MTB (axial, (di)polar, or unipo-lar [7]). Unlike classical experiments however, where the magnetic field is pointing in the opposite direction of the oxygen gradients, we explored different field configurations. In the present experiments, MC-1 cells are introduced in a glass capillary filled with medium-rich solution (see Materials and Methods) that is subsequently sealed. Prior to introduction in the capillary, North seeking cells were selected using a magnet. Thus placing the capillary axis along the converging field direction obtained from Helmholtz coils leads to the formation of a high concentration of bacteria "droplet" at the center of the capillary (see figure 1 A). Few minutes after, a band of bacteria that spontaneously propagates at a velocity around 3µm/s is formed. Due to the high concentration of bacteria in the initial droplet and in the aerotactic band, chemical gradients develop and give rise to a propagation of the band (figure 1 A) as observed and described for other systems [2,12,13]. Once the band is formed, it is possible to rotate the capillary axis and to use the Helmholtz coils to see how the band propagation responds to a uniform magnetic field. Most strikingly, we have measured the band velocity for magnetic fields perpendicular to the propagation direction (the capillary axis). As shown in figure 1 B, the remarkable observation is that the front propagation (i) persists and (ii) exhibits a velocity which is independent of the field magnitude. This observation is different from what was previously observed for Magnetospirillum gryphiswaldense strain MSR-1 [13] where the tilt of 90 • of the magnetic field drastically reduced the magnetoaerotaxis efficiency. This difference likely comes from the different magnetotactic behaviors displayed by these two strains. Indeed, strain MSR-1 has an axial magnetotactic behavior where aerotaxis is prioritized over magnetotaxis, while strain MC-1 is a dipolar MTB where the cells give priority to following the direction indicated by the magnetic field. Very clearly, such a motion associated to magnetochemotaxis cannot fit into the existing framework of chemically-activated back and forth motion along the magnetic field lines. Note that few experiments have also been conducted where the field (i) was set homogeneous and parallel to the band propagation direction, or (ii) was stopped. In all cases, band propagation persisted with velocities not much affected. For instance figure 1 B shows the propagation speed along or against the magnetic field direction. While the field magnitude indeed either accelerates or decelerates the band, the fieldassited contribution never exceeds the propagation velocity found for perpendicular field. Somehow, such an observation is coherent with the fact that magnetotactic cocci related to strain MC-1 can be found in most aquatic environments on Earth, including at the equator, where the magnetic field is horizontal and thus perpendicular to the oxygen concentration gradient in water [10]. However, it requires extending the magneto-aerotactic paradigm to incorporate alternative swimming strategies which could overcome the constraint of the field direction. To collect clues of the underlying mechanism responsible for the cross field-direction propagation, figure 1 C provides a zoom on the front region, where a few individual bacteria trajectories have been superimposed in colors. As marked by the arrow symbols along trajectories, it is possible to distinguish very clearly events of dramatic reorientation -showing up as kinks-which allow the bacteria to swim across field lines. These events are reminiscent of a well-known chemotactic strategy of run-and-tumble performed by numerous bacteria among which E. coli. Such run-and-tumble motility was observed in Magnetospirillum magnetotacticum strain AMB-1 swimming in homogeneous chemical conditions [14]. In the following, we perform complementary experiments to verify this hypothesis of a run-and-tumble strategy which can take over classical magneto-aerotaxis depending on environmental conditions. II. CHEMICALLY-INDUCED TUMBLING OF MAGNETOCOCCUS MARINUS In order to get some insight into the MC-1 swimming strategy, and to characterize the proposed run-andtumble mechanism, we used dilute suspensions of bacteria introduced into a PDMS (polydomthylsiloxane) channel (see Materials and Methods) that had been filled with homogeneous medium, either denoted as rich (containing thiosulfate, the energy source used by MC-1) or poor (without thiosulfate). Note that this poor medium ap-pears closer to environmental conditions. Under these experimental configurations, single individuals dynamics could be followed under magnetic field, with no evolution of the properties over 20 min. A. No field We start by examining the swimming behavior in the absence of -added-magnetic field. Note that the earth magnetic field at our latitude is around 17µT, for which our cultured strains showed no biased directional motion. Figure 2 A shows a typical set of individuals tracks captured for MC-1 in the rich medium. Each track has been analyzed in search for tumbling events, here defined as locations along the tracks which combine high angular and low translational velocities ( [15] and Materials and Methods). The detected events are indicated as red positions on the experimental tracks. In the rich medium, most trajectories were found to be smooth with only a minority containing tumbling events as can be seen in Figure 2 A. On the contrary, when considering tracks recorded in the poor (thiosulfate-free) medium (figure 2 C), tumbling trajectories were found to be dominant with only a minority of remaining smooth trajectories. More quantitatively, figure 2 B gathers the overall percentage of trajectories showing tumbling events in the two different media. The amount of tumbling trajectories evolves markedly from 26±9% in a homogeneous rich medium up to 67 ± 9% in the poor medium (calculated respectively over 1000 and 1500 trajectories). This is consistent with our observation for chemical waves under perpendicularly oriented magnetic field, i.e. strong reorientation events -tumbling-can be promoted by the chemical environment. Concentrating on either smooth or tumbling trajectories alone, we have characterized the auto-correlation function of the swimming orientation. Figure 2 D shows the normalized experimental correlation functions together with their adjustment with an exponential decay cos θ(t) cos θ(t + τ ) = exp(−t/τ ). The effect of the tumbling events on the persistence time of the swimming orientation is evident, with decorrelation times of, respectively, τ = 3.6 ± 0.4 s for smooth trajectories and τ = 1.0 ± 0.2 s for tumbling trajectories. More precisely in the case of the so-called smooth trajectories, we expect the dynamics to correspond to persistent random walk [16,17]. Within this configuration, we expect the decorrelation time for orientation to read τ = τ R /2 with τ R the random noise orientation time scale of expression 8πηR 3 /kT , with η the surrounding viscosity, k the Boltzmann constant, T the temperature and R the bacterium radius. With a typical radius around 1 µm [6,18] this would predict a decorrelation around 3.0 s in perfect agreement with the measured value. Now turning to the tumbling trajectories, beyond the decrease of the correlation time already reported, it is possible to look at the distribution of running times, which separate each tumbling event. As shown in figure 2 E, the measured density probability function is found to be exponential, with a characteristic decay time of 0.6±0.2 s in agreement with the orientation decorrelation time. This is reminiscent of the characteristic Poisson distribution of the run times as found for E.coli [16], and further legitimates the suggested analogy with run-and-tumble trajectories. In some aspects, it echoes recent work on another magnetotactic bacterium, Magnetospirillum magneticum strain AMB-1, which showed that the response to the magnetic field of MTB can incorporate active pathways [19]. Here we do also identify an active response into the magnetoaerotactic bacteria dynamics; however not related to the magnetic field action but to the chemical environment. B. Under homogeneous magnetic field Under magnetic field, as shown in figure 3 A, cell's trajectories continue to exhibit two distinct behaviors: smooth and run-and-tumble. For the smooth swimmers, in figure 3 B we see the polar orientation diagram (inset) together with the probability distribution function (PDF) of orientations. Starting from an isotropic distribution without field, the swimming direction gets peaked around the field direction, with increasing directionality as the magnetic field increases. Under the classical description, we expect the bacteria orientations to be passively set by the magnetic torque exerted on magnetosomes, with the finite width of the distribution arising from the orientation noise attributed to rotational Brownian motion. Indeed the figure 3 B shows that the PDF of smooth swimmers is perfectly fitted by a Langevin paramagnetic prescription of the form PDF∝ exp([M B/kT ] cos θ). Moreover, the fitted distribution yields the value of the reduced energy M B/kT = 8 ± 1. Assuming that the only source of orientational noise comes from thermal Brownian motion, this provides a value of the magnetic momentum carried by each bacterium of M ≈ 1.7 ± 0.2 10 −16 S.I., consistent with direct measurements of the momentum (see Materials and Methods). Accordingly, the average axial velocity (along the field direction) is perfectly fitted by the prediction for paramagnetic orientating swimmers as can be seen in figure 3 D. Note that the magnetic field dependency was also checked and found to agree quantitatively with the paramagnetic approach [3]. Therefore, the smooth swimmers, which dominate the observed dynamics in rich medium, are corresponding to the previously reported response of MTB under field. It further ensures the coherence of our observations with previous reports [20]. Concerning the run-and-tumble dynamics, we first compare the velocity PDFs of smooth and tumbling swimmers in figure 3 C. As can be seen, the swimming activity appears unmodified in tumbling conditions, with a mean velocity essentially unchanged around 100 ± 25 µm/s and marginal changes in the distribution. Indeed, the tumbling PDF differs only by a little skew towards low velocities which is coherent with the fact that tumbling events are associated to locally high rotational velocities and low translational ones. An "active magnetotaxis" model was proposed for Magnetospirillum magneticum strain AMB-1, where the magnetotactic cells would sense the magnetic field thanks to a chemotaxis protein that would interact with the magnetosome chain and transfer the magnetic signal to the flagella [21]. We have measured the tumbling rate as a function of the magnetic field and have seen no dependency which leaves the question of the challenging model of "active magentotaxie" not consistent with our observation in strain MC-1. The ability of bacteria to escape the direction set by the external magnetic field can be quantified through the diffusion coefficient D ⊥ in the direction perpendicular to the imposed field. In practice, this is measured from the correlation function according to the Green-Kubo relationship: Figure 4 shows the cross-diffusivity for smooth and tumbling swimmers as a function of the external field, the two appearing markedly different. Under low fields, smooth swimmers explore unexpectedly more space around the magnetic field line than tumblers. Indeed the bacteria have an effective diffusion coefficient that is proportional to V 2 τ r where τ r is the rotational diffusion time [22]. Tumbling can be seen as a reduction in τ r which implies a reduction in cross-direction explored and thus in the effective diffusion coefficient D ⊥ (figure 4 B). However, the situation gets reversed at higher fields (figure 4 C) where tumbling swimmers become the most efficient at exploring the environment. By allowing initial conditions spanning the all directions, tumbling events allow excursions in the transverse direction which are statistically hardly possible for the smooth trajectories that get more and more confined along the field direction. The range of spatial exploration by the tumbling swimmers does not depend on the applied magnetic field (inset figure 4 A) Overall, we have evidenced that depending on the chemical environment, MC-1 can show very different swimming behaviors. When studied in the rich medium, bacteria display smooth trajectories oriented on average by the magnetic field. These trajectories are illustrative of persistent random walks in external fields, which combine (i) constant velocity swimming, (ii) random noise and (iii) external torques which compete for the swimmers' orientation. Indeed, the classical paradigm for passive magnetotaxis is well recovered under these circumstances, even quantitatively. However, as already pointed out, growth media used in the lab are generally much more concentrated in energy source than what bacteria usually found in their environment [23]. D ⊥ = ∞ 0 v ⊥ (t)v ⊥ (0) dt. While most studies that sought to decipher the magnetotactic behavior analize bacteria in their growth conditions [7], here we show that the use of a poor medium -arguably closer to environmental conditions-conduct to the switch into a run-and-tumble dominated swimming. This newly discovered strategy has a deep impact on the cross-direction diffusivity of bacteria and could thus be an advantage in poor conditions for efficiently finding the ecological niche. III. CONFINEMENT-INDUCED TUMBLING Besides the chemical composition of the environment surrounding the MTB, we show that other environmental constraints can induce changes in their swimming behavior under the form of tumbling events. These constraints are of steric or geometrical type as can be encountered in natural environment such as sediments. Experimentally, these effects have been probed in a model geometry made up from a straight microfluidic channel which posseses "rough" structured walls. Dilute MC-1 cells (volume fraction less than 0.1%) are suspended in the rich medium in order to minimize the intrinsic amount of tumbling. Again, north-seekers have been selected upon filling with a magnet, and we hereafter consider all swimmers as smooth swimmers. Bacteria are then submitted to a homogeneous magnetic field along the channel axis and their (swimming) motion inside the micro-channel is analyzed ( figure 5 A). Note that for comparison purposes, channel walls have been made asymmetric with one flat wall and one sawtooth textured wall. As evidenced in figure 5 B-C, a bacterium that collides with the wall will eventually reorient itself until it swims again away from the wall. This major reorientation in trajectories can again be assimilated to a tumbling event, now occuring in response to a mechanical constraint rather than a chemical one. Interestingly, recent studies have demonstrated close behaviors triggered by wall encounters. For instance Magnetospirillum magneticum strain AMB-1 has been shown to switch from north-seeking swim to south-seeking swim when meeting an interface [24] while cells of strain MO-1 (a magnetotactic cocci related to MC-1) was demonstrated to be able of making U-turns when encountering obstacles [25]. We now look at the magnetic field influence on walltumblers. We observe that, when the field increases, the magnetic reorientation -that follows the tumbling eventis more effective yielding to shorter re-injection length into the main bacteria stream (figure 5 B-C). Increasing the bacteria concentration up to ≈ 1%, we can run Particule Images Velocimetry algorithms to access the velocity profiles of bacteria within the microchannel. Figure 5 D shows the asymmetry of the velocity profiles obtained, which matches the asymmetry of the walls structuring. Indeed the effect of the walls is related to the previously called re-injection length -that is the average exploration length towards the channel center-that the bacteria are able to travel before being reoriented. Experimentally, the length is defined from velocity profiles by fitting them with an exponential relaxation and defining the influence length L as the decay length (figure 5 E). In Figure 5 F, we represent the evolution of the influence length L associated to sawtooth-shaped walls on the velocity profile of the suspension. Starting from a finite length at vanishing magnetic field, this influence length decreases with increasing field. Plotting LB rather than L reveals the existence of a plateau value at high fields, thus suggesting that the influence length decreases as 1/B in this limit. Concentrating first on the small field limit, once a bacterium has tumbled at the wall and is re-injected into the main stream, the wall influence is bounded by the persistence orientation time in dilute- regime and so is the influence length L ∝ V 0 τ R . In the high field limit, the relaxation time towards the field direction becomes shorter than the Brownian persistent time and thus dominates the extent to which wall influence propagates. The ratio between both times writes τ R τ B = 8πηR 3 k B .T 8πηR 3 M.B −1 = M k B .T × B.(1) According to the orientation distribution (Fig. 3), this ratio is around 4 for B = 0.1 mT, for which we do observe the plateau in LB (figure 5 F). This confirms the domination of the magnetic alignment relaxation time in this regime, for which we then expect the L ∝ 1/B dependency according to L ∝ V 0 τ B = V 0 8πηR 3 M.B .(2) Overall, flow profiles close to sawtooth-shaped walls suggest that the effect of an obstacle on bacteria dynamics can be summed up by a wall tumbling followed by an orientation relaxation over a length scale either dominated by a persistent random walk exploration (low field) or a deterministic trajectory under magnetic field. In the environment, obstacles are many and the probability to meet a particle is high particularly in sediment where MC-1 and other MTB are generally found. Such tumbling behavior allows the cells to avoid obstacles and to continue swimming toward their preferred oxygen concentration. IV. CONCLUSION. Our results show that the motility of strain MC-1 is affected by the environment. The magneto-aerotaxis model is not just a passive alignment of the cells while they are swimming. Indeed, whereas the cells are constrained to swim in one direction due to the magnetic forces they are still able to sense their environment and to swim in other directions if the chemotactic machinery of the cells detects the need to do so. In a rich medium bacteria exhibit a smooth motility perfectly described by a Langevin paramagnetic model. In a poor medium, closer to environmental conditions, we show that MC-1 cells can exhibit a run-and-tumble motion outstripping the simple paramagnetic behavior. By changing their direction of motility in poor media bacteria increase the chance to meet their energy source while swimming around the oxic-anoxic interface. Furthermore we show that that tumbling can also occur under geometrical constrains as encountered in crowded environments of sediments for instance. Finally, in a wider perspective, let us note that when under the conditions for a paramagnetic swimming response, magnetotactic bacteria form a remarkable active system of extremely efficient self-propeled individuals whose orientation can be quantitatively controlled by a physical magnetic field [26]. In the blooming context of biological active matter or in the perspective of drivable in-body micro-swimmers for medical applications, this system forms a very appealing model of controllable active system. V. MATERIALS Bacterial strain. Cells of Magnetococcus marinus strain MC-1 were grown in a slightly modified semi-solid medium described in [6].The medium consisted of an artificial seawater (ASW) containing (per litre): NaCl, 16.43 g; MgCl 2 .6H 2 O, 3.49 g; Na 2 SO 4 , 2.74 g; KCl, 0.465 g; and CaCl 2 .2H 2 O, 0.386 g. To this was added (per litre) the following prior to autoclaving: 5 mL modified Wolfe's mineral elixir [2], 0.25 g NH 4 Cl, 2.4 g HEPES, 100 µL0.2% (w/v) aqueous resazurin and 2.0 g agar noble (Difco). The medium was then adjusted to pH 7.0, boiled to dissolve the agar and autoclaved. After the medium had cooled to about 45C, the following solutions were added (per liter) from stock solutions (except for the cysteine, which was made fresh and filter-sterilized directly into the medium): 2.8 mL 0.5 M potassium phosphate buffer, pH 6.9; L-cysteine, to give a final concentration of 0.4 g.L −1 ; 0.5 mL vitamin solution [2]; 3 mL 0.01 M FeSO 4 dissolved in 0.2 M HCl; 3 mL 40% sodium thiosulfate pentahydrate solution (i.e., 10mM of energy source); and 2.7 mL 0.8 M NaHCO 3 (autoclaved dry; sterile water added after autoclaving to make the fresh stock solution). The medium (10 mL) was dispensed into sterile, 15 x 125 mm screw-capped test tubes. All cultures were incubated at 28 • C and, after approximately 1 week, a microaerobic band of bacteria formed at the oxic-anoxic interface (pink-colorless interface) of the tubes. This band contains a majority of north-seeking cells; the rare southseeking cells were eliminated via a magnetic separation that consists in transferring the cells from their semi-solid medium to a liquid "swimming medium". Swimming media. The medium referred as rich medium consisted of ASW to which was added (per litre) the following: 5 mL modified Wolfe's mineral elixir, 0.25 g NH 4 Cl, 2.4 g HEPES. The medium referred as poor medium consisted of the same medium as the rich one devoid of thiosulfate. Both media have a pH adjusted to 7. Microfabrication. Microchannels were fabricated by standard "soft lithography" technique: a mold made of negative photoresist (SU8 3100 Microchem) is obtained by conventional photolithography. Channels are then made of PDMS poured on these molds, at 70 • C for 3 hours, with a typical thickness of 1mm. After cross-linking, the PDMS imprint is pealed of its mold. The channels are cut, the entrance and the outlet are punched, and are finally bonded with a glass slide after an air plasma treatment. The obtained channels are then filled with swimming media by capillarity. The outlet is then sealed with capillary wax. A bacteria drop is then taken from its growing medium and then guided inside the channels with a magnet. Once bacteria are inside, the entrance is sealed with the wax. Coils control. Coils are controlled directly from the computer, with a Labjack U12 device and an homemade current source. They can be set in different configurations, generating Magnetic field parallel or anti-parallel. We use the anti-parallel configuration to concentrate bacteria, and at the same time sort the south seekers from the north seekers. The parallel configuration allows fields ranging from 0 to 2.8mT, considered homogeneous in the observation zone of 1.2 mm. Alignment of channels with the coils is made as follows: coils are designed with a rectangular hole where a Nikon Mir Glass Slide fits exactly: the field and the line of the mir are perpendicular. The camera observation is then aligned on the mir. Then the micro channel is aligned with a ROI crop selection on the screen on the acquisition system. Preparation of the experiment. We used linear silicone polydimethylsiloxane (PDMS) micro-channels (1200µm × 70µm × 5mm). The channels were filled with the swimming medium by capillary suction, then sealed on one side. A drop of a culture of strain MC-1 was disposed at the extremity of the channel and steered inside the micro-channel with a simple stirring magnet. Once the bacteria inside the channel, this latter is sealed on the former entrance. Using a computer controlled Helmholtz coils system we control the magnetic field inside the channel. We concentrate the north seeking bacteria inside the channel by using a configuration where coils generate fields that face one another and then we set the magnetic field to zero. Microscopy and data acquisition. To acquire images we use a Leica DMI 4000B Microscope in transmitted light with a Zeiss 5X long working distance objective, equipped with a Baumer HXC40 4Mpx CCD camera controlled by custom Labview codes, covering a 1200µm× 1200 µm area. We run both rheology and single particle experiments at 35 fps. The long-working distance objective allows a depth of field of ≈100 µm, making possible to follow bacteria and acquire the projection of their trajectories. In single particle experiment, we assume that the problem is axially symmetric, which allows several calculus tricks for the relationships between projected observables and 3D ones. Tracking and trajectory analysis. Image treatment and tracking was performed using standard tracking routine written with Matlab software (Mathworks). The selection of the trajectories was based on duration (¿1.5 s) exploration radius (¿ 50µm) and average instant speed (¿ 40µm.s −1 ). To avoid taking into account the Helical Motion of the bacteria, we smoothed the positions by a gaussian filter on a width of 4 positions, so that the projected orientation of motion we consider in this study is the one of the main component of the motion. The projected observed speed is assumed to be the real one. The Transverse projected MSD is assumed to be half of the real one. The correlation function of the projected motion is assumed to be the same as the one in 3D. Finally, the projection of the Orientation Distribution function are fitted by the I1(ξ.cos(α))+L−1(ξ.cos(α)) 4ξ −1 sinh(ξ) , which is the statistics of the orientation expected of the Langevin Distribution, projected in the focal plane. Motivated by the large persistence lengths of the trajectories (over 200µm for smooth swimmers), we computed D ⊥ (B), the diffusion coefficient perpendicular to the magnetic field, from the Green-Kubo formula, which reads: 2D ⊥ (B) = ∞ 0 V ⊥ (τ )V ⊥ (τ + t) τ dt.(3) We obtain D ⊥ (B) from the correlation functions of the speed in the transverse directions, which necessitates to follow bacteria during a shorter period of time and can be fitted with exponential functions both for the tumblers and the smooth swimmers. Tumble detection. Inspired from the method of [15], we define a function which combines the variation of the speed of the bacteria and their rotational acceleration, both characteristic of tumbling event where a bacteria stops and turns: f α,β (t) = |ω(t)| α |V (t)| β For a smooth swimmer, this quantity cannot increase a lot more than 4 times its median value computed along a trajectory, as the disorientation is well described by a White Gaussian Noise. For a tumbler, the median reflects its smooth swimming behavior as, for a clear majority of its time it is indeed smooth swimming, and the tumbling events are not visible in the computation of the median. These aredetected, and we compare it to 10 times the median of f α,β (t) for each trajectory. This threshold and the exponents (α, β) are adjusted and for our data, a threshold of 10 and (α, β) = (2, 2) are satisfactory. PIV analysis. PIV analysis was performed using the PIVlab OpenSource Matlab toolbox. Images had their background subtracted before PIV was run. The average velocity was done over 400 images, along 15 periods of the waved wall. We have also shown, in a force free rhoelogy experiment, that tumbling at the wall can be induced by roughness which modifies the shape of the velocity profile. Paramagnetic behavior in rich medium. An important physical parameter to estimate for a magnetic bacterium is its magnetic moment. For this we have car-ried out an experiment where non swimming bacteria are submitted to a gradient of magnetic field: living bacteria are suspended in a liquid rich medium with 0.5% BSA and buffered at pH 10. This high pH kills the bacteria while preserving the cell integrity and the BSA prevents them from sticking to the glass capillary in which the suspension is sucked. The capillary is then disposed in the middle of two facing magnets (Supermagnets W-03-N Gottmadingen, Germany) creating a magnetic gradient of ∇ B = 80T.m −1 . Bacteria are dragged across the channel under the magnetic force created by such a gradient of field and the magnetic momentum is estimated by measuring the average velocities of the bacteria: M = 6πηR.V drag ∇ B . We measure a magnetic momentum M = 1.3 × 10 −16 A.m 2 with a standard error 0.5 × 10 −16 A.m 2 . This value is obtained by an overage over 500 trajectories of bacteria coming from three different batches and it is within the range of magnetic momentum obtained for other magnetotactic bacteria [20]. in rich media without magnetic field: few tumbles (in red) are observed B: Percentage of tumblers for rich and poor media. C: Tracks in poor media without magnetic field: more tumbles (in red) are observed. D: Correlation function of the orientation for tumblers (red ) and Smooth Swimmers (blue •) in the poor media without magnetic field. E: log of the probability density function of the running time between the detected Tumbles. The fit gives a running time of 0.6 ± 0.2 seconds under magnetic field magnetic field (poor medium). B: PDF p(θ) of the swimmers orientation under field (B = 0.21 mT) for smooth (blue •) and tumbling (red ) swimmers. Inset: polar diagram for smooth swimmers fitted by paramagnetic Langevin model. The fit for the smooth swimmers by p(θ) ∝ exp(M Bcos(θ)/kT ) gives M = 1.7 × 10 −16 A.m 2 C: Same as in B, for the PDF of velocity magnitude; dashed lines corresponds to the mean velocities (respectively 91 ± 25 µm/s and 104 µm/s). D: Normalized averaged axial velocity as a function of the magnetic field for smooth (blue •) and tumbling (red ) swimmers. The blue line is the fit of this orientation of the smooth swimmers by the Langevin function coth M B kT − kT M B . diffusion coefficient for smooth (blue •) and tumbling (red ) swimmers as a function of the magnetic field. B: Same as in A for correlation function of perpendicular velocities under no field. C: Same as in B under 0.7mT magnetic field. FIG. 5 . 5A: Force-free rheological experiment: bacteria in structured micro-channels. B: Bacteria tumbles upon wall-collision, followed by trajectories during orientation relaxation at 140 µT. C: Same as B for a field of 490 µT. D: Velocity Profile averaged alongside the channel (from tooth extremities up to flat saturated velocity); Solid line correspond to an exponential profile A exp(−x/L), thus yielding the wall influence length L. E Velocity profiles for various magnetic fields. F: Variations of L under various magnetic field B. Inset: LB vs B. Turning to model situations of dilute bacteria in homogeneous arXiv:1603.00490v1 [cond-mat.soft] 1 Mar 20160s 30s 60s 0.5 mm A 90°I I I I B Speed (μm/s) B (mT) V V V V V V V C 300 microns B B B 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 ACKNOWLEDGMENTSWe thank Eleonora Secchi for her help with the data analysis. We thank Veroniquie Utzinger, Guy Condemine and Nicole Cotte-Pattat for their help with the bacterial cultures. We thank INL for access to their clean room facilities and AXA research fund for its financial support. C.T.L. was supported by the French National Research Agency (GROMA: ANR-14-CE35-0018-01) Making sense of it all: bacterial chemotaxis. 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We have developed a frequentist approach for model selection which determines the consistency between any cosmological model and the data using the distribution of likelihoods from the iterative smoothing method. Using this approach, we have shown how confidently we can conclude whether the data support any given model without comparison to a different one. In this current work, we compare our approach with the conventional Bayesian approach based on the estimation of the Bayesian evidence using nested sampling. We use simulated future Roman (formerly WFIRST)-like type Ia supernovae data in our analysis. We discuss the limits of the Bayesian approach for model selection and show how our proposed frequentist approach can perform better in the falsification of individual models. Namely, if the true model is among the candidates being tested in the Bayesian approach, that approach can select the correct model. If all of the options are false, then the Bayesian approach will select merely the least incorrect one. Our approach is designed for such a case and we can conclude that all of the models are false.

10.1088/1475-7516/2022/03/047

2110.10977

2c37a3a393617aa752a2119cba7996e1b2329b23

Bayesian vs Frequentist: Comparing Bayesian model selection with a frequentist approach using the iterative smoothing method 21 Oct 2021 Hanwool Koo [email protected] Korea Astronomy and Space Science Institute (KASI) 776 Daedeok-daero, Yuseong-gu34055DaejeonKorea KASI Campus University of Science and Technology 217 Gajeong-ro, Yuseong-gu34113DaejeonKorea Ryan E Keeley [email protected] Korea Astronomy and Space Science Institute (KASI) 776 Daedeok-daero, Yuseong-gu34055DaejeonKorea Department of Physics University of California Merced 5200 North Lake Road95343MercedCAUSA Arman Shafieloo [email protected] Korea Astronomy and Space Science Institute (KASI) 776 Daedeok-daero, Yuseong-gu34055DaejeonKorea KASI Campus University of Science and Technology 217 Gajeong-ro, Yuseong-gu34113DaejeonKorea Benjamin L&apos;huillier [email protected] Department of Physics and Astronomy Sejong University 209 Neungdong-ro, Gwangjin-gu05006SeoulKorea Bayesian vs Frequentist: Comparing Bayesian model selection with a frequentist approach using the iterative smoothing method 21 Oct 2021Prepared for submission to JCAP We have developed a frequentist approach for model selection which determines the consistency between any cosmological model and the data using the distribution of likelihoods from the iterative smoothing method. Using this approach, we have shown how confidently we can conclude whether the data support any given model without comparison to a different one. In this current work, we compare our approach with the conventional Bayesian approach based on the estimation of the Bayesian evidence using nested sampling. We use simulated future Roman (formerly WFIRST)-like type Ia supernovae data in our analysis. We discuss the limits of the Bayesian approach for model selection and show how our proposed frequentist approach can perform better in the falsification of individual models. Namely, if the true model is among the candidates being tested in the Bayesian approach, that approach can select the correct model. If all of the options are false, then the Bayesian approach will select merely the least incorrect one. Our approach is designed for such a case and we can conclude that all of the models are false. Introduction The concordance model of cosmology, ΛCDM (Λ for the cosmological constant and CDM for the cold dark matter), is facing conflicts and tensions. It has been the most successful model that explained various astronomical observations with remarkable simplicity and no significant change for decades. Though ΛCDM is consistent with low and high-redshift observations individually, it is in conflict with the combination of low and high-redshift data. This is the case with the H 0 tension, a discrepancy between the present expansion rate measured directly from the Cepheid calibration of Type Ia supernova (SN Ia) distances [1] and that rate derived from the cosmic microwave background (CMB) [2]. Using a model independent analysis to reconstruct the expansion history of the universe can be a useful approach to shed light on the current tensions, but the large gap in the data between the low (such as SN Ia and baryon acoustic oscillation) and high redshift observations (CMB) prevents us from having a complete reconstruction of the expansion history all way to the last scattering surface. Accepting our limitations, it is reasonable to look for alternative models that may perform better than the standard model at fitting combinations of cosmological observations, as well as falsifying such models in a robust manner. In our previous paper we introduced a frequentist approach [3] to test the consistency between any individual cosmological model and the SN Ia observations which are one of the most reliable cosmological datasets. This method is based on the non-parametric iterative smoothing method, introduced and improved by [4][5][6][7], which reconstructs the distance modulus in a model-independent way. Our previous paper provides a detailed description about our method, about how to generate and use the likelihood distributions based on the data covariance matrix. We also showed that our method can also be used to perform parameter estimation for each model. In this work we elaborate further on the important subject of model selection and we compare the performance of our proposed method with that of the conventional approach using Bayesian statistics, namely the Bayesian evidence ratio. To perform this analysis, we simulated a mock Nancy Grace Roman Space Telescope [Roman, formerly WFIRST, 8] SN Ia dataset. We will show that, using our approach, not only can we successfully distinguish and make a preference among different alternative models but we can also rule out all alternative candidates if none is the correct model. This is a great advantage over the conventional Bayesian evidence approach which, in the scenario where none of the candidate models is the true one, will still select one of the false models as best and thus not rule it out. We concretely demonstrate these points by considering three different dark energy models. We imagine a case where the true model, the model which generated the data, is an unknown unknown; it is not in the list of models that are considered possibilities. Rather only two false models are considered. We intend to demonstrate that the standard Bayesian model selection approach will be able to tell which of the two models is the preferred or the less bad fit, but will not be able to identify that neither models are true. In turn, we intend to show that our methodology can identify that both false models are indeed false. We introduce the methodology of using the Bayesian evidence for model selection in Section 2, and of using the iterative smoothing method and likelihood distributions in Section 3. We present our results in Section 4 and summary in Section 5. Model selection using Bayesian evidence In this section, we discuss the common method for comparing models by estimating the Bayesian evidence. The Bayesian evidence Z, also called marginal likelihood, is defined as the integral of the product of the likelihood L(θ) and the prior π(θ) taken over the entire parameter space of θ: Z = L(θ)π(θ) dθ. (2.1) The Bayesian evidence is the average of the likelihood of the data over the parameter space. One notable feature of the Bayesian evidence is that it penalizes extended parameter spaces that do not fit the data well. That is, the parameter space that is within the prior but not within the peak of likelihood will decrease the evidence. In other words, the larger the fraction of the parameter space that is occupied by the peak of the likelihood the larger is the evidence that results. Thus when comparing two models' evidences, the evidence gives advantage to the model with less wasted parameter space i.e. the model that is more predictive. The Bayes factor, which is the ratio of the evidence of the two different models, is one of the most reliable model selection manner in Bayesian statistics. If the Bayes factor is larger than unity, or equivalently, if the difference in the log of the evidences of Model 1 (M 2 ) and Model 2 (M 1 ) ∆ log Z = log Z(M 1 ) − log Z(M 2 ) (2.2) is positive, M 1 is supported by the data over M 2 . We can measure the strength of the preference of M 2 over M 1 using the interpretation of ∆ log Z shown in Table 1. This scale, the Kass-Raftery (KR) scale was suggested by [9] and is a more conservative modification of the widely used Jeffreys' scale [10]. 1 Based on this interpretation, we forecast how frequently the analysis using Bayesian evidence will strongly support M 1 when we simulate future SN Ia datasets and calculate the distribution of ∆ log Z. We use mc3 [12] and dynesty [13], which implements a dynamic nested sampling [14] algorithm, to calculate the Bayesian evidence. Nested sampling [15,16] estimates the Bayesian evidence by transforming the integral over the N-dimensional parameter space into a onedimensional integral over the prior volume that is contained within an iso-likelihood surface. Numerically, the evidence is the weighted sum of likelihood values; the weights are determined ∆ log Z Evidence against M 1 0 to 1 Negligible 1 to 3 Positive 3 to 5 Strong > 5 Very strong Model selection using the iterative smoothing method and likelihood distributions In this section, we discuss our frequentist approach for model selection, which uses the likelihood distribution from the iterative smoothing method. Our iterative smoothing technique [4][5][6][7] seeks to reconstruct a smooth function of the distance modulus µ(z) from the data µ i observed at redshifts z i . This method starts from an arbitrary initial guess,μ 0 (z) and perturbs this guess such that the residuals look smoother and more Gaussian. This procedure works iteratively such that (n + 1)th iteration,μ n+1 (z), is calculated bŷ µ n+1 (z) =μ n (z) + δµ n T · C −1 · W (z) 1 T · C −1 · W (z) ,(3.1) where 1 T = (1, · · · , 1), the weight W and residual δµ n are W i (z) = exp   − ln 2 1+z 1+z i 2∆ 2   (3.2) δµ n | i = µ i −μ n (z i ) (3.3) and C −1 indicates the inverse of covariance matrix of the data. The smoothing width is set to ∆ = 0.3 following previous analyses in [4,[17][18][19]. We define the χ 2 value of the reconstruction µ n (z) as χ 2 n = δµ n T · C −1 · δµ n . (3.4) By design, the iterative smoothing method reconstructs a function which at any iteration fits the data better than at the previous iteration. Furthermore, after a large number of iterations the reconstructions converge to a unique solution independent of the choice of the initial guess. In other words, when starting from different initial guesses, the likelihood of the first iterations may be very different, but after a large number of iterations, the final reconstructions converge to the same solution with a unique likelihood. We use the 1000th iteration of the iterative smoothing method, which is large enough to achieve this convergence (that generally occurs after a few hundreds of iterations) [5,7,18,19]. In the previous work [3], we have applied the smoothing procedure to derive the distribution of the difference in χ 2 between the smoothed function and that of the best-fit of the model being tested (∆χ 2 = χ 2 smooth − χ 2 best−fit ). We call this distribution the likelihood distribution, which follows a frequentist statistical approach and shows how frequently the procedure generates better fits by certain amount of improvement. Then we posed the question: "how much better does this improvement have to be in order to be significant?" and answered it by finding the ∆χ 2 value such that only 5% or 1% of the time would the smoothing method randomly achieve a better improvement than this value. In other words, we have determined the ∆χ 2 values that correspond to the 95% and 99% confidence limits (CLs), ∆χ 2 95% and ∆χ 2 99% . We also demonstrated that likelihood distribution and the corresponding 95% and 99% CLs are independent of the assumed cosmological model. If the ∆χ 2 of real data is larger than the 95% (99%) CLs, then we can conclude that the model is inconsistent with the data at 2σ (3σ) significance, even though the true model is unknown. In this work, we perform model selection by deriving the likelihood distribution using 1000 mock realizations from Roman covariance matrix and previously calculated values of ∆χ 2 95% and ∆χ 2 99% . We forecast how frequently each model is ruled out by the simulated future SN Ia datasets and compare the result with that of the conventional Bayesian analysis. Data simulations & implementation of methods SN Ia distance measurements have become one of the most important datasets of modern cosmology. Since they are standardizable candles, we can use them to directly measure the accelerating expansion of the Universe at late times [20,21]. Almost all previous SN Ia compilations including SuperNova Legacy Survey [SNLS,22], Gold [23], Union [24], Constitution [25], Union2 [26], Union2.1 [27], Joint Light-curve Analysis [JLA, 28] and Pantheon [29] have been shown to be consistent with the flat ΛCDM model. However, these consistency tests require some assumptions for parametrization or functional form. There have also been some model-independent analyses that search for systematics and test the internal consistencies in a non-parametric manner [19,30,31]. To generate simulated future SN Ia data, we first need to construct a covariance matrix. We do this by constructing its diagonal term using the forecasted error information of the Roman telescope that the total error in luminosity distance is ∼ 0.5-1% between z = 0.1-1.7 [32]. With a covariance matrix in hand, we can then generate Gaussian random mock datasets based on an assumed fiducial model (here the Transitional Dark Energy (TDE) model [33]). We use the TDE model with the Hubble constant H 0 = 70 km s −1 Mpc −1 and matter density Ω m = 0.3 as the fiducial model and generate 1000 mock realizations. Then, we choose two other models, the ΛCDM and Phenomenologically Emergent Dark Energy (PEDE) [34,35] models, as candidates in the model selection procedures under discussion (the Bayesian evidence ratio approach and our proposed approach based on the iterative smoothing method). The difference between these models (ΛCDM, PEDE, TDE) is the difference between their equations-of-state parameters. The equation-of-state parameter is w(z) = −1 for the w(z) = w 0 + (w 1 − w 0 ) 1 + tanh z−zt ∆z /2 (4.2) where we choose (w 0 , w 1 , z t , ∆ z ) = (−0.8, −2.0, 1.0, 0.2). These values were chosen to make an example model that is very different than PEDE or ΛCDM. We can see these differences in the equation-of-state parameter in Figure 1a. In contrast with PEDE, which is purely phantom, TDE starts out in the quintessence regime at lower redshift (where most of the data is) but then transitions to the phantom regime at higher redshift. We fix the parameters that determine the equation of state in the TDE model to give it the same number of degrees of freedom as the ΛCDM model. Next, we calculate the expansion history of the Universe, h(z) ≡ H(z)/H 0 where H(z) is the Hubble parameter, for all these models via h 2 (z) = Ω m (1 + z) 3 + (1 − Ω m ) exp 3 z 0 1 + w(z ) 1 + z dz . Results from Bayesian analysis We present the results of our Bayesian analysis in Figure 2. There is shown the distribution of ∆ log Z between the ΛCDM and PEDE models. The distribution is over the different One can see that the Bayesian evidence approach strongly favors ΛCDM over PEDE, while in fact both of these models are false models. ∆ log Z > 3 PEDE consistent PEDE ruled-out ΛCDM consistent 6 994 ΛCDM ruled-out 0 0 ∆ log Z > 5 PEDE consistent PEDE ruled-out ΛCDM consistent 89 911 ΛCDM ruled-out 0 0 Table 2: The number of realizations among 1000 realizations that strongly and very strongly supports the ΛCDM model against PEDE model from the analysis using the Bayesian evidence. The data for each of these cases are generated from the TDE model and thus none of the two models in the table are correct models. realizations of mock Roman-like datasets based on the fiducial TDE model. It is a testament to the precision of the mock Roman-like datasets that the Bayesian evidence gives "Positive evidence" that supports ΛCDM over PEDE in all of the cases, "Strong evidence" in 99.4% of the cases and "Very strong evidence" in 91.1% of the cases, as given from Table 2 in the KR scale. In other words, the Bayesian evidence strongly supports ΛCDM over PEDE in almost all of the mock datasets, yet neither of these models is the correct model. In no cases are both incorrect models found to be inconsistent with the data. Both ΛCDM and PEDE have the same degrees of freedom, and as already shown at low redshift in Figure 1, the expansion history of the fiducial TDE model is closer to that of ΛCDM than that of PEDE. This explains why the conventional Bayesian analysis supports the ΛCDM model over PEDE even though neither are correct. Results from frequentist analysis To compare with the results from the Bayesian analysis, we calculate the likelihood distribution (∆χ 2 ) for the ΛCDM and PEDE models. Figure 3 displays these distributions. Also shown is the 95% (∆χ 2 95% ) and 99% (∆χ 2 95% ) CLs of the likelihood distribution. Figure 3: Likelihood distributions of ∆χ 2 = χ 2 smooth − χ 2 best−fit for each of the considered models, ΛCDM (blue) and PEDE (cyan), from the analysis using the iterative smoothing method. Vertical lines show the 95% (dashed) and 99% (dash-dotted) CLs of the likelihood distribution. These limits represent the expected amount of improvement (over-fitting) the smoothing procedure will perform on the true model. The distribution is for 1000 mock realizations of Roman dataset based on the fiducial TDE model. We can see that, using this approach, in a large number of cases, both ΛCDM and PEDE would be ruled out. Table 3: The number of realizations among 1000 realizations within or outside the 95%, 99% CLs for each model best-fits from the analysis using the iterative smoothing method. The data for each of these cases are generated from the TDE model and thus none of the two models in the table are correct models. shows the number of mock realizations among 1000 realizations that exceed or do not exceed the two different CLs for the ΛCDM and PEDE models. The number of realizations that exceeds the 95%, 99% CLs is 916, 793 for ΛCDM and 998, 986 for PEDE. It is clear that using this approach, in 91.6% of the cases both assumed models are ruled out at 95% confidence and in 79.3% of the cases they are both ruled out at 99% confidence. In both results, the data find the ΛCDM model to be favored over the PEDE model. The Bayes factors always support the ΛCDM model over PEDE with ∆ log Z ≥ 2, which is at least "Positive evidence" according to the KR scale, and in our likelihood distribution analysis, Table 3 shows that in none of the cases where ΛCDM is found to be inconsistent with the data the PEDE model found to be consistent. However, the most obvious and important difference between the results is that our frequentist analysis can rule out both false models at 99% confidence around 80% of the time whereas the Bayesian analysis can never do so. We should emphasize here that the conventional Bayesian approach ranks the relative performances of different models with respect to the data but cannot absolutely rule out a model, while it is a very important advantage that our frequentist approach can rule out different candidates without any prior knowledge about the true model. Summary In this paper, we expand upon the frequentist model selection method that previously introduced in [3] and compare the statistical power of our test to the conventional approach based on the Bayesian evidence ratio. Our method works by calculating the likelihood distribution ∆χ 2 and using the values that enclose 95% and 99% of the volume of this distribution as criteria to test whether real data support a given model, independent of how well other models perform. The comparison of these two approaches is done using mock forecasted future Roman data based on TDE model. Using the conventional Bayesian approach, we show that we can rule out the PEDE model in favor of ΛCDM model in 91.1% of the cases with very strong evidence and in 99.4% of the cases with strong evidence. However, there is no mock dataset for which the ΛCDM model can be ruled out by the method. On the other hand, we show that, using our frequentist iterative smoothing approach, we can rule out both competing candidate models (ΛCDM and PEDE) in 79.3% of the cases with more than 99% confidence and in 91.6% of the cases with more than 95% confidence. The method also shows that ΛCDM is closer to the fiducial TDE model than PEDE is since in 19.3% (8.2%) of our mock datasets, PEDE is ruled out while ΛCDM is found to be consistent at the 99% (95%) CL. Further, there is no mock dataset for which the ΛCDM model can be ruled out while PEDE is consistent with the dataset. Taken together, these results serve as a forecast that future Roman data should have enough SN Ia at high redshift to be able to not only distinguish models confidently using a Bayes factor procedure but also rule out individual models using just the data irrespective of how well other models perform. These results indicate that our model selection methodology that uses the likelihood distribution tests individual models and can absolutely rule out both false models. However model selection using the Bayesian evidence can only assess the relative performance of the two models, but is not able to rule out a least-bad but still false model. This is possible since reconstructions generated by the iterative smoothing method provides model-independent information without knowledge of the true model and without the need for comparing both models with one another. It is a clear advantage of our non-parametric and frequentist approach over Bayesian methods for model selection when the true model is unknown as is the case in cosmology. The analysis using realizations of Roman mock datasets can be done in the same way for forecasting results from other future SN Ia compilations, such as the ones from the Ten-year Rubin Observatory Legacy Survey of Space and Time [LSST,36]. These surveys may provide more SN Ia which can help us detect any possible deviation from the concordance ΛCDM cosmological model. Similarly we can use the same approach and philosophy to construct a model independent approach for model falsification, model selection and parameter estimation using other cosmological data. This will be our future work. Figure 1 : 1The equation-of-state parameter w(z) (left) and expansion history h(z) (right) for the TDE (black; fiducial) model and two comparison models, ΛCDM (blue) and PEDE (cyan), with the same values of Ω m = 0.3. ΛCDM model and w(z) = − 1 3 ln 10 (1 + tanh [log 10 (1 + z)]) − 1 (4.1) for PEDE model. For the TDE model, it is parametrized as Figure 1b shows the results of this calculation of the expansion history of the Universe for the ΛCDM, PEDE and fiducial TDE model, with the same values of Ω m = 0.3. It demonstrates how, despite the equation-of-state parameter being very different between the considered models, the expansion histories of the different models are relatively close to each other since they involve integrals of the equation of state. For each of these models, we allow Ω m and H 0 to vary within [0.0, 1.0] and [60, 80] km s −1 Mpc −1 . Figure 2 : 2Distributions of ∆ log Z for each of the considered pairs of models between the ΛCDM and PEDE model. Vertical lines correspond to ∆ log Z = 3 and 5. The distribution is over the different realizations of mock Roman datasets based on the fiducial TDE model. Table 1 : 1TheKass-Raftery scale: a conservative interpretation of ∆ log Z for model selec- tion [9]. by the nested sampling algorithm. The algorithm repeatedly removes the point with the lowest likelihood among multiple 'live' points that represent different iso-likelihood surfaces, and replace it with a new live point with higher likelihood that is chosen from the Monte Carlo samples. For each iteration, live points are used for estimating prior volume of the previous live point, then we numerically calculate the evidence by summing the product of the likelihoods and prior volumes of all previous live points. The algorithm stops when the calculated evidence converges to within a specified error tolerance. It is called dynamic nested sampling when the number of effective live points is variable at any given iteration. This variable number of live points allows for modulating the speed at which the algorithm converges to the integral. 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The ferromagnetism of Au nanograins is analysed based on a two-dimensional itinerant lattice model with on-site Coulomb repulsion, many-body spin-orbit interactions, and holding two hybridized bands, one correlated and one uncorrelated. Using periodic boundary conditions in both directions, an exact ferromagnetic ground state is deduced for this non-integrable system by applying special techniques based on positive semidefinite operators.

10.1142/s0217979221501484

2109.07297

09d94c3334e2f7d4958a20ec234295fe686d8f46

Nanograin ferromagnets from non-magnetic bulk materials: the case of gold nanoclusters 15 Sep 2021 Nóra Kucska Department of Theoretical Physics University of Debrecen Bem ter 18/BH-4010DebrecenHungary Zsolt Gulácsi Department of Theoretical Physics University of Debrecen Bem ter 18/BH-4010DebrecenHungary Nanograin ferromagnets from non-magnetic bulk materials: the case of gold nanoclusters 15 Sep 2021numbers: 7110-w7110Fd7127+a7170Ej7510-b7575-c The ferromagnetism of Au nanograins is analysed based on a two-dimensional itinerant lattice model with on-site Coulomb repulsion, many-body spin-orbit interactions, and holding two hybridized bands, one correlated and one uncorrelated. Using periodic boundary conditions in both directions, an exact ferromagnetic ground state is deduced for this non-integrable system by applying special techniques based on positive semidefinite operators. I. INTRODUCTION A. About Nanomagnetism Nanomagnetism attracts nowadays extreme attention given by both the theoretical challenges and the broad technological application possibilities of the subject 1 . In this field we often encounter situations when magnetic moments present in bulk magnets produce magnetism also at nanosize level, as e.g. in the cases of magnetic nanocomposites 2 or nanomagnets containing magnetic ions 3 , whose properties are usually connected to the behaviour of magnetic domains or domain walls in constrained geometry. But the field of nanomagnetism presents also examples where materials not containing magnetic moments possessing bulk behaviour placed far from magnetism, present magnetic properties at nanoscale. These materials trigger a special chapter of nanomagnetism, in which the magnetic properties are intimately connected to the nanosize of the sample, and this is the subject which attracted our interest and attention during the research that leads to this article. A typical example on this line is the case of gold, which does not contain magnetic atoms, is diamagnetic in macroworld, but becomes ferromagnetic around 2 nm size 4 . We are concentrating below to this property in our aim to explain it in the light of the presently existing experimental data. B. Nanomagnetism of gold nanoparticles It is known that none of the 4d or 5d elements is magnetic in bulk form i.e. does not possess in macroscopic form non-zero magnetic moment in zero applied external field 5 . But since 1999 it is known that Au nanoparticles become ferromagnetic 6 . The surprising observation is real since it has been checked in several laboratories 4,7,8 , see also 5 . When prepared, the nanoparticles are in most cases defended (ligand-coated, functionalized) at the surface 9 , and this fact leads to several theories aiming to explain the source of this type of nanomagnetismn by the ligand present on the surface (as e.g. 10-13 ). But the ferromagnetism appears also without functionalization 7 , remains when one removes from the surface the ligand 7,14 , and it was observed that photochemically prepared nanoparticles which never have contacted any added ligand have similar magnetic properties 14 . Consequently, not the ligand on the surface causes the magnetic properties, so explaining this type of nanomagnetism other reasons must be used for explanation. One underlines here, that the ligand could modify the magnetic moment (as even observed e.g. 15 ), but the source of the magnetization is completely different. We further note that the presence of magnetic impurities (as e.g. Fe, Ni, Co, Mn, etc.) which could cause the appearance of magnetism experimentally can be excluded 5, 16 . Turning back to theory, one often encounters mostly mean-field type of descriptions (e.g. 17 ) which takes into account spin-spin type of interactions without clarifying precisely why, how and from where the magnetic moments come from, and if are present, why should we neglect the correlations (absent in mean-field). In essence one particle description (as e.g. 18 ), similar to mean-field treatments, also lack inter-particle correlations. The DFT descriptions [as e.g. 19 using spin-polarized generalized gradient approximation (GGA)] has as well shortcomings in correctly reproducing the many-body correlations 20 , and finally leads to ferromagnetism considering the nanocluster an icosahedral superatom in which magnetic alignment is caused by a "superatom Hunds rule". This last is deduced from two-particle exchange, whose application in realistic many-body case is questionable (see e.g. 5 ). We underline here the many-body correlations since the current literature strongly stresses that the correlation effects in Au are important [21][22][23] . Properly described correlation effects in describing gold nanoparticles have been considered in 24 . Here a two-dimensional lattice is considered with itinerant spin-1/2 carriers and periodic boundary conditions in both directions as describing a closed surface. Note that for nanograins more than the half of atoms are disposed on the surface, and the itinerant charged carriers, given by the Coulomb repulsion are also disposed on the surface. In describing the inter-particle correlations on-site Coulomb repulsion has been taken into account in the many-body Hamiltonian, for which an exact ground state solution has been deduced. For small samples (e.g. LxL, L = 12 lattice) the ground state turns out to be ferromagnetic. The result does not require rigorous spherical surface: it demands closed surface, (even small) local Coulomb repulsion and many-body quantum mechanics. The deduced solution emerges in the small size limit, and for L → ∞ loses its importance. That is why by increasing the size, the total magnetization decreases, and at L ≫ 1 gradually disappears as observed experimentally 5 . Since the model is non-integrable (one band 2D Hubbard model) a special technique based on positive semidefinite operator properties has been used in order to deduce the exact ground state. The ferromagnetic property arises from the parallel alignment of spins necessary to avoid the Hubbard (on-site Coulomb) repulsion in order to minimize the ground state energy. The fact that the ground state is a coherent quantum many-body state is supported by experi- II. PLACING THE MODEL CLOSER TO THE REAL SYSTEM The model presented in 24 shows a potential possibility, but in fact is a toy model that should be pushed more closely to the real analysed system. First, the published studies show that the multiband character of Au is important to be taken into consideration. Indeed, it was observed (e.g. 22 ) that in gold, the 5d orbital has a prominent contribution to the conduction electron system. The presence of 5d orbitals accentuate that during the description, at least a correlated band, and a non-correlated band should be taken into account, and since direct evidence of 5d magnetism is not present 22 , a hybridized two band system should represent the starting point. At the level of interactions, since we are confined to short distances, the on-site Coulomb repulsion U must represent the starting point, which is able to describe properly the correlation effects in the system. But the published results show that also another interaction, namely the many-body spin-orbit coupling (SOC), plays a major role. This was observed already at the level of one particle descriptions 18 , during the study of the importance of the correlation effects 23 , but nowadays, given by the observation that relativistic aspects must be included in the proper description 23 , the importance of the SOC becomes to be a clear fact [26][27][28][29] . We note that in fact, the large SOC at Au surface is known for many years now 30,31 . On its turn, the many-body spin-orbit interaction is in fact a relativistic effect (relativistic correction to the Schrödinger equation) which shows that if a carrier holding spin ( σ) is moving (i.e. has momentum p), and during this movement it feels a potential gradient ∇V not colinear to p, an interaction of the formĤ SO = λ σ · ∇V × p appears, which represents the many-body spin orbit interaction, λ being its coupling-constant (i.e. strength), (see Fig.1). Since in central field ∇V ∼ r, and r × p provides the orbital moment,Ĥ SO indeed describes the SOC. But more importantly, at a surface (in the present case at the nanograin surface), perpendicular to the surface automatically a ∇V potential gradient appears, sô H SO will be automatically present. The λ value is typically small, of order 10 −3 eV, but in the Au case, one knows 30 that it attains in order 0.1 eV. That is why SOC is important for gold nanograins as well. We must underline that λ ≪ U usually holds, but even in this case λ produces essential effects because breaks the double spin-projection degeneracy of each band. This is the reason why the perturbative treatment ofĤ SO in presence of U > 0 gives erroneous results, explaining why we use exact methods for description. We further note that at second quantized level, the presence of SOC gives rise in fact in the system Hamiltonian to spin-flip type of hopping terms of specific form 32,33 . In our opinion, the main effect of the ligand on the surface is to influence theĤ SO strength. Our aim in this paper is to show that the main conclusions of 24 concerning ferromagnetism in gold nanograins remain true et exact level also in the case of two-band treatment, and presence ofĤ SO in the many-body 2D Hamiltonian containing the on-site Coulomb repulsion in the correlated band. III. THE USED MODEL HAMILTONIAN AND ITS RESULTS A. The positive semidefinite form of the Hamiltonian The Hamiltonian of the system in its original form has the expression H = p,p ′ i,r σ,σ ′ (t p,p ′ ;σ,σ ′ i,i+rĉ † p,i,σĉ p ′ ,i+r,σ ′ + H.c.) + i Un d,i,↑nd,i,↓(1) where the first term represents the kinetic Ĥ kin , while the second the interaction part t p,p;↑,↓ i,i+x 1 = −t p,p;↓,↑ i,i+x 1 = t R p , t p,p;↑,↓ i,i+x 2 = t p,p;↓,↑ i,i+x 2 = −it R p .(2) The startingĤ in (1) i, i + x 1 , i + x 2 , i + x 1 + x 2 , representing the four corner sites of an unit cell. The in-plaquette notation of this four sites is given by the index n = 1, 2, 3, 4 in positive trigonometric circulation direction (i 1 = i, i 2 =, i + x 1 , i 3 = i + x 1 + x 2 , i 4 = i + x 2 ). After this step, on each plaquette we introduce two block operators m = 1, 2 asB 1,i = V (1) i ,B 2,i =V (2) i , which represent a linear combination of fermionic operatorsĉ p,i,σ (p = s, d) acting on each site of the elementary plaquette on both bands: V (1) i = p,n,σ v (1) p,n,σĉ p,in,σ ,V (2) i = p,n,σ v (2) p,n,σĉ p,in,σ ,(3) where p = s, d, one has n = 1, 2, 3, 4 and σ =↑, ↓. The numerical prefactors v (1) p,n,σ and v (2) p,n,σ in (3) are the block operator parameters whose value is deduced in the transformation process. Note that the block operators contain both spin indices in order to be able to reproduce in the positive semidefiniteB m,iB electron present at the lattice site i. We underline that given by the structure of the Hamiltonian presented in (4), and the fact thatP is a positive semidefinite operator, the ground state |Ψ g of the system is given by the relation P |Ψ g = 0,(5) while the eigenvalue connected to |Ψ g (i.e. the ground state energy E g ) is given by the relation E g = C. As underlined previously, (5) provides the exact ground state independent on dimensionality and integrability. (1) and (4) -note that for this the products and sums present inP B andP U in (4) must be explicitly calculated and written -, and b) equating the coefficients of all different operators from (1) to the coefficient of the same operator in (4). For example, let us take from the starting Hamiltonian in (1) the ↑, ↓ spin-flip hopping terms in the p = s band which has the expressionĥ 1 = t s,s,↑,↓ i,i+x 1ĉ † s,i,↑ĉ s,i+x 1 ,↓ . We observe that such contribution in (4) emerges inP D , namely (if one fixes the m=1 index value, hence one considers only theB 1,i operators) in two products, namelyÎ 1 =B 1,iB † 1,i and I 2 =B 1,jB † 1,j where the lattice site j is placed in the lattice just below the site i. Indeed, fromÎ 1 we obtain the termb 1,i = v (1) s,2,↓ v (1) * s,1,↑ĉ s,i+x 1 ,↓ĉ † s,i,↑ , while fromÎ 2 the contribution b 1,j = v (1) s,3,↓ v (1) * s,4,↑ĉ s,i+x 1 ,↓ĉ † s,i,↑ (note that j + x 1 + x 2 = i + x 1 , andj + x 2 = i). One knows thatb 1,i can be also written asb 1,i = −v (1) s,2,↓ v (1) * s,1,↑ĉ † s,i,↑ĉ s,i+x 1 ,↓ , and similarly, b 1,j = −v (1) s,3,↓ v (1) * s,4,↑ĉ † s,i,↑ĉ s,i+x 1 ,↓ , hence one reobtains exactly the operator expression fromĥ 1 . Other operators of the formĥ 1 are not present in (4) if m = 1 holds, but similar two contributions (b 2,i andb 2,j ) one finds in the m = 2 case. Consequently, since from the exact transformation of (1) to (4) (between others), alsoĥ 1 = m=1,2 (b m,i +b m,j ) must hold, the matching equation connecting t s,↑,↓ i,i+x 1 to block operator parameters becomes − t s,s,↑,↓ i,i+x 1 = m=1,2 v (m) * s,1,↑ v (m) s,2,↓ + v (m) * s,4,↑ v (m) s,3,↓ .(6) Similarly, let us take a next nearest-neighbor (unit cell diagonal) spin-flip hopping term in the same band p = s, namelyĥ 2 = t s,s,↑,↓ i,i+x 1 +x 2ĉ † s,i,↑ĉ s,i+x 1 +x 2 ,↓ . One observes that since the starting Hamiltonian in (1) contains only nearest-neighbor hopping terms,ĥ 2 is missing from (1), consequently t s,s,↑,↓ i,i+x 1 +x 2 = 0 holds. Butĥ 2 type of contributions are present in (4), namely inP D , concretely (for the case of fixed m = 1) inÎ 1 =B 1,iB † 1,i . Indeed, effectuating the product inÎ 1 , we find the contribution d 1,i = v (1) s,3,↓ v (1) * s,1,↑ĉ s,i+x 1 +x 2 ,↓ĉ † s,i,↑ = −v (1) * s, 1, ↑ v (1) s,3,↓ĉ † s,i,↑ĉ s,i+x 1 +x 2 ,↓ which (with another numerical prefactor) contains the operatorĥ 2 missing from the starting Hamiltonian (1). The same type ofĥ 2 contribution is obtained from (4) also at m = 2 in the formd 2,i = v (2) s,3,↓ v (2) * s,1,↑ĉ s,i+x 1 +x 2 ,↓ĉ † s,i,↑ = −v (2) * s, 1, ↑ v (2) s,3,↓ĉ † s,i,↑ĉ s,i+x 1 +x 2 ,↓ . Consequently, because otherĥ 2 type of contributions are no more present inP , and since for the exact transformation of (1) to (4), also the equality 0 =ĥ 2 = m=1,2d m,i must hold, a matching equation emerges of the form − t s,s,↑,↓ i,i+x 1 +x 2 = m=1,2 v (m) * s,1,↑ v (m) s,3,↓ = v (1) * s,1,↑ v (1) s,3,↓ + v (2) * s,1,↑ v (2) s,3,↓ = 0.(7) Equation (7) can be usually deduced, and for the present case, are also presented in Appendix A. C. The deduced ground state and its physical properties Now the third step of the technique follows: the deduction of the ground state |Ψ g . This is done from (5), and one finds |Ψ g = N Λ i=1 m=1,2B † m,i D † i R † N 1 |0 ,(8) where |0 is the bare vacuum with no fermions present. One hasD † i =ĉ † d,i,↑ +ĉ † d,i,↓ , whilê R † N 1 >0 = N 1 α=1ĉ † pα,kα,σα andR † N 1 =0 = 1, where 1 ≤ N 1 < N Λ holds,ĉ p,k,σ being the Fourier transform ofĉ p,i,σ , p α is arbitrarily s or d; σ α is arbitrarily ↑, ↓; k α (with the condition that at α = α ′ one has k α = k α ′ ) is an arbitrary k value from the first Brillouin zone. The uniqueness of the ground state can be demonstrated on the line of 43 . For N 1 = 0, (N 1 > 0), the ground state (8) corresponds to 3/4, (above 3/4), system filling. The (5) is satisfied because i)B † m,iB † m,i = 0, henceP D |Ψ g = 0 holds, and ii) one has on each site at least one carrier in |Ψ G hence alsoP U |Ψ g = 0 is fulfilled. Below one considers the N 1 = 0 case, but similar results one finds at N 1 > 0 as well. As shown by Fig.3, (i.e. 83% of the atoms from the system are neglected), our deduced ground state is no more the ground state of the system, hence ferromagnetism dissapears. This surely happens when the grain mass density (for 16 3 FCC cells here ρ = 20.21 g/cm 3 ) approaches from above the bulk density (for gold, ρ = 19.3 g/cm 3 ). The ferromagnetic spin alignment is enforced by the on-site Coulomb repulsion (Hubbard U term) i.e. many-body correlations, since in this case the ground state energy attains its minimum value by avoiding double occupancy in the direct r-space, the k-space itinerant states providing components on all lattice sites. The here deduced ground state type 32 emerges above system half filling, but similar ferromagnetic ground state can be deduced as well below system half filling 33 . Consequently, doping is necessary for the ferromagnetism to occur as observed experimentally 45 . IV. SUMMARY AND CONCLUSIONS After a survey of experimental data and used theoretical descriptions, the ferromagnetism of gold nanoparticles is explained by a deduced 2D exact many-body ground state determined by the common effect of the spin-orbit coupling, Coulomb correlations, hybridized bands and doping on a closed itinerant nanosurface. Besides to help the doping process, the role of the ligand on the surface turns out to influence only the strength of the spin-orbit coupling. A similar description probably works also in the case of Pd nanograins where the importance of similar effects have been pointed out [46][47][48][49][50] . We further note that if an arbitrary non-magnetic and metallic compound becomes ferromagnetic at nanoscales, and presents similar material properties as detailed for Au at the beginning of Section II, the here presented theory would explain this behavior. Besides, the C constant from the transformed Hamiltonian in (4) is given by C = χN − U d N Λ − i m=1,2 {B m,iB † m,i },(A7) where N represents the number of carriers,N Λ the number of lattice sites, while {X,Ŷ } = XŶ +ŶX holds. The solutions of the matching equations In order to solve the matching equations one starts with the homogenous equations with minimum number of components (as e.g. (A3)), going further by gradually using the more complicated equations. First one finds that all v (1) p,n,σ can be expressed in terms of v (2) p,n,σ parameters as v (1) p,n,σ = K n v (2) p,n,σ , where the proportionality constants K n are given by K 1 = − 1 x , K 2 = − 1 y , K 3 = x * , K 4 = y * ,(A9) where x, y are ( = 0, ∞) parameters. Furthermore it results that from the remaining unknown v (2) p,n,σ , those containing p=s band index can be expressed in terms of the p=d components as follows v (2) s,1,σ = xu σ y v (2) * d,3,−σ , v (2) s,2,σ = u σ v (2) * d,4,−σ , v (2) s,3,σ = − u σ x * y v (2) * d,1,−σ , v (2) s,4,σ = − where φ 3 , φ 4 are arbitrary phases. Then one finds v (2) d,2,↑ = √ 2|y|e iγ |(|y| − 1)| v (2) d,1,↓ , v (2) d,2,↓ = √ 2|y|e −iγ |(|y| − 1)| v (2) d,1,↑ ,(A12) where γ is an arbitrary phase, furthermore, for y = |y| e iφy , one has x = |x| e iφy where |x| = (|y| − 1) / (|y| + 1) holds. The last two unknown block operator parameters are deduced from (t s,d,↑,↑ i,i+x 1 e −iη + t s,d,↑,↑ i,i+x 2 ) = A ↑ e iδ ↑ , A ↑ = t s,d,↑,↑ i,i+x 1 e −iη + t s,d,↑,↑ i,i+x 2 , − t s,d,↑,↑ i,i+x 1 e −iη − t s,d,↑,↑ i,i+x 2 = A ↓ e iδ ↓ , A ↓ = t s,d,↑,↑ i,i+x 1 e −iη − t s,d,↑,↑ i,i+x 2 ,(A13) where η = φ 3 + φ y . Starting from this relation, introducing b = (|y| − 1) 2 2 √ 2(|y| 2 + 1) |y| |u| , u = |u| e iφu , where φ u is an arbitrary phase, one has v (2) d,1,↑ = b A ↑ e i(γ+φu+δ ↑) /2 , v (2) d,1,↓ = b A ↓ e i(−γ+φu+δ ↓) /2 . Furthermore |y| |u| = −t d,d,↑,↑ i,i+x 1 A ↑ A ↓ cos δ ↑ −δ ↓ 2 , tang δ ↑ − δ ↓ 2 = t d,d,↑,↑ i,i+x 2 t d,d,↑,↑ i,i+x 1 ,(A16) and |y| remains arbitrary. Note that the presented solution preserves t p,p ′ ,↑,↑ i,j = t p,p ′ ,↓,↓ i,j . FIG. 1 : 1The many-body spin-orbit coupling mental data. Indeed, if one introduces impurities in the system, the quantum coherence is diminished, hence the magnetic moment decreases as observed by introducing Fe impurities in Au nanograins25 . Ĥ int , of the Hamiltonian (on-site Coulomb repulsion U > 0 in the correlated d-band), and p, p ′ = s, d are representing the band indices where s symbolycally represents the noncorrelated band (being here concretely of sp type). Please note that only the d-band is correlated in Eq.(1), the s-band being considered without Coulomb repulsion is uncorrelated. FIG. 2 : 2The plaquette connected to the site i used for defining the block operators B m,i (see text above Eq.(3)). The Bravais vectors are denoted by x 1 , x 2 . The in-plaquette notation of lattice sites is provided by n = 1, 2, 3, 4.The sum over i incorporates all lattice sites, and periodic boundary conditions are taken into account in both directions. InĤ kin the p = p ′ contributions are representing hybridization terms. Denoting by x 1 , x 2 the Bravais vectors of the 2D system, r = x 1 , x 2 describes nearest neighbour hoppings, while r = 0 characterizes on-site potentials. The SOC is taken into account by the Rashba term 34 i.e. incorporates all desired characteristics underlined in the previous chapter: surface many-body quantum physics (i.e. 2D character), two-bands character (i.e. p = s, d), presence of correlations (the on-site Coulomb repulsion U in the correlated d band), and spin-orbit coupling (Rashba term). For a proper treatment of the common influence of the spin-orbit coupling and correlations, we deduce exact ground states on the line of 32 , but using only the Rashba term in SOC. Since the system is non-integrable, a special technique is used based on positive semidefinite operator properties which has been tested in detail in several circumstances 35-37 , being checked as well in the case of nanostructures 38 , multiband structure 39 , specific correlation effects 40 , effects of the confinement 41 , even disordered systems 42 , detailed review papers being also available relating the technique 43,44 . The technique starts with the transformation of the starting Hamiltonian presented in (1) in positive semidefinite form. For this reason we take elementary plaquettes defined on each lattice site i, see Fig.2, one plaquette containing four sites, namely be observed that the starting Hamiltonian in (1) depends on the coupling constants of the Hamiltonian (as t p,p ′ ;σ,σ ′ i,i+r , U, where r = 0, x 1 , x 2 ; p, p′ = s, d; and σ, σ ′ =↑, ↓ ), while the transformed Hamiltonian in (4) depends on the block operator parameters (as v (m) p,n,σ ,where m = 1, 2; p = s, d; n = 1, 2, 3, 4; and σ =↑, ↓). This means that since theĤ in(1)and (4) is exactly the same, interdependences must be present in between the Hamiltonian coupling constants and block operator parameters. These relations are called to be the matching equations.The matching equations are obtained by a) explicitly writing all different contributions in clearly shows why we need two ( B 1,i , B 2,i i.e. m = 1, 2) block operators for the transformation in the positive semidefinite form: the reason is that this helps us in eliminating those contributions in the transformed positive semidefinite Hamiltonian expression (4), which are not present in the starting Hamiltonian (1). Without two block operators used in the present case, expressions like (7) would nullify the uniquely introduced block operator, so would eliminate the possibility of the transformation of the Hamiltonian in positive semidefinite form. In a similar manner all matching equations can be deduced. In the present case their number is relatively high, namely 74, and are presented together with their solution in Appendix A. Writing the matching equations one finishes the first step of the method (the transformation of the Hamiltonian in positive semidefinite form). After this job, the second step of the technique follows, namely the solution of the matching equations. For this system of equations, the Hamiltonian coupling constants are considered known quantities, while the unknown variables are the block operator parameters. The difficulty of this second step is that the matching equations are representing a coupled, non-linear, complex algebraic system of equations containing (in realistic cases describing real materials) a relatively high number of components (and standard numerical softwares for coupled nonlinear system of equations with high number of equations, in the general case, are not known). But, with our background and experience in solving such systems, the solutions FIG. 3 : 3the normalized ground state expectation value of the total spin z component per number of particles increases in function of the SOC coupling in the absence of external magnetic field, signalling ferromagnetic order. When the number of atoms in the volume exceeds considerably the number of atoms on the surface, the here deduced ground state loses its importance and the magnetic state disappears. Indeed, let us consider a toy The ground state expectation value of the normalized S z total spin per number of particles, in function of the Rashba coupling of the correlated band (t R d ) in units of the nearest neighbour hopping (t d ) in the same band exemplification of this aspect by considering FCC cells (in which usually Au crystalizes) disposed in cubic grains whose size is d. Let further introduce the ratio f = N s /N t , where N t is the total number of atoms from the system, and N s is the number of atoms on the surface. For 1 cell (N t = 14 atoms, d=4.14Å), f=100 %. For 8 cells (N t = 63, d=8.28Å) the studied ratio is still f = 80%. But for 16 3 = 4096 cells (17969 atoms, d=66.24Å = 6.62 nm) one has only f = 17%. Our model being a 2D model, takes into account the surface, so the deduced ground state is the ground state of the Au grain only if f is high. For f = 17% σ are new ( = 0, ∞) parameters. After this stage several solution classes are present, from which we exemplify here the u = u ↑ = −u ↓ case (this condition provides zero value for the local spin-flip contributions). In this situation all the remaining v (2) d,n,σ parameters can be expressed in terms of v (2) d,1,σ as follows V. BIBLIOGRAPHY 2 A. F. Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fischer, R. P. 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(A6) † m,i expressions the spin-flip hopping terms introduced in the Hamiltonian by the spin-orbit contributions. We further underline that two block operators per plaquette are needed in order to exclude from the positive semidefinite form the contributions not present in the starting Hamiltonian (1). Concerning the strategy we use, we remember that, ifÔ is a positive semidefinite operator, than for all its arbitrary matrix elements the relation u|Ô |u ≥ 0 holds, hence all its eigenvalues are non-negative. 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In this study, we present experimental and theoretical analyses of double emulsion drop evaporation. After the apparent completion of evaporation of the inner phase of a double emulsion drop, surprisingly, a resurfacing of a daughter droplet is observed. We further investigated to hypothesize this phenomenon which allowed us to obtain a prolonged fixed contact line evaporation for a single phase drop along with similar occurrence of resurfacing as of the double emulsion drops.

1810.10607

1aaee90d61aa8707a5632cffffb03992fb090e4e

Double emulsion drop evaporation and resurfacing of daughter droplet Muhammad Rizwanur Rahman Department of Mechanical Engineering interfacial Science and Surface Engineering Lab (iSSELab) University of Alberta T6G2G8EdmontonAlbertaCanada Prashant R Waghmare Department of Mechanical Engineering interfacial Science and Surface Engineering Lab (iSSELab) University of Alberta T6G2G8EdmontonAlbertaCanada Double emulsion drop evaporation and resurfacing of daughter droplet In this study, we present experimental and theoretical analyses of double emulsion drop evaporation. After the apparent completion of evaporation of the inner phase of a double emulsion drop, surprisingly, a resurfacing of a daughter droplet is observed. We further investigated to hypothesize this phenomenon which allowed us to obtain a prolonged fixed contact line evaporation for a single phase drop along with similar occurrence of resurfacing as of the double emulsion drops. Introduction The importance of drop evaporation can be identified in numerous applications such as ink-jet printing, coating technologies 1,2 ,self cleaning 3 , bio-sensing 4 and droplet based micro-fluidics 5,6 . For the phenomenon being highly sensitive to surface morphology and its chemical composition 7 it sparked numerous researchers across disciplines to conduct theoretical and experimental investigations [8][9][10][11] . Interestingly, the study of evaporation which is crucial for range of applications, from DNA mapping 12 to chip manufacturing 13 , has not been extended for multi-phase droplets, though that of a single phase droplet has been well analyzed and understood [8][9][10][11] . The study of different aspects of multicomponent drop is sparsely attended. Double emulsion droplets, a simplest representation of multicomponent drop, is of paramount importance for their potential in several applications staring from encapsulation technology, drug delivery to the development of micronano scale devices [14][15][16][17][18][19] . Recently, evaporation of a drop of transparent mixture of water, ethanol and anise oilcommonly known as 'Ouzo drop' was studied where four phases of evaporation were observed 20 . This intrigued researchers to look into the previously unexplored evaporation patterns or modes associated with compound or multiphase droplets. The necessity of such studies becomes * [email protected] paramount for technologies that facilitate targeted and encapsulated delivery of drug or active reagents 21,22 where delivery on demand is crucial. Precise control over the disappearance of the outer protective shell can provide a superior passive control on the time stamping. For evaporation of an isolated liquid sphere in an infinite medium, rate of mass transfer follows a linear relationship with radius, as described by Maxwell's equation, where the diffusive flux is used as an analogy to the electrostatic potential 23 . Picknett and Bexon 24 , in their pioneering work, distinguished between the existence of two modes of sessile drop evaporation, namely, constant contact radius (CCR) or fixed three phase contact line (fixed TPCL) and constant contact angle (CCA) or moving TPCL. The chaotic existence of both modes is often observed, particularly at the end of the droplet evaporation until a visual observation permits to measure the contact angle. The measurable end of the evaporation is always identified by reporting the diminishing contact angle which is difficult and erroneous to report below 5 • . In reality the existence of the liquid thin film with finite volume is always ignored. The transition [25][26][27][28][29] from fixed to moving contact line occurs when the evaporating flux at the TPCL dominates over the evaporation through the liquid-vapor interface. With attainment of a critical contact angle, droplet perimeter can no longer remain pinned on the substrate and starts slipping. Thus, the moving TPCL evaporation mode is observed. This is a consequence of the competition between an intrinsic adhesion force that arrested the contact line motion and an exertion of a force due to evaporation flux that attempts to overcome this barrier by con-tracting the droplet through liquid-vapor interface 25 . The stick-slip or stick-jump at the moving contact line is another observation researchers have reported for drop evaporation in number of situations [30][31][32] . The adaptability of the well advanced single phase droplet evaporation theory was studied for drop evaporation on textured non-wetting situations by Dash and Garimella 33 . Quite recently, intermittent stick-jump mode in drop evaporation has been reported 34 where the authors further exploited the stickjump by maneuvering the surface roughness. This paper investigates double emulsion drop evaporation; however, the adaptability of existing theory for single phase drop evaporation is carefully scrutinized before comparing it with that of double-emulsion scenario. This required us to experiment with single phase droplets that are involved in double emulsion study with the same drop liquid and substrate combinations. While the single phase drop evaporation study is well studied, evaporation of a double-emulsion droplet may pose a complicated scenario since there are new or modified interfaces with an additional liquid-liquid interface and two TPCLs. For comparing with the double emulsion drop evaporation, certain modifications are necessary that are developed after confirming the validity of the well-established models used in the literature. We observed surprising appearance of a daughter droplet after the completion of noticeable evaporation of the inner drop. The unexpected occurrence of a daughter drop is coined as 'resurfacing' of daughter droplet. Based on the phenomenological evidence of resurfacing of a tiny drop during double-emulsion drop evaporation, an hypothesis is proposed which is validated with single phase drop evaporation where the occurrence of daughter drop is carefully engineered. Experimental methods Deposition of a double emulsion drop Generation of a double emulsion drop required the designing of a customized concentric coaxial needle with two different inlets. With a smaller diameter (d o = 0.5mm), the inner needle is slightly (0.5mm) protruded outside of the outer needle which has a larger diameter (d o = 1.8mm). For the experiments, the inner needle was connected to the deposition unit of DSA 100E (KRÜSS GmbH, Hamburg, Germany) while a secondary pump was connected to the inlet port of the outer needle. Initially, the outer phase liquid was pumped through the outer drop inlet port. After the generation of the outer drop of known volume, the inner drop deposition unit of DSA 100E dispensed the inner drop liquid at desired flow rate. The immiscibility between two phases and smaller inner drop volume facilitates the successful generation of the double-emulsion droplet at the tip of the co-axial needle. Each experiments were conducted for at least three times. The uncertainty of drop volume was as low as ±0.1µL. The generated double-emulsion drop (Panel A of Fig. 1) at the tip of the co-axial needle was then brought into contact with pre-cleaned substrate to deposit the drop. As the needle was retracted away from the substrates, the drop detached from the needle. Panel B and C schematically identify different distinguished steps of the evaporation. Droplet liquids and substrates An appropriate selection of liquid combination allowed the detachment of the inner drop from needle at the outer drop-air interface. For validation of the proposed modified model for double emulsion drop evaporation, we studied the evaporation of single phase drop of water (saturated vapor concentration, c s = 0.017 kg/m 3 , diffusion coefficient, D = 2.4 × 10 −5 m 2 s −1 ), diiodomethane (c s = 0.018 kg/m 3 , D = 6 × 10 −6 m 2 s −1 ) and toluene (c s = 0.14 kg/m 3 , D = 8×10 −6 m 2 s −1 ). For double-emulsion drop, diiodomethane was used as inner drop while outer drop was of water. Oleophobic substrates (10 cm × 4 cm), pris-tine adhesive surface (5 cm × 5 cm) and acrylic sheets (5 cm × 5 cm) were used as substrates for evaporation studies. The oleophobic substrates were cleaned with deionized water and ethanol prior to each experiment. For cleaning the acrylic sheet, acetylene was used in addition with de-ionized water and ethanol. The average roughness values of the substrates are reported in Table 1. Visualization and Contact angle measurements After careful deposition on a pre-cleaned substrate, the drop was allowed to attain equilibrium configuration. The change in contact angles and base diameter were recorded by the in-built imaging system of DSA 100E. The CMOS camera allowed to optimize the image settings and recording of the entire drying period at 60 frames per second from side. An additional synchronized CMOS camera captured the top view of the evaporating drop. The outer diameter of the needle was used to calibrate the pixels for their transformation into physical dimension. Base diameter and contact angles were analyzed from the side view of the droplet, whereas, the top views further assisted in comprehending the process, specially at the end of drying period as discussed later in this article. To measure the dynamic contact angle from recorded frames, sessile drop technique was employed. This method utilizes the tangent method for contact angle measurement.Though the Young-Laplace equation fitting method is a better option for measuring static contact angle, but its assumption of symmetric drop shape restricts its applicability for dynamic contact angle measurement. Hence, the tangent method was used for contact angle measurement. Results and discussion Double emulsion drop evaporation The so-called double or multiple emulsion drop system, also often termed as emulsion of emulsion, duplex emulsion, multiple emulsion or compound drops interchangeably 35 , can be defined as a drop completely engulfed or encapsulated by another immiscible liquid drop 36 . In this study, a diiodomethane drop is encapsulated inside a water drop and evaporation is studied on a oleophobic sub-strate. Similar to a single phase drop, evaporation starts with fixed TPCL mode for double emulsion drop when the base radius remains unaltered for a considerable time. This consequences in the decrease of contact angle as well as drop height as shown in Fig. 2(a) (Side views -S1, S2 ; Top views -T1, T2). Ensuingly, the outer phase water drop height decreases to that of the inner drop. At one point of time, the two interfaces interact with each other. We have termed the time period − from this interface interaction to the complete exposure of the inner droplet to air − as 'transition regime'. Since the air-water interface is shrinking due to evaporation, eventually the inner drop gets exposed to the air by forming an air-diiodomethane interface. The partially exposed inner drop is shown in S3 and T3 of Fig. 2(a). It is worthwhile to notice that the second mode of evaporation (moving TPCL) for outer drop is obstructed and shortened by the existence of the inner drop. The time required to complete the fixed TPCL evaporation mode for a single phase water drop is denoted as t dry,w,a in the figure. This is longer in comparison with the time observed for same sized water drop cushion in case of double-emulsion drop case. With this observation, one can argue that the majority of the liquid can be evaporated with a fixed TPCL by carefully maneuvering the outer to inner drop volume or contact line radius ratios. In the considered volume ratio of water and diiodomethane drops, for outer drop a very short period of a moving TPCL mode was observed. Once both the liquids compete for the evaporation, a third mode was observed with moving TPCL with changing contact angles, where the outer drop merely existed at the bottom of the inner drop as seen in S4 and T4 (transition regime). The water drop forms a precursor film around the inner diiodomethane drop. A change in the contact angle confirms the visible drying of the water, i.e., outer phase which is shown in the inset of Fig. 2 (b). The change in the base diameter suggests the coexistence of water and diiodomethane in liquid-vapor phase along the TPCL. This coexistence of both phases and corresponding changes in the diameter and contact angle is certainly a function of diffusivity, evaporation flux and all other evaporation dynamics parameters. The detailed parametric analysis is required to comment on the importance of these parameter which needs further attention. After the complete outer drop evaporation, the inner drop liquid-vapour is exposed to the air and thus the dimensions of the inner drop starts to change. The dotted line at ∼ 1000s demarcates the inner drop and outer drop evaporation regime. Once the inner drop was appeared to be completely evaporated the contact angle was almost immeasurable. Most of the studies reported the end of the evaporation up to this phase. But in the case of doubleemulsion droplet, resurfacing of a daughter droplet from the thin film is noticed which further evaporates at a different rate. Quiet interestingly, for complete inner drop evaporation, the fixed TPCL mode is observed as shown in the Fig. 2 until the sudden appearance of a daughter droplet. Hence the apparent completion of evaporation was deceiving due to the limitations of the imaging system which generally allows to measure contact angles as low as 2 − 5 • . We further quantified the evaporation of this resurfaced new drop which is marked as daughter droplet evaporation regime in fig. 2 (b) and (c). This emergence of daughter drop from an invisible (as viewed from the side) thin film motivated us to analyze the evaporation from the top view as depicted in fig. 2 (a). Despite the fact that the contact angles measured from side view (S6) indicate complete evaporation, the top view shows the presence of a thin-film (T6). After a few seconds, a daughter drop appears for which the contact angle and base diameter can be measured until it dries out. We have confirmed this observation three times and one can assure that in case of the water-diiodomethane double-emulsion drop evaporation, the diiodomethane drop gets pinned during evaporation. The pinning of the contact line might be a result of the presence of water vapor along the TPCL which does not allow the drop to change the mode of evaporation from fixed TPCL to moving TPCL. In case of fixed TPCL, with decreasing contact angle, the pinning force gradually increases and the drop experiences stronger resistance against contact line movement 37 . This consequences in the formation of such thin film and at one point the competition between pinning force and the inward contracting force, reported as snapping 38 reach their maxima and suddenly slippage occurs. Such a slip is not a rare occurrence for drop evaporation 34 . When a droplet evaporation shows alternating switch between fixed and moving TPCL mode, the reconfigured drop demonstrate the jump in the contact angle. But in the situation considered here, pinning of TPCL is not a random occurrence, it is prolonged until the end of the fixed contact line evaporation which is very end of drying period. This delayed fixed TPCL mode with significantly higher evaporation at the contact line of pinned thin film results in a quicker slip motion followed by a formation of daughter drop with a moderately larger contact angle. Non-axisymmetrical snapping off of the contact line can be attributed to the nonuniform surface roughness at the contact line. The observed evaporation on outer, inner and daughter droplet is compared with existing models and modifications were required to obtain the appropriate theoretical predictions. Hence, prior to propose modification to the well-established approaches, confirmation of the validity of considered theoretical models for single phase drops is of paramount importance. In the latter section we have validated the theoretical model with that of the single phase drop evaporation using similar drop-substrate configuration. This is then further extended to double emulsion case. At first, the water (outer)) and diiodomethane (inner) drops of different volume were analyzed separately which will be further compared to the double emulsion drop scenario. As in Fig. 3 (a) and (b), water droplets of different initial volumes are seen to evaporate 39,40 in a similar fashion. Theoretical estimations for fixed TPCL mode, discussed later in this paper, are compared with experimental observations as presented in Fig. 4 (a) for a range of droplet volumes with varied volatility and diffusion property. Single phase drop evaporation A special attention is given to the complete evaporation of a 0.25µL diiodomethane droplet since the inner diiodomethane drop for double emulsion drop evaporation study is of same volume. Figure 4 (b) suggests that in case of single phase diiodomethane drop the number of 'stickjump' 34 scenario is observed. To pinpoint this observation, the position of two ends of the base diameter is traced as depicted in Fig. 4 (b). Similar observations were made for larger (1µL) diiodomethane drop except the fact that the increase in volume increases the drying time. Larger contact angle hysteresis is one of the factors that causes such 'stick-jump' behavior 34 . Hence, hysteresis was measured by tensiometer (K100, KRÜSS GmbH, Hamburg, Germany) and goniometer (DSA100E, KRÜSS GmbH). The contact angle hysteresis for oleo-phobic substrate, acrylic sheet and adhesive surface were 36 • , 30 • and 54 • , respectively. The high contact angle hysteresis contributes to the dominance of fixed TPCL mode 41 . Prior to discussing the theoretical aspect of the evaporation, it is worth mentioning that the time fraction required to complete the first mode (fixed TPCL) of evaporation of liquids with different volume and volatility falls around the same time range. In Fig. 5, for all liquid-solid combinations considered in this study, a fixed TPCL mode of evaporation is observed for a time period more than half of the total drying time. For comparing these results with double emulsion drop case, a single phase droplet model is adapted. For evaporation of a sessile droplet, contact line dynamics and surface morphology complicate the scenario where contact angle as well as base diameter of the TPCL 24,42 dictate the dynamics. The diffusion model proposed by Popov 43 for a sessile droplet (of mass M and density ρ) with contact radius, R c and contact angle, θ , takes the form of equation 1 : dM dt = ρ dV dt = −πRD∆c f (θ ) (1) Here, the function f (θ ) is given by the following expression f (θ ) = sinθ 1 + cosθ + ∞ o 1 + cosh(2θ τ) sinh(2πτ) tanh[(π − θ )τ]dτ (2) where, τ is non-dimensional drying time 43 . As the droplet gets pinned during fixed TPCL mode, the loss in mass translates into corresponding decrements in height and contact angle until a critical contact angle is at-tained. Though a droplet would like to evaporate without any additional penalty in its energy, by maintaining equilibrium contact angle, the pinning of the TPCL and dominant evaporation flux across the liquid-air interface create resistance against the smooth decrease in base diameter 44 . But as a critical angle is approached, the evaporation flux at the TPCL becomes large enough to surpass the energy barrier resulting in the change of base diameter. Occasionally, in this second mode of evaporation the stick-slip 45 or stick-jump behaviour of TPCL is noticed 34 . Assuming the spherical cap assumption along with the functional variation of contact angle (Eqn 2), the instantaneous change of droplet contact angle in moving TPCL with fixed contact angle mode can be derived as 33 : dθ dt = − D∆c ρR 2 (1 + cosθ ) 2 f (θ ) (3) The estimations, predicted by Eq. 3, are compared with experimental data in Fig. 4 (a), i.e., for the single phase droplet scenario, which clearly suggests that the selected theoretical model (Eqn. 3) can predict the droplet evaporation dynamics. However, one can detect deviation from theoretical predictions for highly volatile liquid drop as noticed for toluene in Fig. 4 (a). Double drop's disparity from single drop Following to the validation of single phase, the similar model was extended for all three drops (outer, inner and daughter) of double emulsion droplet. For a double emulsion droplet, one might expect that after the complete outer drop evaporation, the inner drop attains the Young's configuration. The inner diiodomethane drop attains the equilibrium inside water medium with contact angle θ DI,w and by the time it is exposed to air, contact angle reduces to θ * . For analyzing the significance of θ * the contact angle of the diiodomethane in saturated water vapor (θ DI,sat )as well as in water medium (θ DI,w ) is also measured and compared. The contact angles of diiodomethane in air, water medium and in saturated water vapor (θ DI,sat ) are provided in Table 2 and Fig. 2(b). The new configuration with contact angle, θ * less than θ DI,air and θ DI,sat , suggests the change in the local surface energy of the solid, i.e., solid-air interfacial energy. The transition from θ DI,w to θ * is due to the evaporation of the outer drop and the appearance of new configuration compared to θ DI,air and θ DI,sat might be due to the adsorption of water as well as diiodomethane vapor. We assume, this change follows similar behaviour as θ w , hence in Fig.2(b) we connect θ DI,w to θ * with a dashed line parallel to the evaporation of single phase water drop evaporation passing through θ DI,a and θ DI,sat . As observed for saturated environment wettability studies 46 , the saturated vapour not only maintains the contact angle closer to theoretically predicated Young's angle, but also circumvent the decrements in the contact angle due to evaporation 46,47 . During the outer water drop evaporation the surrounding medium for the inner drop gets saturated with the water vapour which further gets adsorbed on the solid surface. Once the outer liquid cushion is evaporated, inner drop is suddenly exposed to a modified surface energy interface that results in another marginal decrease in the contact angle, ∆θ ∼ 5 • (inset of Fig. 2(b)) with sudden increase in base diameter, Fig. 2(c)) of the inner drop. ∆Φ ∼ 0.1 mm (inset of It is important to comment on the theoretical modeling of evaporation, validated in Fig. 3 (b), for doubleemulsion droplet case, in particular, for inner and resurfaced daughter drop. It is evident that, double-emulsion drop evaporation follows the same modeling as of single phase until the onset of the transition regime. One can argue that the role of the inner drop is negligible for outer drop evaporation, hence the continuous line, the behavior predicted for a single phase drop, perfectly matches with the experiential results. If we consider a single phase water drop (without inner drop) of the total volume equal to the volume of double emulsion drop, the fixed TPCL evaporation in air can be observed up to t dry,w,a as shown in the Fig.2 (b). But the presence of the inner drop alters the total evaporation time as well as the mode of the evaporation. Since the evaporation of inner and daughter drop is mainly of fixed TPCL, it is worthwhile to validate the single phase drop evaporation models for inner and daughter drop. As presented for outer drop evaporation, the continuous lines in Fig. 2(b) represent the theoretical behaviour predicted by Eq. 3 for inner and daughter drop. It is evident that this approach either over predicts or under predicts for inner and daughter drops. We carefully performed the parametric analysis and concluded that the presented modelling is sensitive to the concentration and diffusion of the phases involved in the evaporation. For a double-emulsion drop evaporation case, it is debatable whether outer or inner drop properties play a role or combined properties need to be considered. The tuning of the theoretical model suggests that while considering the model for inner drop and daughter drop evaporation, volume weighted averaging of the concentration gradient and the diffusion coefficients predicts the behaviour closer to the experimental observations. The dashed lines in Fig. 2 (b) depict the modified theoretical predictions with appropriate averaged proper-ties of liquids. This modification in the diffusion and concentration is attributed to the presence of outer phase, i.e., vapor phase of the outer liquid in the vicinity of the TPCL and the liquidair interface. This might have altered the properties dominating the phenomenon; hence using only single phase properties (continuous lines -blue) over or under predict the experimental observations. This signifies the role of altered surrounding conditions due to the evaporation of two different liquids. Proper quantification of this alteration and the physical explanation may interest researchers for a detail study. Further, the relative solubility of the two associated phases is another aspect which needs to be considered. But in the present study, we have not varied the relative solubility which would be another interesting aspect to investigate. It is worthwhile to note that the evaporation experiments were conducted for different volumes of the inner and outer drops (0.25 and 0.50 µL inner diiodomethane drop in 2, 3 and 5µL outer water drop). Similar evaporation patterns can be observed with the exception of the evaporation rate which depends on the initial volume and contact angle as depicted in Fig. 3 (a) for single phase drop evaporation. After proposing the revised model for double emulsion droplet evaporation, the occurrence of daughter droplet is investigated in the upcoming section. Resurfacing of evaporating drops Interestingly, the occurrence of daughter drop was absent in the case of single phase scenario, i.e., diiodomethane drop on the same substrate. It is important to investigate why such a resurfacing was never observed in the case of single phase droplet evaporation. Therefore, experiments with evaporating water droplet on a number of substrates including acrylic, copper, aluminum sheets, micro textured and adhesive surfaces were performed. Surprisingly, only adhesive coated surfaces demonstrated the resurfacing of water drops as can be seen in Fig. 6 (a). The top two panels depict the top and side views of the evaporating drops. Change in contact angle with corresponding base diameter is shown with filled and empty symbols, respectively. The dominance of fixed TPCL evaporation mode convinced us to conclude that if fixed TPCL evaporation can be significantly prolonged over the drying time, one can observe the resurfacing of the daughter droplet. Fig. 6 (a) mainly focuses on the end of the fixed TPCL evaporation until it reaches the smallest measurable contact angle. Different stages presented in the panel are denoted along the change in the contact angle with roman numbers (I -IV). Care-ful microscopic observation of the adhesives layer suggests that the surface contains micro-nano features ( Fig. 6 (b) -I). Thus the surface facilitates the pinning of the contact line which eventually forces the drop to form a film before the daughter drop formation. In this case, while the daughter drop resurfaces, a big jump in contact angle (from 0 to 30 • ) is noticed as shown in Fig. 6 (a). To view the film and resurfacing of the drop, the camera viewing angle was slightly tilted (∼ 2 • ) which demonstrates thin film (III and IV in side view) corroborating the presence of liquid film in corresponding top views. The pinning of the TPCL can be confirmed by ring like impression similar to 'coffee stain ring' as shown in Fig. 6 (b) -II. This ring acts as a peripheral pinning location that holds the droplet until it converges to a thin film with vanishing contact angle. However, it is well established that, if the evaporating flux at TPCL is significantly larger than the evaporation flux across the liquid-air interface, it surpasses the pinning strength and hence, moving TPCL evaporation can be observed. In case of the evaporation on adhesive surface, the evaporation flux at TPCL is not large enough until the drop attains the form of a thin film. The moment the evaporation across the air-liquid interface of thin film is not dominant enough, resurfacing triggers into the formation of daughter drop as shown in Fig. 6 (a) IV − VI. However, a double emulsion or a single diiodomethane drop do not exhibit such behavior on this particular substrate emphasizing on the dependency or sensitivity of this phenomenon on surface-liquid combinations. With the observation of resurfacing, we identified a critical aspect that dictates the formation of the daughter droplet, i.e., pinning of the contact line for entire evaporation of the droplet. To validate this proposed hypothesis, we artificially engineered a physical barrier by engraving a ring on an acrylic substrate. This artificial ring of 1.5mm diameter is of the same dimension as that of the base diameter of water drop of a given volume. The top view of the engraved acrylic substrate is shown in Fig. 6 (b) -III. For comparison, we initially studied the water drop evaporation on an acrylic substrate without any ring as shown in Fig. 6(c) which clearly demonstrates the usual modes of the evaporation. Top and side views at three different time instants also depicts the movement of the TPCL. Since there is no pinning of the three phase contact line, we cannot expect a thin film phase and subsequent resurfacing. Discordantly, when evaporation of water drop is observed on the same substrate with a ring (with micro metric depth of ∼ 100µm), we observe the prolongation of fixed TPCL mode over almost the entire drying period as shown in Fig. 6 (d). The drop remains pinned along the TPCL (S I, S II ; T I, T II) until the contact angle reaches zero and forms a thin film (T III; S III). This is followed by the resurfacing of a daughter droplet (S IV, S V ; T IV, T V) as hypothesized, with a jump in contact angle as depicted in the plot along with associated decrease in base diameter. Thus, by employing our hypothesis i.e., forcefully pinning the TPCL, resurfacing of a daughter droplet is demonstrated on a regular substrate which otherwise doesn't behave similarly. Conclusions The evaporation of single phase drops well agree with existing theoretical model, however significant deviations have been observed for the double drop case. A modified theoretical approach agrees with the observed evaporation modes for the double emulsion drops. Evaporation of such droplet exhibits the commonly observed modes of evaporation with two new regimes in its drying time. The transition regime from outer to inner drop constitutes a sudden spreading of the inner droplet which results in a wetting scenario that is different from the theoretically expected equilibrium configuration for similar liquid-solidvapour combination. The sudden change in the contact angle imprints the complete drying of the outer drop liquid and can be attributed to complete exposure of the inner droplet to environment. A resurfacing of a daughter droplet is witnessed after the commonly identified completion of the evaporation. This observation is critically investigated and attributed to the pinning of the three phase contact line. Later, we forcefully pinned the three phase contact line of a single phase droplet by carefully engineering a substrate and a mechanism of daughter droplet resurfacing from thin film is established. Fig. 1 1Evaporation of double emulsion drop (images are not drawn to scale): Panel A: (I) generation of double emulsion drop at the tip of the co-axial needle (II) deposition of double emulsion drop (III) evaporating double emulsion drop ; Panel B: Schematic of the double emulsion drop evaporation process:(I) evaporation of the outer phase , (II) disappearing outer phase drop, (III) evaporation of the inner phase (IV) resurfacing of daughter droplet ; Panel C: experimentally observed four stages (I-IV) of double emulsion drops. Synthetic colouring is used to distinguish between inner and outer drop phase. Fig. 2 2Evaporation of water-diiodomethane double emulsion droplet (a) Side views (S1-S4) and corresponding top views (T1-T4) of an evaporating double emulsion drop : evaporation of the outer phase followed by simultaneous evaporation of both the drops in a transition phase (S1-S4 ; T1-T4). Later (S5 ; T5), the outer phase is completely dried and inner drop evaporates until a film phase (S6 ; T6) with nearly zero contact angle is achieved. S7 and T7 show the resurfacing of the film into a small daughter droplet. (b) Dynamic variations in the contact angle of the three evaporating drops -outer, inner and daughter. Dynamic variations in the contact angle of the three evaporating drops -outer, inner and daughter. Symbol representing the experimental observation, dotted and continuous lines are theoretical predictions with appropriate models. Inset shows the transition regime where the inner drop attains θ * -it is different from (θ DI,a ), (θ DI,w ) or (θ DI,sat ) on the same substrate. (c) Corresponding change in the base diameter depicting the same phenomenon magnified in the inset. Fig. 3 3Evaporation of single phase droplets: (a)Contact angles and (b) base diameter of evaporating water droplets of different volumes demonstrate both fixed and moving TPCL during the drying time. Fig. 4 4(a) Comparison between experimental data and theoretical model 33 for fixed TPCL mode for single phase droplet evaporation (b) evaporation of a diiodomethane droplet − fixed TPCL mode for the first half of the drying time is followed by intermittent moving TPCL mode with a number of stick-slips. Fig. 5 5Droplets of liquids with varying volatility follows fixed TPCL mode more than half of its drying period and the onset of transition between two modes of evaporation occurs approximately at same time fraction. Fig. 6 6Evaporation and resurfacing of droplet. (a) Evaporation on adhesive surface incorporates thin film phase and resurfacing into a smaller droplet. The grey circles with numbers refer to the corresponding top and side views. Experimental data for contact angles and base diameter (non dimensional) are presented with filled and empty symbols. (b) I − microscopic view of the adhesive surface shows the micro-nano features of the surface (b) II − a ring like impression creates local pinning sites along the TPCL (b) III − a micro metric ring is machined on an acrylic sheet to dummy the ring effect (c) water drop evaporation on a acrylic sheet without ring shows no film phase or resurfacing (d) with the ring on the acrylic sheet distinct thin film phase followed by resurfacing of a daughter droplet is seen. Table 1 1Surface roughness of the substratesSubstrate Measurement technique Roughness (water drop) (nm) acrylic Optical Profilometry 4-5 oleophobic Atomic Force Microscopy 10-30 adhesive Optical Profilometry 130-135 Table 2 2Contact angles of different drop liquids on the substrates used for experimentDrop liquid Liquid boiling point Substrate Medium Contact Angle ( • C) ( • ), ±3 water 100 oleophobic air 80 water 100 adhesive air 105 water 100 acrylic air 78 water 100 acrylic with ring air 85 diiodomethane 181 oleophobic air 70 diiodomethane 181 oleophobic water 120 diiodomethane 181 oleophobic water vapor (sat.) 65 toluene 110.6 oleophobic air 38 AcknowledgmentThe authors thank Natural Sciences and Engineering Research Council (NSERC) for the financial support in the form of Grant No. RGPIN-2015-06542. The authors highly acknowledge Dr. Xuehua Zhang for her suggested edits to enhance the representation. 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Let ∆ be a (d−1)-dimensional pure f -simplicial complex over vertex set [n]. In this paper, it is proved that ∆ being minimal CM implies d ≥ 3 and n = 2d. It is also indicated that the recent work of[6]implies that shellable condition on a pure simplicial complex ∆ is identical with existence of a full series of CM subcomplexes of ∆.

2103.16078

bc5cd765701a6e7b1fc5a71d93b9aafdc727ea6a

Simplicial complexes which are minimal Cohen-Macaulay * 1 Feb 2022 Yanyan Wang School of Mathematical Sciences Shanghai Jiao Tong University Tongsuo Wu School of Mathematical Sciences Shanghai Jiao Tong University Simplicial complexes which are minimal Cohen-Macaulay * 1 Feb 2022Simplicial complexCohen-Macaulayshellableshelled overf -simplicial complex 2020 AMS Classification: Primary: 13F5505E45Secondary: 13H1055U10 Let ∆ be a (d−1)-dimensional pure f -simplicial complex over vertex set [n]. In this paper, it is proved that ∆ being minimal CM implies d ≥ 3 and n = 2d. It is also indicated that the recent work of[6]implies that shellable condition on a pure simplicial complex ∆ is identical with existence of a full series of CM subcomplexes of ∆. Introduction For a natural number n and 1 < d < n, let where 2 [n] is the power set of [n]. A simplicial complex ∆ over a vertex set [n] is a subset of 2 [n] , which has the hereditary property under inclusion and is such that {i} ∈ ∆ holds for all i ∈ [n]. Recall that a facet of ∆ is a maximal element with respect to inclusion, and the facet set of ∆ is denoted as F (∆). The dimension dim ∆ of ∆ is the maximal number |F | − 1, where F ∈ F (∆) runs over all facets of ∆. If dim ∆ equals to |F | − 1 for each facet F , then ∆ is said to be pure. Let ∆ (i) = {F ∈ ∆ | |F | ≤ i + 1} be the i'th skeleton of ∆. Cohen-Macaulay (abbreviated as CM) property is one of the central research topics in commutative algebra and the rich and deep homological achievements have fruitful applications in combinatorial aspects of commutative rings ( [16,7,3,17,14,10]). In combinatorial commutative algebra, shellable and pure simplicial complexes are the main source of CM simplicial complexes. In a most recent work [6], Dao, Doolittle and Lyle discovered a new important combinatorial property of a CM simplicial complex ∆, i.e., ∆ F ∩ F is pure of dimension |F | − 2 for any facet F of ∆, where F (∆ F ) = F (∆) {F }. Based on the property, the notion of a minimal CM simplicial complex ∆ is introduced and studied. To be more precisely, ∆ is called minimal CM if ∆ is CM but no ∆ F is CM for any facet F of ∆. Acyclic behavior of a minimal CM ∆ is studied and, sufficient conditions are provided for a complex to be minimal CM . Many interesting examples of minimal CM complexes are also exhibited. Recall also from Zheng [18] the other important combinatorial property of a CM simplicial complex, i.e., CM simplicial complexes are connected in codimension one, i.e., for any distinct facets F and G, there is a sequence F = F 0 , F 1 , . . . , F r = G of facets such that |F i ∩ F i+1 | = |F i+1 | − 1 for all i = 0, 1, . . . , r − 1. In this paper, we use [6, Lemma 3.1] to study the exact relation of shellable and CM properties for a pure complex ∆, and we study the condition for a minimal CM f -simplicial complex to be acyclic. In Section 2, we recall some work of [6] and, give a brief survey on f -simplicial complexes. In Section 3, we first indicate that shellable condition on a pure complex ∆ is identical with CM properties of a full series of subcomplexes of ∆, and then we use this observation to construct nontrivial examples of pure shellable complexes by taking advantage of CoCoA in an algorithmic approach, after applying Eagon-Reiner theorem. In Section 4, we compute the dimension of the subspace ker(∂ r ) in a reduced chain complex of the simplex [n] , and apply it to deduce that a minimal CM f -simplicial complex exists in [n] d implies n = 2d. Preliminaries For a (d − 1)-dimensional simplicial complex ∆, there is a related chain complex of K-spaces: C : 0 −→ C d−1 ∂ d−1 −→ C d−2 ∂ d−2 −→ · · · −→ C 1 ∂ 1 −→ C 0 −→ 0, where C i is a free K-module with basis set {σ ∈ ∆ | |σ| = i + 1}, while for any 1 ≤ k 1 < k 2 < · · · < k r+1 ≤ n, ∂ r (k 1 k 2 . . . k r+1 ) = r+1 i=1 (−1) i−1 k 1 . . . k i−1ki k i+1 . . . k r+1 . For each i, recall that im ∂ i+1 ⊆ ker ∂ i holds, and the quotient K-spacẽ H i (∆) =: ker ∂ i /im ∂ i+1 is called the i th homology group of ∆. IfH i (∆) = 0 holds for all i, then ∆ is said to be acyclic. Clearly, ∆ is acyclic if and only if the corresponding chain complex C is an exact sequence. Recall that a cone is always acyclic (see, e.g., [17]), where ∆ is called a cone if there exists a vertex such that all facets contain it as an element, and note that dimH 0 (∆) + 1 is the number of connected components of ∆ ([17, Proposition 5.2.3]). Throughout, let K be a field and let S = K[x 1 , . . . , x n ] be the polynomial ring over K. Throughout, unless otherwise specifically stated, let ∆ be a (d − 1)-dimensional pure simplicial complex with vertex set [n], where ∆ = ∅, {∅} and ∆ is not a simplex. We first recall some work of [6] on minimal Cohen-Macaulay simplicial complexes. For a facet F of a simplicial complex ∆, let ∆ F = G | G ∈ F (∆) {F } . ∆ is called a shelling move of ∆ F if ∆ F ∩ F is pure of codimension 1, i.e., ∆ F ∩ F is generated by some nonempty subset of ∂F . Lemma 2.1. ([6, Lemma 3.1]) If a simplicial complex ∆ is CM with |F (∆)| ≥ 2, then ∆ is a shelling move of ∆ F for any facet F of ∆. A CM simplicial complex ∆ is called minimal CM, if either it is a simplex, or else |F (∆)| ≥ 3 and, no ∆ F is CM for any facet F of ∆. Here the definition of minimal CM is slightly different from that of [6], since we are mainly interested in the nonempty simplicial complexes. Then, the shelling move property implies the following: Theorem 2.2. ([6, Theorem 3.2]) Let ∆ be a (d − 1)-dimensional CM simplicial complex, which is not minimal. Then there exists a minimal CM subcomplex Γ and a series of facets F j , . . . , F 1 of ∆, such that each Γ ∪ F i , . . . , F 1 is CM and, each Γ ∪ F 1 , . . . , F i+1 is a shelling move of Γ ∪ F 1 , . . . , F i . In [6], ∆ is said to be shelled over Γ. Clearly, shelled over is a kind of generalization of shellable for a simplicial complex. Next, we record the following result, which is needed in this paper: (1) dimH i−1 (∆ F ) =    dimH i−1 (∆), if 0 ≤ i < d dimH i−1 (∆) − 1, if i = d . ( 2) f k−1 (∆ F ) =    f k−1 (∆), if 0 ≤ i < d f k−1 (∆) − 1, if i = d . (3) depth ∆ = depth ∆ F . Surely, this theorem together with Reisner theorem imply that a minimal CM simplicial complex is acyclic. Now we give a brief survey on some related established results on f -simplicial complexes. For any square-free monomial ideal I of S, let G(I) be the set of minimal monomial generators, and let sm(I) be the set of square-free monomials. For the ideal I, recall that there exist two related simplicial complexes, i.e., the nonface simplicial complex δ N (I) =: { F ∈ 2 [ n ] | X F ∈ sm(S) sm(I) } of I and the facet simplicial complex δ F (I) =: F ∈ 2 [ n ] | X F ∈ G(I) of the clutter G(I). If they possess a same f -vector, then the ideal I is called an f -ideal. For a simplicial complex ∆, if its facet ideal I(∆) =: {X F | F ∈ F (∆)} is an f -ideal, then ∆ is called an f -simplicial complex. A graph G is said to be an f -graph, if the edge ideal I(G) is an f -ideal. Note that in defining an f -graph G, G is regarded as a simplicial complex of dimension no more than 1, although we do have I(G) = I Ind(G) , where Ind(G) is the independence simplicial complex of the graph G. Refer to [1,13,9,12,11] for further related studies. For a simplicial complex ∆ on the vertex set [n], let ∆ c =: {F | F c ∈ F (∆)} i.e. = {[n] G | G ∈ F (∆)} . The definition of an f -simplicial complex seems to be reasonable with hindsight, due to the following two theorems on f -ideals. Note that G(I) is said to be an L-set (U-set, respectively) if the set of all degree d − 1 factors of elements of G(I) has exactly n d−1 elements (respectively, the set of degree d + 1 square-free monomials extended from elements of G(I) has cardinality n d+1 ). If G(I) is both a U-set and an L-set, then G(I) is an LU-set. With the bijection from X α to {i ∈ [n] | i ∈ α} (e.g., x 1 x 3 x 4 → {1, 3, 4}), one obtains the notion of an L-set (U-set, LU-set respectively) for the facet set F (∆). Recall also the following recently discovered result: Theorem 2.6. ([4, Theorem 4.1]) ∆ is an f -simplicial complex, if and only if ∆ c is an f - simplicial complex. Equivalently, a square-free monomial ideal I of S is an f -ideal if and only if the Newton complement dual idealÎ = x 1 x 2 · · · x n /u | u ∈ G(I) of I is an f -ideal. It is clear that Theorem 2.6 follows easily from Theorem 2.5 for a pure simplicial complex ∆. Recall that for an f -graph G, it is proved that the complement graph G is bipartite, thus Ind(G) = G. Recall that all f -graphs are pure shellable as a graph, i.e., the independence complex Ind(G) is pure and shellable ([8, Theorem 6.5]), while the definition of an f -graph is actually an f -simplicial complex of dimension less than or equal to 1. Thus [8, Theorem 6.5] may be re-stated as the following: If ∆ is an f -simplicial complex of dimension less than or equal to 1, then the homogeneous complement simplicial complex ∆ ′ =: [n] 2 F (∆) is pure shellable. Based on this observation, it is natural to ask the following question: Question 2.7. For a pure f -simplicial complex ∆ of dimension d − 1, is the homogeneous complement simplicial complex ∆ ′ =: σ | σ ∈ [n] d F (∆) of ∆ shellable? We do not know counterexample in [5] 3 and in [6] 3 . But it fails in [8] 4 . We will give a negative answer in Example 3.5. We remark that there exist a lot of pure f -simplicial complexes which are not CM when d − 1 ≥ 2. Finally, we claim that there exist f -simplicial complexes which are minimal CM: constructed in [2]. It is noticed in [11] that ∆ (hence, ∆ c ) is an f -simplicial complex. Then we take advantage of Eagon-Reiner theorem ([10, Theorem 8.1.9]) and CoCoA ( [5]) to check that both simplicial complex ∆ and its complement ∆ c are minimal CM. Example In particular, neither ∆ nor ∆ c is shellable, which is hard to check without the notion of minimal CM (refer to [2, Example 7.7] for a general theoretical treatment). Note that a permutation on the set [6] may produce a new simplicial complex ∆ 1 , which has the same property with ∆. For example, the permutation (1, 2, 3, 4, 5, 6) acts on F (∆) and produces Pure shellable versus Cohen-Macaulay We begin with the following immediate consequence of Theorem 2.5: Corollary 3.1. Let ∆ be a (d − 1)-dimensional pure f -simplicial complex with vertex set [n]. Then we have depth(∆) =    d, if ∆ is CM d − 1, if ∆ is not CM. Proof. Assume that ∆ is not CM. Since ∆ is a (d − 1)-dimensional pure f -simplicial complex, F (∆) is an L-set, thus ∆ (d−2) = [n] ddepth(∆) = 1 + max{i | the i ′ th skeleton ∆ (i) is CM.}. If the minimal subcomplex Γ in Theorem 2.2 is a simplex, say, Γ = F 0 , then it follows by Lemma 2.1 that the following is a shelling of ∆, thus ∆ is shellable: F 0 , F 1 , F 2 , . . . , F j . To be more precisely, we have Theorem 3.2. For a pure simplicial complex ∆, the following statements are equivalent: (1) ∆ is shellable. (2) There exists a full sequence of subcomplexes ∆ i such that all ∆ i are CM, i.e., there is a total order F j , F j−1 , . . . , F 1 , F 0 of all facets of ∆ such that each ∆ i =: F 0 , F 1 , . . . , F i is CM for j ≥ i ≥ 1, or equivalently, each ideal I(∆ c i ) has a linear resolution. Proof. (1) =⇒ (2) : Let F 0 , F 1 , . . . , F j be a shelling of ∆ and let ∆ i = F 0 , F 1 , . . . , F i . Then for any i with 1 ≤ i ≤ j, ∆ i =: F 0 , F 1 , . . . , F i is pure and shellable, thus is CM. (2) =⇒ (1) : Let F j , F j−1 , . . . , F 1 , F 0 be a full sequence of facets of ∆ such that each ∆ i =: F 0 , F 1 , . . . , F i is CM for j ≥ i ≥ 1. Then by Lemma 2.1, ∆ j−1 ∩ F j , ∆ j−2 ∩ F j−1 , . . . ∆ 1 ∩ F 1 are all pure of dimension dim ∆ − 1. By definition, F 0 , F 1 , . . . , F j is a shelling of ∆, thus ∆ is a shellable simplicial complex. The rest statement follows from Eagon-Reiner theorem, and is convenient for checking by applying CoCoA. Clearly, Theorem 3.2 shows the exact relation between the conditions of shellable and CM for a pure simplicial complex. It also exhibits the importance of Lemma 2.1. As is well-known, it is in general a hard work to check if a pure simplicial complex is shellable. It seems that the new concept shelled over could open an algorithmic gate on attacking this problem, based on the algebraic characterization of a CM simplicial complex by Eagon-Reiner theorem. Refer to Examples 2.8, 3.4 and 3.5 for concrete operations and calculations. When considering the condition of connected in codimension 1 ([18, Proposition 1.12]), we have the following easy observation: In the following, we consider simplicial complexes of kind (8, 4) and apply Theorem 3.2 and Example 2.8 to the following construction: Example 3.4. We start from the set A = {1345, 1347, 1358, 1367, 1368, 1456, 1468, 1478, 1567 is tested via CoCoA to have linear resolution. Note that ∆ 1 is not an f -simplicial complex since 127 is not in the lower set of F (∆ 1 ), i.e., F (∆ 1 ) is not an L-set. We do not know if ∆ 1 is shellable. (2) Inspired by the previous construction, we now construct an f -simplicial complex Note that for the same CM simplicial complex ∆ 2 , the first minimal CM subcomplex is F 0 and it has only one facet, while the second minimal CM subcomplex is Γ and it has ten facets. This is the end of Example 3.4. ∆ 2 = {F | F ∈ A ∪ C} , where C = {1235, Finally, note that Example 2.8 provides a very well-distributed simplicial complex, i.e., each number r in [6] appears 5 times in the facets. Motivated by this observation, we now construct a very well-distributed simplicial complex whose facet set contains 34 elements in [8] {1234, 1235, 1246, 1247, 1258, 1345, 1358, 1367, 1368, 1378 We checked the following: (1) D is very well-distributed, i.e., it has type 1 17 2 17 3 17 4 17 5 17 6 17 7 17 8 17 . (2) D is an LU-set over [8], so that adding any element from [8] 4 D can generate a pure f -simplicial complex Γ c . (3) The ideal I(∆ c ) has the following linear resolution, thus ∆ is not CM: 0 −→ R(−8) 2 −→ R(−7) 21 ⊕ R(−8) −→ R(−6) 68 ⊕ R[−7] 2 −→ R(−5) 81 ⊕ R[−6] −→ R(−4) 34 −→ R. Note that ∆ c has the same properties. Among the 36 f -simplicial complexes Γ obtained in (2) Deleting any element will result in a homogeneous complement of some Γ c in (2). We use CoCoA to calculate the 36 I((Γ c ) ′ ) and, find 15 CM simplicial complexes Γ ′ . The following are all elements when one of which is deleted, the corresponding I((Γ c ) ′ ) has linear resolution, thus Γ ′ is CM: Note that (Γ c ) ′ = (Γ ′ ) c always holds true. It also gives a negative answer to Question 2.7. This is the end of the example. Note that in many examples of CM simplicial complexes ∆, we have ∆ F ∩ F = G | G ∈ ∂F holds true for most of the facets F , but not in all cases, as the following example shows: Minimal Cohen-Macaulay f -simplicial complexes In this section, we study properties of f -simplicial complexes which are minimal CM. For this, we need the following: holds true. Since C r−2 is a free K-module with basis [n] r−1 , we have got a system of homogeneous linear equations, which consists of n r−1 equations with n r variable x i 1 i 2 ...ir . We write these x i 1 i 2 ...ir as well as i 1 i 2 . . . i r−1 in lexicographic order, and consider the rank of the coefficient matrix M ( n r−1 )×( n r ) . Clearly, the first n−1 r−1 columns, i.e., the coefficients of x 1i 2 ...ir , are linearly independent. It can be checked that each other column is a linear combination of them. Furthermore, for i 1 i 2 . . . i r with 1 ∈ {i 1 , . . . , i r }, note that Lemma 4.1. Let C : 0 −→ C n−1 ∂ n−1 −→ C n−2 ∂ n−2 −→ · · · ∂ 2 −→ C 1 ∂ 1 −→ C 0 −→ 0∂ r−1 (i 1 i 2 . . . i r ) = i 2 . . . i r − i 1 i 3 . . . i r + · · · + (−1) r−1 i 1 . . . i r−1 , The exact details are essentially the same with the verification of the fact that a cone is acyclic, refer to [17,Proposition 5.2.5]. Finally, we proved that the dimension of the vector space ker ∂ r−1 is the following n r − n − 1 r − 1 = n − 1 r . Note that the key to calculate the kernel of general ∂ i−1 is the equality n i = n − 1 i + n − 1 i − 1 . This is the end of the verification. Remark. We illustrate the proof in computational way in two particular cases. The first case is [6] 3 , and we check that ker ∂ 1 has dimension n−1 2 , where n = 6. In fact, let 1≤i<j≤6 x ij {i, j} ∈ ker ∂ 1 , we have 0 = 1≤i<j≤6 x ij ({j} − {i}). Since C 0 in the chain complex is a free K-module with basis {1}, {2}, . . . , {6}, we get the following system of linear equations:                          6 i=1 −x 1i = 0 x 12 − 6 i=2 x 2i = 0 x 13 + x 23 − 6 i=3 x 3i = 0 3 i=1 x i4 − x 45 − x 46 = 0 4 i=1 x i5 − x 56 = 0 5 i=1 x i6 = 0. The coefficient matrix is           −1 −1 −1 −1 −1 1 −1 −1 −1 −1 1 1 −1 −1 −1 1 1 1 −1 −1 1 1 1 1 −1 1 1 1 1 1           , hence, the solution of the system has exactly 10 = 5 2 = 6 2 − 5 1 free variables, they are all k ij except these {i, j} including 1. Note that for a general n, dim ker(∂ 1 ) = n−1 2 is verified in an exactly same way. Note also that in the coefficient matrix, we have column vector relation v 23 = v 12 − v 13 . 12 13 14 15 16 17 18 23 24 25 26 27 28 34 35 36 37 38 45 46 47 48 56 57 58 67 68 78 123 1 -1 1 Since C 1 in the chain complex is a free K-module with basis set {ij | 1 ≤ i < j ≤ 8}, we get a system of linear equations, which consists of 8 2 linear equations with 8 3 variables x ijk . We write out the coefficient matrix M in Table 4.1 and it is clear that the row (in the table) rank of the matrix is not less than 7 2 . Actually, after doing Gaussian elimination via excel, it is calculated that the matrix has rank 7 2 =: 21. We also checked the rank by taking advantage of CoCoA ([5]) 5.3.3. Besides, all x ijk except x 1jk 's can be chosen as free variables. Certainly, there is an alternative explanation as appeared in Table 4.1. This shows dim ker(∂ 2 ) = 8 3 − 7 2 = 7 3 , as is claimed. Note that for a general n, dim ker(∂ 2 ) = n−1 3 is verified in a completely same way. Note also that in Table 4.1, we have row vector relation v T 234 = v T 123 − v T 124 + v T 134 , which is a particular case of (1). We get n(n − 1) · · · (n − d + 1) 2 · d! = (n − 1)(n − 2) · · · (n − d + 1) (d − 1)! , since a minimal CM simplicial complex is always acyclic by [6]. Thus we have n = 2d. By Mayer-Vietoris long exact sequence theorem, we get Proof. For any facet F of ∆, let ∆ 1 = ∆ F , ∆ 2 = F , ∆ 3 = ∆ 1 ∩ ∆ 2 =: ∆ F ∩ F . Then ∆ = ∆ 1 ∪ ∆ 2 . By [15, Theorem 25.1, page 142], we have the following long exact sequence of K-spaces: 0 −→H d−1 (∆ 3 ) −→H d−1 (∆ 1 ) ⊕H d−1 (∆ 2 ) −→H d−1 (∆) ∂ d−1 −→H d−2 (∆ 3 ) −→H d−2 (∆ 1 ) ⊕H d−2 (∆ 2 ) −→H d−2 (∆) ∂ d−2 −→ · · · · · · · · · ∂ 2 −→ H 1 (∆ 3 ) −→ H 1 (∆ 1 ) ⊕ H 1 (∆ 2 ) −→ H 1 (∆) ∂ 1 −→H 0 (∆ 3 ) −→H 0 (∆ 1 ) ⊕H 0 (∆ 2 ) −→H 0 (∆) −→ 0. Note that Lemma 2.1 implies that ∆ 3 is pure of dimension d − 2, while it follows from Theorem 4.3 thatH i (∆) = 0 holds for all i, andH i (∆ 2 ) = 0 holds true clearly. Then 0 −→H i (∆ 3 ) −→H i (∆ 1 ) −→ 0 is an exact sequence for every i. [n] =:{1, 2, . . . , n}, [n] d =: {A ∈ 2 [n] | |A| = d}, Corollary 2. 4 . 4Let ∆ be a minimal CM complex over vertex set [n] with dimension d−1. Theñ H d−1 (∆ F ) = 0 holds true for any facet F of ∆. Proof. This is an immediate consequence of Lemma 2.1, Theorem 2.3 and [15, Theorem 25.1, page 142]. Theorem 2. 5 . 5([8, Theorem 2.3]) Let S = K[x 1 , . . . , x n ], and let I be a square-free monomial ideal of S with the minimal generating set G(I), where all monomials of G(I) have a same homogeneous degree d. Then I is an f -ideal if and only if, the set G(I) is an LU-set and, |G(I)| = 1 2 n d holds true. Clearly, both ∆ 1 and ∆ c 1 are f -simplicial complexes and minimal CM. Proposition 3. 3 . 3Let ∆ be a pure simplicial complex of dimension d−1, which is not a simplex. Consider the following conditions:(1) For each face σ of ∆ such that dim lk ∆ (σ) > 0, lk ∆ (σ) is connected.(2) For each facet F of ∆, ∆ is a shelling move of ∆ F . Then (1) implies(2).Proof. This follows from the proof of [6, Lemma 3.1]. The converse does not hold true in general. For example, the simplicial complex ∆ =: 1234, 1235, 1278, 1279 is a shelling move over ∆ F for each facet F , but lk ∆ (12) = 34, 35, 78, 79 and it is disconnected. For a simplicial complex ∆, recall from [10, Lemma 1.5.3] that I ∆ ∨ = I(∆ c ) holds, where ∆ c = [n] F | F ∈ F (∆) and, ∆ ∨ is the Alexander dual complex of ∆. Recall that ∆ is said to be CM, if the Stanley-Reisner ideal I ∆ of ∆ is CM. Recall also the Eagon-Reiner theorem ([10, Theorem 8.1.9]), i.e., a simplicial complex ∆ is CM if and only if the Stanley-Reisner ideal I ∆ ∨ of ∆ ∨ has linear resolution. Thus ∆ is CM if and only if the monomial ideal I(∆ c ) has linear resolution. These results together with CoCoA are crucial to our next work. F 2 = 4678, F 1 = 2347 of facets are found to make the CM simplicial complex ∆ 1 shelled over the minimal CM simplicial complex Γ, where each of the 24 monomial ideals I( (∆ c 1 ) {F c 25 } ), I( F (∆ c 1 ) {F c 25 , F c 24 } ), . . . , I( F (∆ c 1 ) {F c 25 , . . . , F c 2 } ) 4 . Example 3. 5 . 5Let F (∆ c ) be the set D consisting of 34 elements, where D = Example 3. 6 . 6Let F (∆ c 3 ) = D∪{1236}, in which D is taken as in Example 3.5. Let F c = 4578 and consider Γ =: ∆ F ∩ F . Then we have 123 ∈ F (Γ), thus F (Γ) is a proper subset of ∂F =: {123, 126, 136, 236}. Notice the following 30 = 5 × 6 = 10 × 3, 34 × 4 = 8 × 17 = 136 < 140 = 35 × 4. Notice the following fact: (1) For any odd number d ≥ 3, d × 1 2 2d d is divided by 2d, i.e., 4 | 2d d holds true. Based on the examples and Theorem 2.5, we now pose the following: Conjecture 3.7. (a) There exist in [2d] d very well-distributed f -simplicial complexes which are minimal CM, if one of the following conditions holds true:(1) d ≥ 3 and d is an odd number.(2) d ≥ 4 and d is an even number such that 4 | 2d d .(b) In [8] 4 ,there exists no very well-distributed f -simplicial complex which is minimal CM. Note that 4 ∤ 8 16 also holds. Thus in [16] 8 , the pure simplicial complexes may perhaps behave just like the pure simplicial complexes in [8] 4 . be the chain complex of the simplex [n] over a field K. Then the K-subspace ker(∂ r ) has dimension n−1 r+1 .Proof. For 1 < r ≤ n − 1, let 1≤i 1 <i 2 <···<ir≤n x i 1 i 2 ...ir · i 1 i 2 . . . i r ∈ ker ∂ r−1 ,where the second i 1 i 2 . . . i r denotes the subset {i 1 , i 2 , . . . , i r } of [n] and x i 1 i 2 ...ir are elements of the base field K. Then 1≤i 1 <i 2 <···<ir≤n x i 1 i 2 ...ir r j=1 (−1) j−1 i 1 i 2 . . . i j−1îj i j+1 . . . i r = 0 in the i 1 . . . i r -th column vector v i 1 ...ir of M, the i 1 . . .î j . . . i r -th component is (−1) j−1 (1 ≤ j ≤ r) and, all other components are zero. Thus we have v i 1 ...ir = v 1i 2 ...ir − v 1i 1 i 3 ...ir + · · · + (−1) r−1 v 1i 1 ...i r−1 . Corollary 4. 2 . 2Let ∆ = {σ | σ ∈ [n] r } . Then we have dimH i (∆ Theorem 4. 3 . 3Let ∆ be a simplicial complex over vertex set [n] with dim ∆ = d − 1 ≥ 0. If ∆ is an f -simplicial complex and it is minimal CM, then d ≥ 3 and n = 2d. Proof. It is known that connected simplicial complexes of dimension 1 are shellable and pure. On the other hand, if ∆ is not connected, then it is not CM. So, there exists no minimal CM f -simplicial complexes of dimension 1. Now let ∆ be an f -simplicial complex of dimension d − 1, which is minimal CM. ThenH d−1 (∆) = 0 by Theorem 2.3, which means that ∂ d−1 is injective. Since ∆ is an f -simplicial complex, F (∆) is an L-set, hence ∆ (d−2) = [n] d−1 holds true, thus,H i (∆) = 0, ∀0 ≤ i ≤ d − 3. Hence ∆ is acyclic if and only ifH d−2 (∆) = 0, and the latter holds true if and only if dim K ker(∂ d−2 4.1, we have dim K ker(∂ d−2 ) = n−1 d−1 . Corollary 4. 4 . 4If ∆ is an f -simplicial complex generated by a subset of [2d] d and it is minimalCM, thenH i (∆ F ) ∼ =H i ( F ∩ ∆ F ) holds for all integer i, where F is any facet of ∆ and, F ∩ ∆ F is pure of dimension d − 2. 2.8. ([2,11]) Consider a simplicial complex ∆ with facet setF (∆) = {123, 125, 136, 145, 146, 234, 246, 256, 345, 356} −1 holds. Clearly, [n] d−1 is pure shellable, thus it is CM. Then the result follows from the fact that , 1578} . 1578}By Example 2.8, it generates a minimal CM simplicial complex Γ, whereΓ = {F | F ∈ A} . and let ∆ 1 = {F | F ∈ A ∪ B .Then it is checked that ∆ 1 is a CM simplicial complex via CoCoA ([5]), and the following sequenceF 25 = 1246, 1258, 1235, 1357, 3478, 1568, 1234, 2345, 2356, 2457, 2468, 2578, 2678,(1) Let B = {1234, 1235, 1246, 1258, 1357, 1458, 1568, 2345, 2347, 2346, 2356, 2457, 2468, 2578, 2678, 2467, 2456, 2567, 2367, 3456, 3478, 3678, 4567, 4578, 4678}, 2567, 2456, 3678, 3456, 2346, 1458, 4578, 4567, 2367, 2467, We checked that A ∪ C is an LU-set, thus ∆ 2 is indeed an f -simplicial complex. We checked that ∆ 2 is CM via CoCoA. Furthermore, we claim that the complex ∆ 2 is shellable with the shelling F 0 , F 1 , . . . , F 34 , where and the answer is yes. The following sequenceG 25 = 1247, G 24 = 1357, 1237, 2347, 2348, 2345, 2356, 1235, 1236, 1268, 1258, 2578, 2678, 2468, 2467, 2457, 3478, 3678, 1568, 4678, 3456, 3467, 4567, G 2 = 4578, G 1 = 1458 of facets are found to make the CM simplicial complex ∆ 2 shelled over the minimal CM simplicial complex Γ, where all the 24 facet monomial ideals I( F (∆ c1236, 1237, 1247, 1268, 1258, 1357, 1458, 1568, 2345, 2347, 2348, 2356, 2457, 2467, 2468, 2578, 2678, 3456, 3467, 3478, 3678, 4567, 4578, 4678}. F 34 = 1247, F 33 = 1237, 1357, 1358, 1368, 2347, 2348, 2356, 1236, 1235, 1268, 1258, 1345, 1347, 1367, 1567, 1568, 1578, 1458, 1456, 1468, 1478, 3478, 2345, 2457, 2467, 2468, 2578, 2678, 3456, 3467, 3678, 4567, F 1 = 4678, F 0 = 4578. In fact, we use CoCoA to show that each I( F c 0 , . . . , F c i ) has linear resolution (∀34 ≥ i ≥ 1), thus all F 0 , F 1 , . . . , F i are Cohen-Macaulay simplicial complexes. Then it follows from Theorem 3.2 that ∆ 2 is shellable with F 0 , F 1 , . . . , F 34 as a shelling. It is natural to ask if ∆ 2 is shelled over Γ constructed in (1)? We tried this via CoCoA, 2 ) {G c 25 , . . . , G c r } ) are tested to have linear free resolutions. CM simplicial complexes is minimal CM. Furthermore, all Γ are shelling moves of Γ F 2 , where F c 2 = 1235. Finally, we consider D ′ =: [8] 4 D, which consists of 36 elements, as follows: D ′ = {1236, 1237, 1238, 1245, 1248, 1256, 1257, 1267, 1268, 1278, 1346, 1347, }, 7 are CM. In fact, F c 35 can be chosen as anyone of the following: 1236, 1238, 1268, 1346, 1348, 1468, 2368 such that I( D c ∪ {F c 35 } ) has linear resolution. Note that unfortunately, none of the 7 1348, 1356, 1357, 1458, 1468, 1567, 2345, 2347, 2357, 2368, 2457, 2458, 2568, 2578, 3458, 3467, 3468, 3478, 3567, 3678, 4567, 4568, 4678, 5678}. Table 4 . 41 Transpose M T of the coefficient matrix M The second case is [8] 4 , and we check that ker ∂ 2 has dimension n−1 3 , where n = 8. In fact, let 1≤i<j<k≤8 x ijk {i, j, k} ∈ ker ∂ 2 , we have 0 = 1≤i<j<k≤8 x ijk ({j, k} − {i, k} + {i, j}). F -Ideals of degree 2. Algebra Colloq. G Q Abbasi, S Ahmad, I Anwar, W A Baig, 19G.Q. Abbasi, S. Ahmad, I. Anwar and W.A. Baig. F -Ideals of degree 2. Algebra Colloq. 19(2012) 921 − 926. Shellable nonpure complexes and posets. I. A Björner, M L Wachs, Trans. Amer. Math. Soc. 348A. Björner, M.L. Wachs. Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., 348(1996) 1299 − 1327. Cohen-Macaulay Rings. W Bruns, J Herzog, Cambridge University PressNew York; Cambridge, RevW. Bruns, J. Herzog. Cohen-Macaulay Rings. New York: Cambridge University Press, Cambridge, Rev. Ed., 1998. Newton complementary duals of f -ideals. S Budd, A Van Tuyl, arXiv:1804.00686v1Canad. Math. Bull. 622S. Budd, A. Van Tuyl. 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F -ideals and f -graphs. Comm. Algebra 45 : 8 (2017) 3207 − 3220. On the (n, d) th f -ideals. J Guo, T S Wu, J. Korean Math. Soc. 52J. Guo, T.S. Wu. On the (n, d) th f -ideals. J. Korean Math. Soc. 52 : 4(2015) 685−697. J Herzog, T Hibi, Monomial Ideals. GTM 260 London. Springer-Verlag London LimitedJ. Herzog and T. Hibi. Monomial Ideals. GTM 260 London: Springer-Verlag London Limited, 2011. The Cohen-Macaulay property of f -simplicial complexes. A-M Liu, J Guo, T S Wu, PreprintA-M. Liu, J. Guo and T.S. Wu. The Cohen-Macaulay property of f -simplicial com- plexes. Preprint 2020. On the connectedness of f -simplicial complexes. H Mahmood, I Anwar, M A Banyamin, S Yasmeen, J. Algebra Appl. 1569H. Mahmood, I. Anwar, M.A. Banyamin and S. Yasmeen. On the connectedness of f -simplicial complexes. J. Algebra Appl. 15 : 6(2016) 1750017, 9pp. Construction of Cohen-Macaulay f -Graphs. H Mahmood, I Anwar, M K Zafar, J. Algebra Appl. 1367H. Mahmood, I. Anwar and M.K. Zafar. 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Comm. in Algebra 32 : 6(2004) 2301−2324.

We have computed linear non-adiabatic oscillations of luminous red giants using a nonlocal and anisotropic time-dependent theory of convection. The results show that loworder radial modes can be self-excited. Their excitation is the result of radiation and the coupling between convection and oscillations. Turbulent pressure has important effects on the excitation of oscillations in red variables.

10.1093/mnras/sty2014

1808.09620

e4e9cdf6107a89b14635348375a27891d9777b55

Turbulent convection and pulsation stability of stars -III. Non-adiabatic oscillations of red giants 2018 D R Xiong Purple Mountain Observatory Chinese Academy of Sciences 210008NanjingChina L Deng Key Laboratory of Optical Astronomy National Astronomical Observatories Chinese Academy of Sciences 100101BeijingChina C Zhang Key Laboratory of Optical Astronomy National Astronomical Observatories Chinese Academy of Sciences 100101BeijingChina Turbulent convection and pulsation stability of stars -III. Non-adiabatic oscillations of red giants MNRAS 0002018Accepted 2018 July 16. Received 2018 July 12; in original form 2017 July 24Preprint 30 August 2018 Compiled using MNRAS L A T E X style file v3.0stars:late-type -stars: oscillations -stars: variables: general -Magellanic Clouds -convection We have computed linear non-adiabatic oscillations of luminous red giants using a nonlocal and anisotropic time-dependent theory of convection. The results show that loworder radial modes can be self-excited. Their excitation is the result of radiation and the coupling between convection and oscillations. Turbulent pressure has important effects on the excitation of oscillations in red variables. INTRODUCTION In the H-R diagram, there are a lot of pulsating red variables in the low-temperature area to the right of the Cepheid instability strip. They have the largest number among all known types of pulsating variables, and yet we know little about them. In the General Catalogue of Variable Stars (Kholopov et al. 1985(Kholopov et al. -1988, luminous red variables, usually known as long-period variables (LPVs), are divided into three types according to the regularity of their light curves: Miras, semi-regular variables (SRVs) and irregular variables. The study of red variables has made considerable progress in the past two decades with the help of photometric observations from projects like MACHO, OGLE and 2MASS. Wood et al. (1999) and Wood (2000) found 5 ridges (sequences A-E) in the period-luminosity (PL) diagram of luminous red variables in the Large Magellanic Cloud (LMC). OGLE observations showed more complex structures in the PL diagram, and at least 14 ridges could be defined (Kiss & Bedding 2003;Wray et al. 2004;Soszyński et al. 2004Soszyński et al. , 2005Soszyński et al. , 2007. Soszyński et al. (2004) found that stars with the primary periods on sequence A rarely had their second and third dominant periods on sequences C and C , and stars with the primary periods on sequence C rarely had their second and third dominant periods on sequence A. This indicates that stars with their primary periods on sequence A are a special type of red variables. They are named OGLE Small Amplitude Red Giants (OSARGs), but their origin is still unclear. At first Wood et al. (1999) thought that variable luminous red giants in MACHO observations were asymptoticgiant-branch (AGB) stars, but later Kiss & Bedding (2003) and Ita et al. (2004) found evidence showing that there were luminous red variables on both the AGB and the red-giant branch (RGB). Their PL sequences were parallel with slight offsets. Takayama et al. (2013) compared observed period ratios of luminous red giants with theoretical ones and concluded that OSARGs were radial and non-radial p modes. Wood (2015) reached a similar conclusion, and identified the stars on sequences C and C as predominantly radial pulsators; the radial pulsation modes associated with A , A, B, C and C were the fourth, third, second and first overtones and the fundamental modes, respectively. However, Trabucchi et al. (2017) found that both sequences B and C corresponded to first-overtone pulsation. For a long time, convection has been considered to be a pure damping mechanism of stellar oscillations. The excitation mechanism of pulsating red variables has been a long-standing theoretical problem. There are two different opinions regarding the excitation mechanism of OSARGs. The prevailing opinion is that they are stochastically excited, just like solar-like oscillations (Christensen-Dalsgaard et al. 2001;Soszyński et al. 2007;Takayama et al. 2013). However, Xiong & Deng (2013) came to the conclusion that there were no essential differences between the excitation mechanisms of oscillations in OSARGs and Miras. They were both the results of radiation and convective coupling. In this paper, we aim at probing the excitation mechanism for luminous red giants observed by MACHO and OGLE. In section 2, we briefly introduce theoretical stellar models and the scheme of computation. The results of computation of linear nonadiabatic oscillations are given in section 3, and the excitation mechanism of oscillations is discussed in section 4. We summarize our results in section 5. THEORETICAL MODELS AND SCHEME OF COMPUTATION It is well known that traditionally there are 4 equations for stellar structure: the conservation of mass, momentum and energy, and the equation of radiative transfer. Convection is treated with the mixing-length theory (MLT; Böhm-Vitense 1958). MLT is a phenomenological theory based on the analogy between turbulent convection and kinetic theory of gas molecules. In fact turbulence is much more complex than motions of gas molecules. Most of the time, gas molecules are free except when they collide. Their mean free path is far less than the characteristic length of the average fluid field change. Therefore, the molecule transport process (molecular viscosity, molecular heat conduction, molecular diffusivity, etc.) can be treated with a local theory. On the contrary, turbulent elements are in continuous interaction, and the characteristic length of turbulent elements and the average field change are comparable. Therefore, the convective transport process is a non-local phenomenon. Nonlinearity and non-locality of fluid motions are two main difficulties in the study of convection. The fundamental shortcoming of the local MLT is that it is not a dynamic theory following the hydrodynamic equations, therefore it cannot correctly describe the dynamic behaviours of turbulent convection. This shortcoming is very serious, or even intolerable, in dealing with dynamic problems of non-local and time-dependent convection. To solve this problem, we have developed a nonlocal and anisotropic time-dependent theory of convection based on hydrodynamic equations and turbulence theory (Xiong 1989;Xiong et al. 1997;Deng et al. 2006). There are 8 partial differential equations in our complete set of equations for chemically homogeneous stellar structure and oscillations. 4 of them are the equations of mass conversation, momentum conversation, energy conversation, and radiative transfer. When convection sets in, the second-order auto-correlation of turbulent velocity (the Reynolds stress) emerges in the momentum conversation equation, and the second-order cross-correlation of turbulent velocity and temperature (the convective enthalpy flux) appears in the energy conversation equation. The other 4 equations are the dynamic equations of the auto-and cross-correlations of turbulent velocity and temperature. There are 3 convective parameters (c1, c2, c3), which are connected to dissipation, diffusion and anisotropy of turbulent convection, respectively. These parameters can be calibrated by using the structure of the solar convection zone, the turbulent velocity and temperature fields of the solar atmosphere, the solar lithium abundance and numerical simulations of hydrodynamics. From a pure theoretical point of view, these parameters cannot be a group of constants. They depend not only on stellar luminosity and effective temperature, but also on depth (radius) inside stars. Fortunately, our study shows that adopting a group of parameters (c1, c2, c3)=(0.64, 0.32, 3.0) as calibrated using the Sun, we can reproduce almost all of the instability strips of pulsating stars in the H-R diagram. Moreover, within a wide range of these parameters stellar pulsation stability is not sensitive to the change of parameters, and is almost independent of the specific values of (c1, c2, c3) (Xiong et al. 2015, ; here after Paper I). We have calculated both the radial and nonradial oscillations of luminous red giants. In the present paper we dis- cuss only the radial oscillations. The nonradial oscillations will be studied in detail in future work. For stellar radial oscillations, the effects of the core area can be neglected except for its gravity field. Therefore, envelope models are enough for our analysis. The number of equations for static structures and linear radial non-adiabatic oscillations in term of completely non-local and anisotropic theory is 12, while the number of equations for linear nonradial non-adiabatic oscillations is 16. In Paper I we gave the complete set of equations for stellar structure and oscillations as well as a brief description of our non-local and anisotropic time-dependent theory of convection. For detailed derivations of the equations please refer to our previous work (Xiong 1989;Xiong et al. 1997;Deng et al. 2006). Computing envelope models in the completely non-local and anisotropic convection theory is very difficult and timeconsuming. We use quasi-anisotropic non-local convection models as equilibrium models for pulsation calculations. Our study shows that not only quasi-anisotropic non-local convection models approach the completely anisotropic ones very well in terms of the T − P structure and turbulent velocity-temperature fields, but also their results of pulsation stability are consistent (see Paper I). RESULTS OF THEORETICAL COMPUTATION Adopting the scheme described in last section, we computed the radial and non-radial linear non-adiabatic oscillations of RGB and AGB models. The evolutionary models were from the Padova code (Bertelli et al. 2008(Bertelli et al. , 2009, as shown in Figure 1. The initial chemical abundances of LMC were Y =0.26, Z=0.008. Considering the lack of models in the thermal pulsating AGB (TP-AGB) phase and the uncertainty of mass loss, AGB models only included those from the horizontal branch to the beginning of the TP-AGB. The high-luminosity AGB stars with L/L > ∼ 3.3 were extrapolated into the TP-AGB phase without mass loss. Therefore the results of oscillation computations of high-luminosity AGB stars are not very reliable, and can only be used as reference. We computed 150 non-local convective envelope models along each evolutionary track, then calculated their linear non-adiabatic oscillations for luminous RGB and AGB models of 0.8-1.8 M . The surface boundary of static envelop models and oscillation computations was at τ = 10 −3 , where τ is the optical depth. The bottom boundary was set deep enough: the bottom temperature T b ∼ 8 × 10 6 K or fractional radius r b /R0 ∼ 0.01. A slightly modified version of the Mihalas-Hummer-Däppen (MHD) equation of state Mihalas et al. 1988;Däppen et al. 1988), and OPAL tabular opacity (Rogers & Iglesias 1992) complemented by low-temperature tabular opacity (Alexander & Ferguson 1994) were adopted. All the computations of static models and oscillations followed the Henyey's algorithm (Henyey et al. 1964). In order to compare with observations, stellar parameters were converted to U BV RIJHK magnitudes using the transformation program provided by Worthey & Lee (2011). The JHK colors were then converted to the 2MASS system using transformations given by Carpenter (2001). The adopted LMC distance module was 18.5 mag (Pietrzyński et al. 2013). The observed sequences of luminous red variables in the LMC are shown in Figure 2 using data from the OGLE-III catalogue of LPVs in the LMC (Soszyński et al. 2009). The abscissas are the logarithmic period (in days) log P . The ordinate in panel a is the reddening-free Wesenheit index WI , defined as WI = I − 1.55(V − I). (1) The ordinate in panel b is the 2MASS magnitude KS. The catalogue contains 79200 OSARGs, 11128 SRVs and 1667 Miras. In Figure 2 only the primary period of each star is plotted. Stars on sequences A , A, B, C , and C are shown as cyan, blue, green, orange and red points, respectively. The subscripts O and C in panel a are used to distinguish between oxygen-rich and carbon-rich SRVs and Miras. The theoretical PL relations of low-order radial modes for M = 1.0 M and 1.8 M evolutionary models are shown in Figure 2 as filled and open symbols, respectively. Circles, triangles, inverse triangles, squares, and diamonds represent the pulsationally unstable radial fundamental modes, the first, second, third, and fourth overtones, respectively. In our computation of non-adiabatic radial oscillations, almost all of the p1-p39 modes of all the RGB and AGB models converged successfully without difficulty. Only for high-luminosity AGB models, the non-adiabatic oscillations of the fundamental modes were often nonconvergent, leading to the missing modes in Figure 2. We are still not very clear whether the non-convergence of the fundamental modes came from numerical calculations or had other causes. If the reason is the former, it it very difficult to explain why the non-convergence only existed in the cal- Figure 2. Sequences of LMC LPVs in the W I −log P plane (panel a) and K S − log P plane (panel b). Stars on sequences A , A, B, C , and C are shown as cyan, blue, green, orange, and red points, respectively. Sequence D is not shown. The filled and open black symbols show the theoretical PL relations for 1.0 M and 1.8 M evolutionary models, respectively. Circles, triangles, inverse triangles, squares, and diamonds are the pulsationally unstable radial fundamental modes, the first, second, third, and fourth overtones, respectively. culations of the fundamental modes, while the other modes converged exceptionally well. As can be seen in Figure 2, the theoretical period ranges of low-order radial oscillations are approximately consistent with the observed period ranges of the LPV sequences. However, there are systematic differences between the observed and theoretical PL relations. The main reasons are as follows. (i) Uncertainties in the transformation from theoretical stellar parameters (M , L, Te) to observed magnitudes (V , I, KS). The uncertainties are more significant for luminous RGB and AGB stars, which have extended dust envelopes. By comparing Figures 2a & b, it is can be seen that the differences between theoretical and observed PL relations in KS are clearly larger than those in WI . It seems that the long-wavelength end is more affected by the dust envelopes. (ii) Uncertainties in stellar evolutionary models. Compared with normal-mass stars in their early evolutionary stages, the evolutions of RGB and AGB stars are subject to uncertainties from mass loss, convection treatment, and low-temperature opacity. (iii) Mass variations. The luminous red variables observed by MACHO and OGLE are a group of stars with certain mass, age, and metallicity distributions, while our theoretical stellar models shown in Figure 2 are RGB and AGB models with M = 1.0 M and Z = 0.008. Therefore differences between theory and observation are expected. Using a modified Petersen diagram, Wood (2015) found that the stars on sequences A -C have different masses. EXCITATION MECHANISM We have long known that Cepeheid and Cepeheid-like variables are driven by radiative κ mechanism, and the existence of the red edge of the Cepheid instability strip is due to convective damping. Therefor for a long time convection is believed to be a pure damping mechanism of oscillations. However, why are there still various types of red variables in the low-temperature area beyond the Cepheid instability strip? Xiong et al. (1998) studied non-adiabatic oscillations of luminous red variables, and pointed out the existence of a Mira instability strip in the low-temperature area beyond the Cepheid instability strip. In low-temperature red variables, convection replaces radiation as the dominant means of energy transport. Radiation is still an important driving and damping mechanism of oscillations in red variables, but it works together with convection, instead of being the dominant mechanism in high-temperature variables. Radiative transfer is a wellknow physical process, while the coupling between convection and oscillations remains a major difficulty in the study of red variables (Houdek & Dupret 2015). Therefore it is fair to say that convection is the key to solve the problem of the excitation of oscillations in red variables. Unfortunately, so far there has not been a widely accepted stellar convection theory. Therefore, controversies are unavoidable about the excitation mechanism of oscillations in red variables. The following discussions represent the theoretical interpretations of this problem within the framework of our non-local and time-dependent convection theory (Xiong et al. 2015) Accumulated work is a convenient and useful way in studying the excitation mechanisms of pulsating variables. It not only gives quantitatively estimate of the plusational amplitude growth rate, but also shows the locations of the driving and damping. The total accumulated work can be written as W = WP g + WP t + Wvis,(2) where WP g , WP t , and Wvis are the gas pressure component, the turbulent pressure component, and the turbulent viscosity component, respectively. WP g = − π 2E k M 0 0 Im P ρ δP P δρ * ρ dMr,(3)WP t = − π 2E k M 0 0 Im ρx 2 δx x δρ * ρ dMr,(4)Wvis = − π 2E k M 0 0 Im ρχ 1 1 δχ 1 * 1 χ 1 1 d d ln r δr * r dMr, (5) where E k = 1 2 M 0 0 ρω 2 δrδr * dMr,(6) and ω is the circular frequency of the oscillation mode. ρg ij x 2 and ρχ ij are the isotropic and anisotropic components of the turbulent Reynolds stress and are expressed in tensor form with g ij being the metric tensor (see Paper I and Zhang et al. (2017) for related definitions and discussions). Convection drives mixing of material and transport and exchange of energy and momentum in stellar interiors, and therefore affects the structure, evolution, and pulsation stability of stars. Equations 4 and 5 are the work contributions by the Reynolds stress by means of momentum exchange, i.e. the dynamic coupling between convection and oscillations. Due to the inertia of turbulent motions, the variation of the turbulent pressure ρx 2 in stellar oscillations always lags slightly behind the variation of density. As a result, a positive Carnot cycle is formed in the pressure-volume (Pt − V ) diagram, converting irregular turbulent kinetic energy into regular kinetic energy of stellar oscillations. Therefore, turbulent pressure is always a driving mechanism of oscillations. It has important effects on the excitation of stellar oscillations (Xiong & Deng 2013;Xiong et al. 2016;Antoci et al. 2014). The anisotropic component χ ij represents shear motions of the fluid, converting regular kinetic energy of oscillations into irregular turbulent kinetic energy. This process takes place at low wave numbers (large-scale turbulence) of the turbulent energy spectrum. The energy is then gradually shifted to high wave numbers (small-scale turbulence) as a result of turbulence cascade, and is eventually converted into thermal energy by molecular viscosity. Therefore, turbulent viscosity is always a damping mechanism against oscillations. The gas pressure component of the accumulated work in equation 3 includes the contributions from both the radiative and convective energy transfer. We can separate the contributions from the radiative flux, the turbulent thermal flux, and the turbulent kinetic energy flux one by one in the average energy equation (Xiong & Deng 2010. They are closely coupled to each other; variation in one causes compensating variations in the others. In the deep interior of the convection zone, the convective flux dominates. Our study shows that the variation of the turbulent thermal flux is usually synchronized with the variation of density (with a slight phase lag). This means that in stellar oscillations thermal convection takes out more energy at high-temperature highdensity phase, and blocks in more energy at low-temperature low-density phase. This mechanism works like a refrigerating machine, converting pulsation kinetic energy into thermal energy. Therefore, turbulent thermal convection is usually a damping mechanism against oscillations. Figure 3 is a sketch showing the turbulent pressure component WP t , the turbulent viscosity component Wvis, and the turbulent thermal convection component WP gc as a function of ωτc, where τc is the dynamic time scale of convective motions (Xiong & Deng 2013). As stellar parameters (M , L, Te, X, Y , Z) and oscillation frequencies vary, the relative sizes of the contributions from radiation, turbulent pressure, turbulent thermal convection, and turbulent viscosity also vary. The combined effect is that the total accumulated work W sometimes is positive, and sometimes is negative, corresponding to unstable and stable oscillations, respectively. Therefore a star shows different oscillation characteristics in different evolutionary stages. Figure 4 and 5 show the accumulated work W (solid lines) and the components WP g (dashed lines) WP t (long-dashed lines) Wvis (dotted lines) of the fundamental modes (panels a) and the ninth overtones (panels b) as a function of depth for a low-luminosity red giant and a high-luminosity red giant, respectively. The fractional radiative flux log Lr/L (dashed-dotted lines) and the hydrogen and helium ionization zones (thick horizontal lines) are also plotted. In the low-luminosity red giant, the fundamental mode (Figure 4a) is stable due to the damping mainly from turbulent viscosity, but the ninth overtone ( Figure 4b) is unstable with the driving contributed by both the gas pressure and turbulent pressure components. However, we see the opposite results in the high-luminosity red giant. The fundamental mode (Figure 5a) is pulsationally unstable due to the dominating driving effect from turbulent pressure; while the ninth overtone (Figure 5b) becomes stable because the damping effects from both turbulent thermal convection and turbulent viscosity overtake the driving effect from turbulent pressure. Figure 6 shows the numerical results of linear nonadiabatic oscillations for 1.0M stellar models. The small dots are pulsationally stable modes, and the open circles are unstable modes. The sizes of the open circles are proportional to the logarithms of the amplitude growth rates. It can be clearly seen that low-order modes are pulsationally stable in low-luminosity red giants, while the mid-to high-order modes are pulsationally unstable. Therefore low-luminosity red giants show characteristics of solar-like oscillations. As luminosity increases, pulsationally unstable modes shift to lower radial orders. In high-luminosity red giants, only a few low-order modes are pulsationally unstable, while all the mid-to high-order modes become pulsationally stable. Therefore high-luminosity red giants show typical characteristics of Mira-like oscillations. SUMMARY We have computed the linear non-adiabatic oscillations of evolutionary models of low-mass red giants. The main results can be summarized as follows. (i) Sequences A-C of luminous red variables in LMC of MACHO and OGLE observations are low-order radial modes of low-mass RGB and AGB stars. (ii) Oscillations in Miras and OSARGs can be self-excited as a result of radiation and the coupling between convection and pulsation. Turbulent pressure has important effects on the excitation of oscillations in low-temperature red variables. (iii) Our study shows that for low-luminosity red giants, the low-order modes are stable, while the intermediate-and high-order modes are unstable. These stars show characteristics of solar-like oscillations. As the luminosity increases, unstable modes move towards lower orders. In luminous red giants, only a few low-order radial modes are unstable, while all of the intermediate-and high-order modes become stable. These stars show characteristics of Mira-like oscillations. Figure 1 . 1RGB (black solid lines) and AGB (red dotted lines) evolutionary tracks with M = 0.8, 1.0, 1.4 and 1.8 M in the H-R diagram. Pluses mark the positions of the M = 1.0 M equilibrium models in Figure 2. Figure 3 . 3Sketch of the frequency dependence of the effects of turbulent pressure (solid line), turbulent thermal flux (dashed line), and turbulent viscosity (dotted line). Figure 4 . 4Accumulated work and its components as a function of depth log T for a low-luminosity red giant. The solid lines are the total accumulated work W . The dashed, long-dashed, and dotted lines are the gas component W Pg , turbulent pressure component W P t , and turbulent viscosity component W vis , respectively. The dash-dotted lines are the fractional radiative flux log Lr/L. The horizontal lines indicate the locations of the ionization regions of hydrogen and helium. Panel a: p0 mode. Panel b: p9 mode. Figure 5 . 5The same asFigure 3but for a high-luminosity red giant. Figure 6 . 6Pulsationally stable (small dots) and unstable (open circles) radial modes in the log L/L − nr plane for AGB models. The sizes of the open circles are proportional to the logarithms of the amplitude growth rates of the modes. D. R.Xiong et al. MNRAS 000, 1-7 (2018) ACKNOWLEDGEMENTSThis work was supported by National Natural Science Foundation of China (NSFC) through grants 11373069, 11473037, and 11403039.This paper has been typeset from a T E X/L A T E X file prepared by the author. . D R Alexander, J W Ferguson, ApJ. 437879Alexander D. R., Ferguson J. W., 1994, ApJ, 437, 879 . V Antoci, M Cunha, G Houdek, ApJ. 796118Antoci V., Cunha M., Houdek G., et al., 2014, ApJ, 796, 118 . 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C Zhang, L Deng, D R Xiong, Raa, 1729Zhang C., Deng L., Xiong D. R., RAA, 17, 29

The scaled-particle theory equation of state for the two-dimensional hard-disk fluid on a curved surface is proposed and used to determine the saddle-splay modulus of a particle-laden fluid interface. The resulting contribution to saddle-splay modulus, which is caused by thermal motion of the adsorbed particles, is comparable in magnitude with the saddlesplay modulus of a simple fluid interface.

10.1209/0295-5075/85/56001

0812.1133

aef3fec8587b3f3b3ddce0583e5fae36fd46d8c5

Saddle-splay modulus of a particle-laden fluid interface 5 Dec 2008 S V Lishchuk Materials and Engineering Research Institute Sheffield Hallam University S1 1WBSheffieldUnited Kingdom Saddle-splay modulus of a particle-laden fluid interface 5 Dec 2008 The scaled-particle theory equation of state for the two-dimensional hard-disk fluid on a curved surface is proposed and used to determine the saddle-splay modulus of a particle-laden fluid interface. The resulting contribution to saddle-splay modulus, which is caused by thermal motion of the adsorbed particles, is comparable in magnitude with the saddlesplay modulus of a simple fluid interface. Introduction The surface free energy density of fluid interfaces depends upon their curvature. This dependence affects the nucleation in liquids [1,2,3], and has important role in determining the structure and dynamics of the systems with complex fluid interfaces, such as membranes or surfactants [4]. For small curvature of the interface, the dependence of the surface free energy f upon the geometry of the interface is conveniently described by the Helfrich curvature expansion [5] f = σ + 2κ(H − H 0 ) 2 +κK (1) In this equation the geometry of the interface is characterized by mean curvature H = 1 2 (1/R 1 + 1/R 2 ) and Gaussian curvature K = 1/(R 1 R 2 ), R 1 and R 2 being principal radii of curvature of the interface. The surface tension σ, bending modulus κ, spontaneous curvature H 0 , and saddle-splay modulus (or Gaussian rigidity)κ are the material parameters of the interface. By virtue of Gauss-Bonnet theorem, the contribution of the last term in Eq. (1) into the total free energy of the system depends on the topology of the system. Indeed, the value of the saddle-splay modulus affects the processes which involve changes in the topology of fluid interfaces [6,7,8,9,10,11]. An interesting example of a system which can be macroscopically viewed as a complex fluid interface is the fluid interface laden with colloidal micro-or nanoparticles. To minimize total interfacial energy, particles suspended in a bulk fluid self-assemble on the fluid interface [12]. This process, first observed by Ramsden in 1903 [13], has recently attracted significant scientific attention [14,15,16,17]. It has also potential for a range of novel applications [18,19,20,21,22]. On the scale large compared to the size of the adsorbed particles, a particleladen interface may be viewed as continuous. If the interface is isotropic on this scale, the interfacial free energy can be described by Eq. (1), and the interface can be characterized by the material parameters σ, κ, H 0 , andκ. The present letter is devoted to the study of the saddle-splay modulusκ of a particle-laden fluid interface at low surface concentration of the adsorbed particles. In this case we can represent the interface as a two-dimensional fluid on a curved surface. The main contribution to the interaction between particles at low concentration comes from the excluded volume (different particles cannot occupy the same space). Hence we approximate the system by a two-dimensional hard-disk fluid on a curved surface. Hard-disk fluids in curved geometry were used before to study packing of disks [23,24,25,26], ordering phase transition [27], topological defects [28], and as a model of glass-forming liquids [24,25,29]. Several equations of state were proposed for hard-disk fluids in spherical [30,31] and hyperbolic [32,26,33] geometries. In the present work we shall use scaled-particle theory (SPT) [34] to derive the equation of state of two-dimensional hard-disk fluid on a curved surface. We shall then use the resulting equation of state to determine saddle-splay modulus κ for particle-laden fluid interface at low concentration of the adsorbed particles. Saddle-splay modulus In accordance with Eq. (1), the saddle-splay modulus is given by the derivative of the surface free energy density with respect to Gaussian curvature, κ = ∂f ∂K K=0 .(2) Using the expression for the excess free energy βF ex N = ρ 0 Z − 1 ρ dρ,(3) where Z ≡ βP ρ(4) is the compressibility factor, ρ = N/A is the number density (number of particles per unit area), P is pressure, β = 1/k B T is the inverse temperature, we may represent Eq. (2) in the form κ = ρ β ρ 0 1 ρ ∂Z ∂K K=0 dρ,(5) where the derivative is taken at constant particle density ρ. Equation (5) can be used to calculate saddle-splay modulus of the interface from the curvature dependence of the compressibility factor, which is generally given by the equation of state of the system. We shall use the SPT equation of state for a hard-disk fluid on a curved surface, which is derived in the following sections. SPT equation of state for hard disks Scaled-particle theory was originally developed by Reiss et al [34] and further improved afterwards [35,36,37,38,39]. Applied to the case of hard disks on a 2D plane, SPT leads to a particularly simple equation of state which is nevertheless in good agreement with computer simulation results throughout most of the fluid range of densities [40,41]. SPT for two-dimensional hard-disk fluids in its simplest form can be summarized as follows (see textbook [42] for more details). The reversible work W (R 0 ) is considered which is required to create a circular cavity of radius R 0 in the fluid of hard disks of radius R. The assumption is made that for R 0 > 0, W (R 0 ) is given by a polynomial in R 0 W (R 0 ) = w 0 + w 1 R 0 + S(R 0 )P, R 0 ≥ 0.(6) The last term S(R 0 )P (S(R) being the area of the disk of radius R), which is dominant for large cavities (R 0 ≫ R), follows from thermodynamics. For small cavities (0 ≤ R + R 0 ≤ R), W (R 0 ) can be written in form W (R 0 ) = −k B T ln [1 − ρS(R 0 + R)] , −R ≤ R 0 ≤ 0.(7) The coefficients w 0 and w 1 are then determined by requiring the work W (R 0 ) and its derivative W ′ (R 0 ), given by Eqs (6) and (7), to be continuous at R 0 = 0. The explicit expression for the excess chemical potential of the fluid, µ ex = W (R), can be determined from Eq. (6), and subsequently used to write the SPT equation of state. In the case of the flat surface, the area of the disk is S(R) = πR 2 .(8) The corresponding values of the coefficients w i are given by βw 0 = − ln(1 − η), βw 1 = 2πρR 1 − η ,(9) where η = πR 2 ρ is the hard-disk packing fraction. The chemical potential of the fluid, µ, is given by βµ = ln Λ 2 ρ − ln(1 − η) + 2η 1 − η + βP η ρ ,(10) where Λ is de Broglie thermal wavelength. The SPT equation of state is then obtained from Eq. (10) and the thermodynamic relation ∂P ∂ρ = ρ ∂µ ∂ρ ,(11) and has the form reported by Helfand et al [40]: Z = 1 (1 − η) 2 .(12) 4 SPT equation of state for hard disks on a curved surface SPT equation of state for a hard-disk fluid on a curved surface can be obtained in the same way as in the flat case described above. The difference is that expression (8) for the area of the disk of radius R is no longer valid on a curved surface. For small Gaussian curvature (K ≪ 1/R 2 ) we shall replace it by the formula for the area of a geodesic disk on two-dimensional Riemannian manifold, obtained by Bertrand and Diguet in 1848 [43], S(R) = πR 2 (1 − ξ) + o(ξ),(13) where we have introduced the dimensionless quantity ξ ≡ KR 2 12 .(14) Requiring work W (R 0 ) and its derivative W ′ (R 0 ), as given by Eqs (6) and (7), to be continuous at R 0 = 0, we obtain the following expressions for the coefficients w i , βw 0 = − ln[1 − πR 2 ρ(1 − ξ)],(15)βw 1 = 2πRρ(1 − 2ξ) 1 − πR 2 ρ(1 − ξ) ,(16) and the chemical potential, βµ = ln Λ 2 ρ − ln[1 − η(1 − ξ)] + 2η(1 − 2ξ) 1 − η(1 − ξ) + βP η(1 − ξ) ρ ,(17) Equations (17) and (11) lead to the following form of the SPT equation of state for hard-disk fluid on a curved surface: Z = 1 − ηξ [1 − η(1 − ξ)] 2 .(18) Saddle-splay modulus from SPT equation of state The expression for saddle-splay modulus is obtained by substituting the compressibility factor given by the equation of state, Eq. (18), into formula (5). The result isκ SP T = −k B T η 2 (3 − 2η) 12π(1 − η) 2 .(19) Note that although using the truncated series in R given by the formula (13) for the area of the large disk in the expression (6) is generally not justified, it is still suitable for our purpose of calculating saddle-splay modulus since we are interested in the limit K → 0. The dependence of the saddle-splay modulusκ upon the disk packing fraction η, given by Eq. (19), is presented in Figure 2. The value of the saddle-splay modulus for particle-laden interfaces appears to be smaller than the values |κ| ∼ 10k B T typical for lipid monolayers [44]), but is comparable to the value |κ| ≈ Conclusion The main message of this letter is that the thermal motion of the particles adsorbed on a fluid interface contributes to the saddle-splay modulus of the interface. This result may have implications in the structure and dynamics particleladen systems that allow topological changes, for example, fusion of particles in Pickering emulsions, or structural reorganization in particle-stabilized foams. The simplest version of the scaled-particle theory allows construction of a rather simple equation of state for a hard-disk fluid on a curved surface. In order to improve the formula obtained for the saddle-splay modulus of a particle-laden fluid interface, it seems reasonable to attempt to construct the equation of state that gives more accurate dependence of the compressibility factor with respect to Gaussian curvature of the interface, which can be verified by using the virial expansion on the curved surface or the computer modelling of the system. The result can also be extended by taking into account the influence on the value of saddle-splay modulus of other contributions to the interparticle interaction, such as capillary, electrostatic, van der Waals etc, as well as the role of particles' anisotropy. The prediction of the elastic properties of the interfaces with large concentration of particles, in which two-dimensional solid structure forms, presents another interesting and more complicated problem. Figure 1 1demonstrates satisfactory agreement of the compressibility factor Z calculated from Eq. (18) with the Monte Carlo results for hard disks on a sphere reported by Giarritta et al[27]. In the case of zero Gaussian curvature (ξ = 0) Eq. (18) coincides with Eq.(12). Figure 1 : 1Compressibility factor Z as a function of reduced number density of the fluid ρσ 2 , where σ ≡ 2R is particle diameter. Circles represent Monte Carlo results for N = 400 hard disks on a sphere[27], line corresponds to SPT equation of state(18). 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This article proposes the first known algorithm that achieves a constant-factor approximation of the minimum length tour for a Dubins' vehicle through n points on the plane. By Dubins' vehicle, we mean a vehicle constrained to move at constant speed along paths with bounded curvature without reversing direction. For this version of the classic Traveling Salesperson Problem, our algorithm closes the gap between previously established lower and upper bounds; the achievable performance is of order n 2/3 .

cs/0603010

236ccc5964ac49a4485c77b824c57e471ac7b68c

Asymptotic constant-factor approximation algorithm for the Traveling Salesperson Problem for Dubins' vehicle 2 Mar 2006 Ketan Savla Emilio Frazzoli Francesco Bullo Asymptotic constant-factor approximation algorithm for the Traveling Salesperson Problem for Dubins' vehicle 2 Mar 2006 This article proposes the first known algorithm that achieves a constant-factor approximation of the minimum length tour for a Dubins' vehicle through n points on the plane. By Dubins' vehicle, we mean a vehicle constrained to move at constant speed along paths with bounded curvature without reversing direction. For this version of the classic Traveling Salesperson Problem, our algorithm closes the gap between previously established lower and upper bounds; the achievable performance is of order n 2/3 . I. INTRODUCTION The Traveling Salesperson Problem (TSP) with its variations is one of the most widely known combinatorial optimization problems. While extensively studied in the literature, these problems continue to attract great interest from a wide range of fields, including Operations Research, Mathematics and Computer Science. The Euclidean TSP (ETSP) [1], [2] is formulated as follows: given a finite point set P in R 2 , find the minimum-length closed path through all points in P . It is quite natural to formulate this problem in context of Dubins' vehicle, i.e., a non-holonomic vehicle that is constrained to move along paths of bounded curvature, without reversing direction. The focus of this article is the analysis of the TSP for Dubins' vehicle; we shall refer to it as DTSP. Exact algorithms, heuristics as well as polynomial-time constant factor approximation algorithms are available for the Euclidean TSP, see [3], [4], [5]. It is known that nonmetric versions of the TSP are, in general, not approximable in polynomial time [6]. Furthermore, unlike most other variations of the TSP, it is believed that the DTSP cannot be formulated as a problem on a finite-dimensional graph, thus preventing the use of well-established tools in combinatorial optimization. On the other hand, it is reasonable to expect that exploiting the geometric structure of Dubins' paths one can gain insight into the nature of the solution, and possibly provide polynomial-time approximation algorithms. A fairly complete picture is available for the minimumtime point-to-point path planning problem for Dubins' vehicle, see [7] and [8]. However, the DTSP seems not to have been studied as extensively. In [9], some results for the worst case tours of DTSP were provided. A lower bound on the expected cost of a stochastic DTSP visiting randomly generated points was provided in [10]. Here, we shall Ketan specifically concentrate on the case when the target points in the environment are generated stochastically according to a uniform distribution. We shall refer to such a problem as stochastic DTSP. In this context the first algorithm with asymptotic sub-linear cost was proposed in [11]; an improved algorithm was proposed in [12]. The motivation to study the DTSP arises in robotics and uninhabited aerial vehicles (UAVs) applications, e.g., see [13], [14], [15]. In particular, we envision applying our algorithm to the setting of an UAV monitoring a collection of spatially distributed points of interest. Additionally, from a purely scientific viewpoint, it appears to be of general interest to bring together the work on Dubins' vehicle and that on TSP. Some concrete results along these lines have been obtained in [9] and in [11] where an algorithm to guarantee sub-linear cost for the stochastic DTSP was proposed. UAV applications also motivate us to study the Dynamic Traveling Repairperson Problem (DTRP), in which the aerial vehicle is required to visit a dynamically changing set of targets. This problem was introduced by Bertsimas and van Ryzin in [16] and then decentralized policies achieving the same performances were proposed in [13]. However, as with the TSP, the study of DTRP in context of Dubins' vehicle has eluded attention from the research community. The contributions of this article are twofold. First, we propose an algorithm for the stochastic DTSP through a point set P , called the RECURSIVE BEAD-TILING ALGORITHM, based on a geometric tiling of the plane, tuned to the Dubins' vehicle dynamics, and a strategy for the vehicle to service targets from each tile. Second, we obtain an upper bound on the stochastic performance of the proposed algorithm and thus also establish a similar bound on the stochastic DTSP. The upper bound on the performance of the RECURSIVE BEAD-TILING ALGORITHM belongs to O(n 2/3 ) with high probability, and it is known that the lower bound on the achievable performance belongs to Ω(n 2/3 ). The algorithm we introduce in this article is the first known algorithm providing a provable constant-factor approximation to the DTSP optimal solution. Notation Here we collect some concepts that will be required in the later sections. A Dubins' vehicle is a planar vehicle that is constrained to move along paths of bounded curvature, without reversing direction and maintaining a constant speed. Accordingly, we define a feasible curve for Dubins' vehicle or a Dubins' path, as a curve that is twice differentiable almost everywhere, and such that the magnitude of its curvature is bounded above by 1/ρ, where ρ > 0 is the minimum turn radius. Let P = {p 1 , . . . , p n } be a set of n points in a compact region Q ⊂ R 2 and P n be the collection of all point sets P ⊂ Q with cardinality n. Let ETSP(P ) denote the cost of the Euclidean TSP over P , i.e., the length of the shortest closed path through all points in P . Correspondingly, let DTSP ρ (P ) denote the cost of the Dubins' TSP over P , i.e., the length of the shortest closed Dubins' path through all points in P . In what follows, ρ ∈ R + is take constant, and we study the dependence of DTSP ρ : P n → R + on n. For f, g : N → R, we say that f ∈ O(g) (respectively, f ∈ Ω(g)) if there exist N 0 ∈ N and k ∈ R + such that |f (N )| ≤ k|g(N )| for all N ≥ N 0 (respectively, |f (N )| ≥ k|g(N )| for all N ≥ N 0 ). If f ∈ O(g) and f ∈ Ω(g), then we use the notation f ∈ Θ(g). II. THE STOCHASTIC DTSP In [9], a simple heuristics, the ALTERNATING ALGO-RITHM for the Dubins' TSP for a given point set was proposed. The length of tour generated by this algorithm was also characterized and it was shown that it belongs to Ω( √ n) and O(n). It was also shown that this simple policy performs well when the points to be visited by the tour are chosen in an adversarial manner. However, it is reasonable to argue that this algorithm might not perform very well when dealing with a random distribution of the target points. In particular, one can expect that when n points are chosen randomly, the cost of the DTSP increases sub-linearly with n, i.e., that the average length of the path between two points decreases as n increases. In this section, we consider the scenario when n target points are stochastically generated in Q according to a uniform distribution. A novel algorithm, the BEAD-TILING ALGORITHM was proposed in [11] to service these points in such a way that its tour length grew sub-linearly with the number of points asymptotically with high probability, where an event is said to occur with high probability if the probability of its occurence approaches 1 as n → +∞. Here, we present a novel version of this strategy in the form of the RECURSIVE BEAD-TILING ALGORITHM and characterize its performance. We assume that the environment Q is a rectangle of width W and height H; different choices for the shape of Q affect our conclusions only by a constant. In what follows we select a reference frame whose two axes are parallel to the sides of Q. Let n target points be generated stochastically according to uniform distribution in the region Q. Let P = (p 1 , . . . , p n ) be the locations of these target points. A. A lower bound First, we summarize a result from [10], that provides a lower bound on the expected length of the stochastic DTSP. Theorem 2.1: (Lower bound on stochastic DTSP) For all ρ > 0, the expected cost of a stochastic DTSP visiting a set P of n uniformly-randomly-generated points in Q, E[DTSP ρ (P )] belongs to Ω(n 2/3 ). B. A constructive upper bound In [11], a novel algorithm, the BEAD-TILING ALGO-RITHM, to compute Dubins' path through a point set in the region Q was proposed. In this section, we design the RECURSIVE BEAD-TILING ALGORITHM based on the ideas of the BEAD-TILING ALGORITHM. We will show that the proposed algorithm provides a constant factor approximation to the optimal DTSP with high probability. We start by describing some useful geometric objects. 1) The basic geometric construction: Consider two points p − = (−l, 0) and p + = (l, 0) on the plane, with l ≤ ρ, and construct the region B ρ (l) as detailed in Figure 1. In the following, we will refer to such regions as beads. The region B ρ (l) enjoys the following asymptotic properties as (l/ρ) → 0 + : (P1) The maximum "thickness" of the region is equal to ρ 2l p − p + B ρ (l)w(l) = 4ρ 1 − 1 − l 2 4ρ 2 = l 2 2ρ + o l 3 ρ 3 . (P2) The area of B ρ (l) is equal to Area[B ρ (l)] = lw(l) = l 3 2ρ + o l 4 ρ 4 . (P3) For any p ∈ B ρ , there is at least one Dubins' path γ p through the points {p − , p, p + }, entirely contained within B ρ , and such that its length is at most Length(γ p ) ≤ 4ρ arcsin l 2ρ = 2l + o l 2 ρ 2 . These facts are verified using elementary planar geometry. 2) Periodic tiling of the plane: An additional property of the geometric shape introduced above is that the plane can be periodically tiled by identical copies of B ρ (l), for any l ∈ (0, 2ρ]. (Recall that a tiling of the plane is a collection of sets whose intersection has measure zero and whose union covers the plane.) Let µ(l) = Area[B ρ (l)] Area[Q] . Consider a bead B entirely contained in Q; the probability that the i-th point is sampled in B is equal to µ. Furthermore, the probability that exactly k out of the n points are sampled in B has a binomial distribution, i.e., indicating with n B the total number of points sampled in B, Prob[n B = k|n samples] = n k µ k (1 − µ) n−k .(1) Choose µ as a function of n, in such a way that ν = nµ(n) is a constant. In such a case, the limit for large n of the binomial distribution (1) is the Poisson distribution of mean ν, that is, lim n→∞ Prob[n B = k|n samples] = ν k k! e −ν(2) C. The algorithm Consider a tiling of the plane such that Area[B ρ (l)] = W H/(2n); in such a case, µ = 1/(2n), and ν = 1/2. (Note that this implies that n must be large enough that l < 2ρ.) Furthermore, the tiling is chosen is such a way that it is aligned with the sides of Q, see Figure 2. Sketch of tiling of the region before the first phase of the RECURSIVE BEAD-TILING ALGORITHM. The proposed algorithm will consist of a sequence of phases; during each of these phases, a Dubins tour (i.e., a closed path with bounded curvature) will be constructed that "sweeps" the set Q. In the first phase, a Dubins tour is constructed with the following properties: (i) it visits all non-empty beads once, (ii) it visits all rows 1 in sequence top-to-down, alternating between left-to-right and right-to-left passes, and visiting all non-empty beads in a row, (iii) when visiting a non-empty bead, it services at least one target in it. In order to visit the outstanding targets, a new phase is initiated. In this phase, instead of considering single beads, 1 A row is a maximal string of beads with non-empty intersection with Q. we will consider "meta-beads" composed of two beads each, as shown in Figure (3), and proceed in a similar way as the first phase, i.e., a Dubins tour is constructed with the following properties: (i) the tour visits all non-empty meta-beads once, (ii) it visits all (meta-bead) rows in sequence top-todown, alternating between left-to-right and right-to-left passes, and visiting all non-empty meta-beads in a row, (iii) when visiting a non-empty meta-bead, it services at least one target in it. This process is iterated at most log 2 n + 1 times, and at each phase meta-beads composed of two neighboring meta-beads from the previous phase are considered; in other words, the meta-beads at the i-th phase are composed of 2 i−1 neighboring beads. After the last phase, the leftover targets will be visited using, for example, a greedy strategy, or the Alternating Algorithm. We have the following result, which we prove using a technique similar to that developed in [17]. Lemma 2.2: Let P ∈ P n be uniformly randomly generated in Q. Then, the number of unvisited targets after the last phase of the RECURSIVE BEAD-TILING ALGORITHM over P belongs to O(log n) with high probability. Proof: Associate a unique identifier to each bead, e.g., integers between 1 and 2n; call such a set of identifiers I. Let b(t) ∈ I be the identifier of the bead in which the t-th target is sampled, and let h(t) ∈ N be the phase at which the t-th target is visited. Without loss of generality, we will assume that if b(t 1 ) = b(t 2 ), and t 1 < t 2 , then h(t 1 ) < h(t 2 ). Indicate with v i (t) the number of beads that contain unvisited targets at the inception of the i-th phase, computed after the insertion of the t-th target. Furthermore, let m i be the number of i-th phase meta-beads (i.e., meta-beads containing 2 i−1 neighboring beads) with a non-empty intersection with Q. Clearly, v i (t) ≤ v i (n), m i ≤ 2m i+1 , and v 1 (n) ≤ n ≤ m 1 /2 with certainty. The t-th target will not be visited during the first phase if it is sampled in a bead that already contains other targets. In other words, Pr[h(t) ≥ 2|v 1 (t)] = v 1 (t) m 1 ≤ v 1 (n) 2n ≤ 1 2 . Similarly, the t-th target will not be visited during the i-th phase if (i) it has not been visited before the i-th pass, and (ii) it belongs to a meta-bead that already contains other targets not visited before the i-th phase: Pr [h(t) ≥ i + 1|(v i (t − 1), v i−1 (t − 1), v 1 (t − 1))] = Pr[h(t) ≥ i + 1|h(t) ≥ i, v i (t − 1)] · Pr[h(t) ≥ i|(v i−1 (t − 1), . . . , v 1 (t − 1))] ≤ v i (t − 1) m i Pr[h(t) ≥ i|(v i−1 (t − 1), . . . , v 1 (t − 1))] = i j=1 v j (t − 1) m j ≤ i j=1 2 j−1 v j (n) 2n = 2 i−3 2 n i i j=1 v j (n). (3) Fig. 3. Sketch of "meta-beads" at successive phases in the recursive bead tiling algorithm. For a fixed i ≥ 1, define a sequence of binary random variables Y t = 1, if h(t) ≥ i + 1 and v i (t − 1) ≤ β i n; 0, otherwise.(4) In other words, Y t is equal to 1 if the t-th target is not visited within the first i phases despite the fact that the number of beads still containing unvisited target at the inception of the i-th phase is less than β i n; the values {β i } will be defined shortly. Even though the random variable Y t depends on the targets generated before the t-th target, the probability that it takes the value 1 is bounded by Pr[Y t = 1|b(1), b(2), . . . , b(t − 1)] ≤ 2 i(i−3) 2 i j=1 β j =: p i , regardless of the actual values of b(1), . . . , b(t − 1). It is known (e.g., see [17]) that if the random variables Y t satisfy such a condition, the sum Y t is stochastically dominated by a binomially distributed random variable, namely, Pr n t=1 Y t > k ≤ Pr[B(n, p i ) > k]. In particular, Pr n t=1 Y t > 2np i ≤ Pr[B(n, p i ) > 2np i ] < 2 −npi/3 ,(5) where the last inequality is obtained using Chernoff's bound. Let us define the sequence {β i } through the recursion β 1 = 1, β i+1 = 2p i = 2 i(i−3) 2 +1 i j=1 β j = 2 i−2 β 2 i , which leads to β i = 2 1−i .(6) With the above definition in mind, (5) can be rewritten as Pr n t=1 Y t > β i+1 n ≤ Pr[B(n, p i ) > β i+1 n] < 2 −βi+1n/6 = 2 − n 3·2 i which is less than 1/n 2 for i ≤ i * (n) := ⌊log 2 n − log 2 log 2 n − log 2 6⌋ ≤ log 2 n. Note that β i ≤ 12 log 2 n n ∀i > i * (n).(7) Let E i be defined as the event that v i (n) ≤ β i n. Note that if E i is true, then v i+1 (n) ≤ n t=1 Y t : the right hand side represents the number of targets that will be visited after the i-th phase, whereas the left hand side counts the number of beads containing such targets. We have, for all i ≤ i * (n): Pr [v i+1 > β i+1 n|E i ]·Pr[E i ] ≤ Pr n t=1 Y t > β i+1 n ≤ 1 n 2 , that is , Pr [¬E i+1 |E i ] ≤ 1 n 2 Pr[E i ] , and thus (recall that E 1 is true with certainty): Pr [¬E i+1 ] ≤ 1 n 2 + Pr[¬E i ] ≤ i n 2 . In other words, for all i ≤ i * (n), v i (n) ≤ β i n with high probability. Let us turn our attention to the phases such that i > i * (n). The total number of targets visited after the i * -th phase is dominated by a binomial variable B(n, 12 log 2 n/n); in particular, dealing with conditioning as before, we get Pr [v i * +1 > 24 log 2 n|E i * ]·Pr[E i * ] ≤ PrPr [v i * +1 > 24 log 2 n] ≤ 1 n 12 + Pr[¬E i * ] ≤ 1 n 12 + log 2 n n 2 . In other words, the number of targets that will be left after the i * -th phase will be bounded by a logarithmic function of n with high probability. D. A bound on the length of the solution What we know at this point is that after a sufficiently large number of phases, almost all targets will be visited, with high probability. The key point is to recognize that the length of each phase is decreasing at such a rate that the sum of the lengths of all the phases remains bounded. We first state and prove the following result which characterizes the length of Dubins' path required to execute the RECURSIVE BEAD-TILING ALGORITHM. Lemma 2.3: (Length of path for the RECURSIVE BEAD-TILING ALGORITHM) Let P ∈ P n be uniformly randomly generated in Q. Then the length of Dubins' path required to execute log n phases of the RECURSIVE BEAD-TILING ALGORITHM over P belongs to O(n 2/3 ). Proof: Let L i denote the upper bound on the length of the path for the i th phase. Then one can see that log 2 (n) i=1 L i ≤ 3 ⌈ log 2 (n) 2 ⌉ j=1 L 2j−1 . Let us first compute the length of a pass, in either direction. The number of beads traversed will be no more than W 2 j l n = c 1 2 j n 1 3 ,(8) where c 1 = W 3 √ ρW H is a constant. The length of Dubins' path contained entirely within a meta-bead at the (2j − 1)-th phase is less than 2 j−1 2l n + o(l 2 n ) . Hence, the total path length per pass will be bounded by: L pass,2j−1 ≤ 2 j−1 2l n + o(l 2 n ) c 1 2 j n 1 3 + 1 = c 1 l n n 1 3 + 2 j l n + c 1 2 n 1 3 o(l 2 n ) + 2 j−1 o(l 2 n )(9) as l n → 0 + . The cost of a U-turn, i.e., the length of the path needed to reverse direction and move to the next row of beads, is bounded by L u−turn,2j−1 ≤ 7 3 πρ+2 j−2 w(l n ) = 7 3 πρ+2 j−2 l 2 n 2ρ +o(l 3 n ) .(10) The total number of passes will be at most N pass,2j−1 = H 2 j−2 w(l n ) ≤ ρH 2 j−3 l 2 n + 1.(11) The cost of closing the tour is bounded by a constant, say L closure,2j−1 ≤ 4(W + Hπρ). Concluding, the total path length will be bounded by L 2j−1 = N pass,2j−1 (L pass,2j−1 +L u−turn,2j−1 )+L closure,2j−1 .(13) Substituting eqns. (9), (10), (11) and (12) in eqn. (13), one can find constants k 1 , k 2 and k 3 such that L 2j−1 ≤ k 1 1+n 2 3 o(n − 2 3 )+n 1 3 +k 2 2 j n − 1 3 + o(n − 2 3 ) + k 3 2 −j 1 + no(n − 2 3 ) + n 2 3 . From this expression for the length of path during odd phases, one can conclude that log(n) i=1 L i ≤ 3 ⌈ log(n) 2 ⌉ j=1 L 2j−1 ∈ O(n 2/3 ). Based on the results obtained so far, we are now ready to state an upper bound on the length of the path traveled by Dubins' vehicle to service all the n targets while executing the RECURSIVE BEAD-TILING ALGORITHM followed by the ALTERNATING ALGORITHM; let L RecBTA,ρ (P ) represent the corresponding quantity. Theorem 2.4: (Upper bound on the length of the total path) Let P ∈ P n be uniformly randomly generated in Q. For all ρ > 0, L RecBTA,ρ (P ) ∈ O(n 2/3 ) with high probability. Proof: By Lemma 2.3 the length of path to execute the RECURSIVE BEAD-TILING ALGORITHM belongs to O(n 2/3 ). From Theorem 2.2, the number of targets remaining at the end of the RECURSIVE BEAD-TILING ALGORITHM belongs to O(log n) with high probability. These remaining points can be serviced by any greedy policy or some heuristics (e.g., ALTERNATING ALGORITHM [9]) in O(log n) time. The statement of the theorem follows immediately. Combining results from Theorem 2.1 and Theorem 2.4, one can conclude that the RECURSIVE BEAD-TILING AL-GORITHM is a constant factor approximation to the optimal DTSP with high probability. III. CONCLUSIONS In this article, we have studied the TSP problem for vehicles that follow paths of bounded curvature in the plane. For the stochastic setting, we have obtained upper bounds that are within a constant factor of the lower bound established in literature [10]; the upper bounds are constructive in the sense that they are achieved by two novel algorithms. It is interesting to compare our results with the Euclidean setting (i.e., the setting in which curves do not have curvature constraints). For a given compact set and a point set P of n points, it is known [1], [2] that the ETSP(P ) belongs to Θ( √ n). This is true for both stochastic and worst-case settings. In this article, we showed that, given a fixed ρ > 0, the stochastic DTSP ρ (P ) belongs to Θ(n 2/3 ) with high probability. It is known [9] that the worst-case DTSP ρ (P ) belongs to Θ(n). In the future, we plan to perform extensive simulations to support the results obtained in this article. Future directions of research include study of centralized and decentralized versions of the DTRP and general task assignment and surveillance problems for various non-holonomic vehicles. Fig. 1 . 1Construction of the "bead" Bρ(l). The figure shows how the upper half of the boundary is constructed, the bottom half is symmetric. Fig. 2 . 2Fig. 2. Sketch of tiling of the region before the first phase of the RECURSIVE BEAD-TILING ALGORITHM. ≤ Pr[B(n, 12 log 2 n/n) > 24 log 2 n] ≤ 2 −12 log 2 n ; Savla and Francesco Bullo are with the Center for Control, Dynamical Systems and Computation, University of California at Santa Barbara, [email protected], [email protected] Emilio Frazzoli is with the Mechanical and Aerospace Engineering Department, University of California at Los Angeles, [email protected] ACKNOWLEDGMENT This material is based upon work supported in part by ONR YIP Award N00014-03-1-0512 and AFOSR MURI Award F49620-02-1-0325. The authors would like to thank John J. Enright for helpful discussions. The shortest path through many points. J Beardwood, J Halton, J Hammersly, Proceedings of the Cambridge Philosophy Society. the Cambridge Philosophy Society55J. Beardwood, J. Halton, and J. Hammersly, "The shortest path through many points," in Proceedings of the Cambridge Philosophy Society, vol. 55, pp. 299-327, 1959. Probabilistic and worst case analyses of classical problems of combinatorial optimization in Euclidean space. J M Steele, Mathematics of Operations Research. 154749J. M. Steele, "Probabilistic and worst case analyses of classical prob- lems of combinatorial optimization in Euclidean space," Mathematics of Operations Research, vol. 15, no. 4, p. 749, 1990. On the solution of traveling salesman problems. D Applegate, R Bixby, V Chvátal, W Cook, Proceedings of the International Congress of Mathematicians, Extra Volume ICM III. the International Congress of Mathematicians, Extra Volume ICM IIIBerlin, GermanyDocumenta MathematicaD. Applegate, R. Bixby, V. Chvátal, and W. Cook, "On the solu- tion of traveling salesman problems," in Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, (Berlin, Germany), pp. 645-656, Aug. 1998. Proceedings of the International Congress of Mathematicians, Extra Volume ICM III. Nearly linear time approximation scheme for Euclidean TSP and other geometric problems. S Arora, Proc. 38th IEEE Annual Symposium on Foundations of Computer Science. 38th IEEE Annual Symposium on Foundations of Computer ScienceMiami Beach, FLS. Arora, "Nearly linear time approximation scheme for Euclidean TSP and other geometric problems," in Proc. 38th IEEE Annual Symposium on Foundations of Computer Science, (Miami Beach, FL), pp. 554- 563, Oct. 1997. An effective heuristic algorithm for the traveling-salesman problem. S Lin, B W Kernighan, Operations Research. 21S. Lin and B. W. Kernighan, "An effective heuristic algorithm for the traveling-salesman problem," Operations Research, vol. 21, pp. 498- 516, 1973. P-complete approximation problems. S Sahni, T Gonzalez, Journal of the Association of Computing Machinery. 233S. Sahni and T. Gonzalez, "P-complete approximation problems," Journal of the Association of Computing Machinery, vol. 23, no. 3, pp. 555-565, 1976. On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents. L E Dubins, American Journal of Mathematics. 79L. E. Dubins, "On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents," American Journal of Mathematics, vol. 79, pp. 497- 516, 1957. Classification of the Dubins set. A M Shkel, V J Lumelsky, Robotics and Autonomous Systems. 34A. M. Shkel and V. J. Lumelsky, "Classification of the Dubins set," Robotics and Autonomous Systems, vol. 34, pp. 179-202, 2001. On the point-to-point and traveling salesperson problems for Dubins' vehicle. K Savla, E Frazzoli, F Bullo, American Control Conference. Portland, ORK. Savla, E. Frazzoli, and F. Bullo, "On the point-to-point and traveling salesperson problems for Dubins' vehicle," in American Control Conference, (Portland, OR), pp. 786-791, June 2005. UAV routing in a stochastic time-varying environment. J J Enright, E Frazzoli, Electronic Proceedings. Prague, Czech RepublicIFAC World CongressJ. J. Enright and E. Frazzoli, "UAV routing in a stochastic time-varying environment," in IFAC World Congress, (Prague, Czech Republic), July 2005. Electronic Proceedings. On traveling salesperson problems for Dubins' vehicle: stochastic and dynamic environments. K Savla, F Bullo, E Frazzoli, IEEE Conf. on Decision and Control. Seville, SpainK. Savla, F. Bullo, and E. Frazzoli, "On traveling salesperson problems for Dubins' vehicle: stochastic and dynamic environments," in IEEE Conf. on Decision and Control, (Seville, Spain), pp. 4530-4535, Dec. 2005. On the average length of the stochastic TSP with Dubin's metric. S Itani, M A Dahleh, Submitted to the American Control ConferenceS. Itani and M. A. Dahleh, "On the average length of the stochastic TSP with Dubin's metric." Submitted to the American Control Con- ference, 2005. Decentralized algorithms for vehicle routing in a stochastic time-varying environment. E Frazzoli, F Bullo, IEEE Conf. on Decision and Control. Paradise Island, BahamasE. Frazzoli and F. Bullo, "Decentralized algorithms for vehicle routing in a stochastic time-varying environment," in IEEE Conf. on Decision and Control, (Paradise Island, Bahamas), pp. 3357-3363, Dec. 2004. Coordinated target assignment and intercept for unmanned air vehicles. R W Beard, T W Mclain, M A Goodrich, E P Anderson, IEEE Transactions on Robotics and Automation. 189R. W. Beard, T. W. McLain, M. A. Goodrich, and E. P. Anderson, "Coordinated target assignment and intercept for unmanned air vehi- cles," IEEE Transactions on Robotics and Automation, vol. 18, no. 9, pp. 911-922, 2002. Combinatorial motion planning for a collection of Reeds-Shepp vehicles. S Darbha, tech. rep., ASEE/AFOSR SFFP, AFRL, EglinS. Darbha, "Combinatorial motion planning for a collection of Reeds- Shepp vehicles," tech. rep., ASEE/AFOSR SFFP, AFRL, Eglin, Aug. 2005. A stochastic and dynamic vehicle routing problem in the Euclidean plane. D J Bertsimas, G J Van Ryzin, Operations Research. 39D. J. Bertsimas and G. J. van Ryzin, "A stochastic and dynamic vehicle routing problem in the Euclidean plane," Operations Research, vol. 39, pp. 601-615, 1991. Balanced allocations. Y Azar, A Z Broder, A R Karlin, E Upfal, SIAM Journal on Computing. 291Y. Azar, A. Z. Broder, A. R. Karlin, and E. Upfal, "Balanced allocations," SIAM Journal on Computing, vol. 29, no. 1, pp. 180- 200, 1999.

Aims. Several authors have found a statistically significant excess of galaxies with higher redshifts relative to the group centre, socalled discordant redshifts, in particular in groups where the brightest galaxy, identified in apparent magnitudes, is a spiral. Our aim is to explain the observed redshift excess. Methods. We use a semi-analytical galaxy catalogue constructed from the Millennium Simulation to study redshift asymmetries in spiral-dominated groups in the Λcold dark matter (ΛCDM) cosmology. We create two mock catalogues of galaxy groups with the Friends-of-Friends percolation algorithm to carry out this study. Results. We show that discordant redshifts in small galaxy groups arise when these groups are gravitationally unbound and the dominant galaxy of the group is misidentified. About one quarter of all groups in our mock catalogues belong to this category. The redshift excess is especially significant when the apparently brightest galaxy can be identified as a spiral, in full agreement with observations. On the other hand, the groups that are gravitationally bound do not show a significant redshift asymmetry. When the dominant members of groups in mock catalogues are identified by using the absolute B-band magnitudes, our results show a small blueshift excess. This result is due to the magnitude limited observations that miss the faint background galaxies in groups. Conclusions. When the group centre is not correctly identified it may cause the major part of the observed redshift excess. If the group is also gravitationally unbound, the level of the redshift excess becomes as high as in observations. There is no need to introduce any "anomalous" redshift mechanism to explain the observed redshift excess. Further, as the Friends-of-Friends percolation algorithm picks out the expanding parts of groups, in addition to the gravitationally bound group cores, group catalogues constructed in this way cannot be used as if the groups are purely bound systems.

10.1051/0004-6361:200810795

0811.3968

0e56129bc9ad0a2beebc5da5eaf9c778a2846afd

The origin of redshift asymmetries: How ΛCDM explains anomalous redshift 10 Dec 2008 December 10, 2008 S.-M Niemi [email protected] Department of Physics and Astronomy University of Turku Tuorla Observatory Väisäläntie 20PiikkiöFinland Nordic Optical Telescope Apartado 474, E−38700 Santa Cruz de La PalmaSanta Cruz de TenerifeSpain M Valtonen Department of Physics and Astronomy University of Turku Tuorla Observatory Väisäläntie 20PiikkiöFinland The origin of redshift asymmetries: How ΛCDM explains anomalous redshift 10 Dec 2008 December 10, 2008Released 2008Astronomy & Astrophysics manuscript no. document c ESO 2008Galaxies: clusters: general -Galaxies: distances and redshifts -Methods: N-body simulations -Methods: statistical - cosmology: large-scale structure of Universe Aims. Several authors have found a statistically significant excess of galaxies with higher redshifts relative to the group centre, socalled discordant redshifts, in particular in groups where the brightest galaxy, identified in apparent magnitudes, is a spiral. Our aim is to explain the observed redshift excess. Methods. We use a semi-analytical galaxy catalogue constructed from the Millennium Simulation to study redshift asymmetries in spiral-dominated groups in the Λcold dark matter (ΛCDM) cosmology. We create two mock catalogues of galaxy groups with the Friends-of-Friends percolation algorithm to carry out this study. Results. We show that discordant redshifts in small galaxy groups arise when these groups are gravitationally unbound and the dominant galaxy of the group is misidentified. About one quarter of all groups in our mock catalogues belong to this category. The redshift excess is especially significant when the apparently brightest galaxy can be identified as a spiral, in full agreement with observations. On the other hand, the groups that are gravitationally bound do not show a significant redshift asymmetry. When the dominant members of groups in mock catalogues are identified by using the absolute B-band magnitudes, our results show a small blueshift excess. This result is due to the magnitude limited observations that miss the faint background galaxies in groups. Conclusions. When the group centre is not correctly identified it may cause the major part of the observed redshift excess. If the group is also gravitationally unbound, the level of the redshift excess becomes as high as in observations. There is no need to introduce any "anomalous" redshift mechanism to explain the observed redshift excess. Further, as the Friends-of-Friends percolation algorithm picks out the expanding parts of groups, in addition to the gravitationally bound group cores, group catalogues constructed in this way cannot be used as if the groups are purely bound systems. Introduction Groups of galaxies contain a large fraction of all galaxies in the Universe (Holmberg 1950;Humason et al. 1956;Huchra & Geller 1982;Geller & Huchra 1983;Nolthenius & White 1987;Ramella et al. 2002). These density enhancements in the sky and in the redshift space are important cosmological indicators of the distribution of matter in the Universe, and may provide important clues for galaxy formation. Groups of galaxies are, in general, divided into a large number of different classes, for example, loose groups (e.g. Ramella et al. 1995;Tucker et al. 2000;Einasto et al. 2003), poor groups (e.g. Zabludoff & Mulchaey 1998;Mahdavi et al. 1999), compact groups (e.g. Shakhbazyan 1973;Hickson 1982;Hickson et al. 1989;Focardi & Kelm 2002) and fossil groups (e.g. Ponman et al. 1994;Jones et al. 2003;D'Onghia et al. 2005;Santos et al. 2007). However, from the observational point of view, groups of galaxies and their member galaxies are not extremely well defined. In recent years a number of grouping algorithms have been developed and applied (e.g. Turner & Gott 1976;Materne 1978;Huchra & Geller 1982;Botzler et al. 2004; Goto et al. 2002;Kim et al. 2002;Bahcall et al. 2003;Koester et al. 2007; Yang et al. 2007) to identify real groups. Despite the vast number of grouping algorithms, the Friends-of-Friends (FoF; Huchra & Geller 1982, hereafter HG82) percolation algorithm remains the most frequently applied. The FoF algorithm or slightly modified versions of it are widely used even for modern day galaxy surveys. Several authors have presented group and cluster catalogues that applied the FoF algorithm based on the SDSS (Sloan Digital Sky Survey) data (e.g. Merchán & Zandivarez 2005;Berlind et al. 2006;Tago et al. 2008) and on the 2dF (The Two Degree Field) data (e.g. Eke et al. 2004;Tago et al. 2006). Few studies (see Niemi et al. 2007, and references therein) have argued that grouping algorithms may not always return true groups; a significant number of groups can be spurious and contain interlopers. Niemi et al. (2007) (hereafter Paper I) have shown that the FoF algorithm produces a significant fraction of groupings which are not gravitationally bound systems, but merely groups in a visual sense. This can introduce various errors when these groups are studied in detail and treated as gravitationally bound structures. This may even be true for compact groups of galaxies. In this case, and also in general, extended X-ray emission can be used to distinguish real, gravitationally bound groups from spurious ones. The analysis of X-ray data suggests that errors are rather common (Ostriker et al. 1995, and references therein). Unfortunately, groups with low mass and a spiral-dominated main galaxy in general do not show any extended X-ray emission. Therefore this method is not useful for small and loose groups of galaxies. An excess of higher redshift galaxies was discovered by Arp (1970Arp ( , 1982 and it was studied in detail by Jaakkola (1971). Since then many authors have found a statistically significant excess of high redshift companions relative to the group centre. Bottinelli & Gouguenheim (1973) extended the study of Arp (1970) to nearby groups of galaxies in which the magnitude difference between the companion and the main galaxy was greater than 0.4 mag. Sulentic (1984) found a statistically significant excess of positive redshifts while studying spiral-dominated (i.e. the central galaxy is a spiral galaxy) groups in the catalogue of galaxy groups by HG82, and derived the redshift excess Z = 0.21 for spiral-dominated groups while the E/S0 dominated (i.e. the central galaxy is an E/S0 galaxy) groups showed a blueshift excess Z = −0.13. Girardi et al. (1992) found discordant redshifts while studying nearby small groups identified by Tully (1988) in the Nearby Galaxy Catalogue. However, the conventional theory holds that the distribution of redshift differentials for galaxies moving under the gravitational potential of a group should be evenly distributed. Even systematic radial motions within a group would be expected to produce redshift differentials that are evenly distributed. Multiple theories have been suggested to explain the observed redshift excess. Sulentic (1984) listed some possible origins for the observed redshift excess. Byrd & Valtonen (1985) and Valtonen & Byrd (1986) argued that this positive excess is mainly due to the unbound expanding members and the fact that the dominant members of these groups are sometimes misidentified. Girardi et al. (1992) argued that the positive excess may be explained if groups are still collapsing and contain dust in the intragroup medium. Hickson et al. (1988) ran Monte Carlo simulations and concluded that the random projection can explain discordant redshifts. Iovino & Hickson (1997) found that projection effects alone can account for the high incidence of discordant redshifts. However, studies by Hickson et al. (1988) and Iovino & Hickson (1997) dealt only with Hickson's compact groups of galaxies. Tully (1987) analyzed his catalogue of nearby groups of galaxies and did not find evidence of redshift asymmetries in galaxy groups. However, unlike in earlier work, his reference system was not the apparently brightest group member, but the unweighted average velocity of members. Zaritsky (1992) studied asymmetric distribution of satellite galaxy velocities with Monte Carlo simulations and concluded that observational biases partially explain the observed redshift asymmetry. Despite the number of explanations none of these explanations are satisfactory. Even new physics has been suggested for the solution (see e.g. Arp 1970). However, there may be a simple explanation of redshift asymmetries which does not require modifications of well accepted physics, now that the ΛCDM model can be counted as generally accepted. This was first proposed by Byrd & Valtonen (1985) who pointed out that redshift asymmetries should arise in nearby groups of galaxies such as HG groups if a large fraction of the group population is unbound to the group. They argued that the redshift asymmetry explains the need for "missing matter", the dark matter that was at the time supposed to exist at the level of the closing density of the universe in groups of galaxies. If the group as a whole is not virialized, there is no need for excessive amounts of binding matter. It has now become possible to test this assumption quantitatively, and this is the main focus of this paper. Independent evidence has recently appeared of unbound outlying populations of galaxies around the Local Group and a few other nearby groups, as one would expect in the ΛCDM cosmology (Teerikorpi et al. 2008). Thus in principle the redshift asymmetry explanation of Valtonen & Byrd (1986) should work; whether it works quantitatively is a question to be answered in this paper. In this paper we study redshift asymmetries theoretically in small groups of galaxies by taking advantage of the largest cosmological N−body simulation conducted so far: the Millennium Run (Springel et al. 2005). We create two 'mock' catalogues of groups of galaxies from the semi-analytical galaxy catalogue (Lucia & Blaizot 2007) of the Millennium Simulation by mimicking observational methods. For the creation of mock catalogues we apply the Friends-of-Friends percolation algorithm developed by HG82. The two mock catalogues differ in the values of free parameters of the FoF grouping algorithm. We show that the excess of positive redshifts is mainly due to wrong identification of the dominant galaxy in a group and, at the same time, the group being gravitationally unbound. We also show that groups that show a large excess of positive redshifts are more often gravitationally unbound than groups that do not show any significant excess. These errors in the identification of the dominant galaxy result from our current inability to measure the relative distances inside groups of galaxies, except for a few of the nearest ones (Karachentsev et al. 1997;Jerjen et al. 2001;Rekola et al. 2005a,b;Teerikorpi et al. 2008). Due to peculiar motions of group members we cannot transform the apparent magnitudes of galaxies in group catalogues into absolute magnitudes precisely. If it were possible and the dominant group members were correctly identified, would it lead to a small blueshift excess, which is due to the magnitude limited observations that cause some of the background galaxies to be invisible and thus excluded from groups. This paper is organized as follows. In Section 2, we discuss our sample of galaxy groups, the Millennium Simulation data and the grouping algorithm adopted. We present our findings and results in Section 3. Finally, we summarize our results in Section 4. Throughout this paper we adopt a parametrized Hubble constant: H 0 = 100h km s −1 Mpc −1 . Unless explicitly noted, we adopt h = 1.0 for convenient comparison with older literature. The sample of galaxy groups The two mock catalogues of groups of galaxies from Millennium Simulation data were constructed so that they would be comparable to the real observational group catalogues HG82 and UZC-SSRS2 (Ramella et al. 2002) as much as possible. Both of these catalogues are produced with the FoF algorithm. However, slightly different values for the free parameters of the algorithm have been used. We show in Section 3.1 that our groups from the Millennium Simulation are comparable to observed groups in a statistical sense. We also discuss briefly how our results compare to works of other authors. Millennium Simulation data The Millennium Simulation (MS; Springel et al. 2005) is a cosmological N−body simulation of the ΛCDM model performed by the Virgo Consortium. The MS was carried out with a customized version of the GADGET2 code developed by Springel et al. (2001b). The MS follows the evolution of 2160 3 particles from redshift z = 127 in a box of 500h −1 Mpc on a side. The cosmological parameters of the MS simulation are: Ω m = Ω dm + Ω b = 0.25, Ω b = 0.045, h = 0.73, Ω Λ = 0.75, n = 1, and σ 8 = 0.9 (for a detailed description of the MS see Springel et al. 2005). The galaxy formation modeling of the MS data is based on merger trees built from 64 individual snapshots. Properties of galaxies in MS data are obtained by using semi-analytic galaxy formation models, where the star formation and its regulation by feedback processes is parametrized in terms of analytical physical models. A detailed description of the creation of the MS galaxy catalogue can be found in Lucia & Blaizot (2007), see also Croton et al. (2006). The MS galaxy database does not directly give a morphology for galaxies. We have used a method which takes an advantage of bulge-to-disk ratios to assign a morphology to every galaxy. Simien & de Vaucouleurs (1986) found a correlation between the B-band bulge-to-disc ratio, and the Hubble type T of galaxies. The mean relation may be written: < ∆m(T ) > = 0.324x(T ) − 0.054x(T ) 2 + 0.0047x(T ) 3 ,(1) where ∆m(T ) is the difference between the bulge magnitude and the total magnitude and x(T ) = T + 5. We classify galaxies with T < −2.5 as ellipticals, those with −2.5 < T < 0.92 as S0s, and those with T > 0.92 as spirals and irregulars. Galaxies without any bulge are classified as type T = 9. These classification criteria are the same as proposed and adopted by Springel et al. (2001a). Group catalogues Our mock catalogues of groups of galaxies are generated with the FoF percolation algorithm developed by HG82. Even though new algorithms have been developed, the FoF still remains the most applied one. The FoF algorithm uses only two criteria for finding group members: position and redshift. It essentially finds density enhancements in position and in redshift space above a set threshold factor. This threshold depends on a chosen value of the free parameter D 0 , the apparent magnitude limit of the search and the Schechter (1976) luminosity function. Density enhancement relative to the mean number density can be calculated from the equation: δρ ρ = 3 4πD 3 0 M lim −∞ Φ(M)dM −1 − 1,(2) where D 0 is the projected separation in Mpc chosen at some fiducial redshift, M lim = m lim − 30 and Φ(M) is the Schechter (1976) luminosity function. For a more detailed description of the FoF algorithm, see Paper I and the references therein, especially HG82. We produce two mock catalogues with different choices of free parameters. Both group catalogues are generated from five independent volumes of the Millennium Simulation galaxy catalogue. Each of the cubes used have a side length of 250h −1 Mpc, and they do not overlap. The observation point inside each volume was chosen to be in the centre of the particular cube. No additional criteria were applied for the selection of observation points. Reasonable statistical agreement, as shown in the next section between mock catalogues and observed group catalogues shows that our method of choosing the origin without any further criteria is strict enough in a statistical study of galaxy groups. Both of our catalogues from simulations, Mock1 and Mock2, contain groups of galaxies whose lower limit on the number of members, n, is 3. All groups containing n > 2 members are considered when group properties are studied in Section 3.1. In Section 3.2 we limit the number of group members to 2 < n < 11, comparable to S84. Our first mock catalogue, named Mock1, is generated with the same values ( m lim = 13.2 [in the Zwicky or de Vaucouleurs B(0) magnitude system], D 0 = 0.60 h −1 Mpc and V 0 = 400 km s −1 ) of the free parameters as the original HG82 catalogue. This choice guarantees that we can compare results found in S84 directly to our simulated catalogue and we can be sure that different choices of parameter values do not effect the results. Even though the Mock1 catalogue uses the same values of parameters as HG82 it contains over 10 times more groups than the original HG82 catalogue. Thus it provides significantly better statistics. Our Mock2 catalogue has an apparent B-band magnitude limit of m lim = 14.0, while V 0 = 200 km s −1 and D 0 = 0.37 h −1 Mpc were adopted for the free parameters of the FoF algorithm, corresponding to the space density enhancement of ∼ 68. Despite the use of more strict parameters, Mock2 contains almost 10 times more groups than Mock1 due to the fainter apparent B-band magnitude limit adopted. Because of the greater number of groups, Mock2 is used for comparison and for better statistics. It should also contain groups which are more often gravitationally bound due to the higher density enhancement of groups. Table 1 shows the values of free parameters used in creating our mock catalogues. It also shows parameter values of various group catalogues based on modern redshift surveys. All catalogues shown in Table 1 have been generated with the FoF algorithm. However, catalogues based on the SDSS and the 2dF data have taken advantage of modified versions of the original FoF. The most noticeable modifications include the use of dark matter mock catalogues and group re-centering. There are large differences between the Mock1 and Mock2 catalogues. The most obvious difference is in the total number of groups. Mock1 contains 1601 groups in total while Mock2 contains 13786 groups. Note that five different 'observation' points inside the MS are used, and none of these observation points overlap each other. The difference in the number of groups is due to the difference in the adopted apparent B-band magnitude limit. If we compare the fraction of gravitationally bound groups between the two catalogues, (i.e. T kin /U < 1, where T kin is the kinetic energy and U is the absolute value of the potential energy of the group; for a detailed description see Paper I) the differences are not great. The fraction of gravitationally bound groups is surprisingly low in both mock catalogues. In the Mock1 catalogue the fraction of bound groups is 52.64 ± 9.21% while in Mock2 it is only 50.19 ± 1.15%. The error limits are standard deviation errors between the five observation points. It is an indication of the reliability of the group-finding algorithm that the same relative number of bound groups are found in spite of the fact that the Mock2 catalogue has about three times higher density enhancement than Mock1. In Paper I we found that the fraction of gravitationally bound groups of dark matter haloes is ∼ 30% when a ΛCDM model has been adopted. The fraction of gravitationally unbound groups is significant in all these mock group catalogues, which suggests that group catalogues based on the FoF algorithm contain a significant fraction of groups that are not gravitationally bound systems. Analysis of groups Comparison with observations In this subsection we briefly show that our mock catalogues are comparable to real observational group catalogues. We compare The group catalogue used, m lim is the apparent magnitude limit of the search (B-band for mock catalogues, r-band for the SDSS and b j for the 2dF, note however that different redshift limits have been applied in individual papers), D 0 is the projected separation in h −1 Mpc chosen at some fiducial redshift, V 0 is the velocity difference in km s −1 , δρ ρ is the density enhancement and α, M * , and φ * parametrize the Schechter (1976) function. The Schechter parameters for the Mock2 sample are derived from the Millennium Galaxy Catalogue by Driver et al. (2007). The Schechter parameters for Ramella et al. (2002) refer to the CfA (SSRS2) groups. our simulated mock catalogues to UZC-SSRS2 and to HG82 catalogues. We only show comparisons in velocity dispersion and in the 'observable' mass of groups. Even though our mock catalogues are comparable to observations, we find some differences. We also find differences between the two mock catalogues. Before discussing these differences we briefly review the comparison catalogues and parameters. The HG82 group catalogue was derived from a whole sky catalogue of 1312 galaxies brighter than m B = 13.2 (in Zwicky or de Vaucouleurs B(0) magnitude system) with complete redshift information. The velocity of each galaxy has been corrected for a dipole Virgo-centric flow. The catalogue of groups was obtained with D 0 = 0.60 h −1 Mpc (corresponding to a density enhancement of ∼ 20) and with V 0 = 400 km s −1 . Only groups containing more than two members have been included in the final catalogue. The UZC-SSRS2 group catalogue was derived from a magnitude-limited redshift sample of galaxies. A compilation of 6846 galaxies with the apparent magnitude limit of m lim ≤ 15.5 was used for the creation of the UZC-SSRS2 catalogue, which contains, in total, 1168 groups. The group catalogue covers 4.69 sr, and the parameter values of V 0 = 350 km s −1 and D 0 = 0.25 h −1 Mpc have been adopted for the creation of the UZC-SSRS2 catalogue. These values correspond to a density contrast threshold ∼ 80. Only groups containing more than two members have been included in the final catalogue. Our mock catalogues were discussed in the previous Section. However, we would like to point out that our Mock1 (Mock2) catalogue is comparable to HG82 (UZC-SSRS2) in the choice of parameters. The Mock2 catalogue does not adopt exactly the same parameters as the UZC-SSRS2, even though the density enhancement is comparable. The reason for not adopting exactly the same values is that volumes inside the MS are not large enough. Adopting m lim = 15.5 would have introduced errors in groups and their properties due to edge effects and missing group members. We wish to avoid this, as our purpose is to study distribution of group members and possible redshift asymmetries. We have shown in Paper I that cosmological N−body simulations can produce groups of galaxies which are comparable to observations. However, in Paper I, we were greatly limited by the volume of our simulation boxes, causing comparisons to be less conclusive. Moreover, In Paper I we compared properties of dark matter haloes to real observed galaxies. As Millennium Simulation offers a larger volume and the proper-ties of the galaxy data are derived with semi-analytical models, the comparison between mock catalogues and observational catalogues (UZC-SSRS2 and HG82) is now more robust. We use the same definitions and equations for velocity dispersion and 'observable' mass of a group as in Paper I. The velocity dispersion σ v of a group is defined as: σ v = 1 N G − 1 N G i=1 (v i − < v R >) 2 ,(3) where N G is the number of galaxies in a group, v i is the radial velocity of the ith galaxy and < v R > is the mean group radial velocity. In observations, the group masses can be estimated by various methods. In the HG82 and in the UZC-SSRS2 catalogues the total mass of a group is estimated with a simple relation: M obs = 6.96 × 10 8 σ 2 v R H M ⊙ ,(4) where σ v is the velocity dispersion, and R H is the mean harmonic radius: R H = π < v R > H 0 sin          1 2          N G (N G − 1) 2         N G i=1 N G j>i θ i j         −1                   ,(5) where θ i j is the angular separation of the ith and jth group members. To compare abundances of groups in magnitude-limited samples we weight each group according to its distance (Moore et al. 1993;Diaferio et al. 1999). This weighting is necessary as there is no 'total' volume of a galaxy sample in magnitude-limited group catalogues. After weighting each group individually we can scale abundances of groups in Figs. 1 and 2 to the comoving volume of the sample. We include all galaxies with cz > 500 km s −1 . This lower cut-off avoids including faint objects that are close to the observation point as these groups could contain galaxies fainter than in real magnitudelimited surveys. Therefore we consider only groups with mean radial velocity, < cz >, greater than 500 km s −1 in Mock1, Mock2, HG82 and UZC-SSRS2 catalogues. Figs. 1 and 2 show that cosmological N−body simulations can produce groups of galaxies whose statistical properties are similar to observed ones. The agreement is, in general, within show that the total density of the Mock1 and HG82 catalogue is lower than of Mock2 and UZC-SSRS2. This result is due to the lower apparent magnitude limit of these catalogues, even when weighting is applied. Mock1 and the HG82 are missing more faint galaxies than Mock2 and UZC-SSRS2 due to the lower magnitude limit. Because of this, these catalogues are missing groups that are built only from relatively faint galaxies i.e. these catalogues miss small groups with small velocity dispersions in comparison to Mock2 and UZC-SSRS2. Fig. 1 shows that groups of galaxies in our mock catalogues are similar to observed ones in a statistical sense when velocity dispersions are studied. However, both mock catalogues show an excess of high velocity dispersion groups in comparison to observations. Despite these differences the K−S test is approved (at level of 0.01) for Mock1 when the comparison is to HG82 and for Mock2 when the comparison is to the UZC-SSRS2 catalogue. It is also noteworthy that the Mock1 catalogue shows a small excess of groups around ∼ 500 km s −1 in comparison to Mock2. This difference is probably due to the larger value of V 0 in Mock1 that allows a greater difference between group members in redshift space. Qualitatively better agreement in velocity dispersion is observed when mock catalogues are compared to the UZC-SSRS2 catalogue. There are also some differences between the HG82 and UZC-SSRS2 catalogue, especially when the abundance of high velocity dispersion groups are considered. The discrepancy between the mock catalogues and the HG82 group catalogue is relatively large when large velocity dispersions are considered. This difference is due to the low number of groups (92) in HG82. Also the volume of the HG82 catalogue is relatively small. Thus, the HG82 catalogue lacks high velocity dispersion groups and clusters as the Virgo cluster is the only big cluster within the visible volume of the catalogue. The quartile values of velocity dispersion of the Mock1 The above differences show that the selection of V 0 is important, especially for velocity dispersions of groups. From Fig. 2 it is clear that our mock catalogues show an excess in the abundance of heavier groups. Despite the differences the K−S test supports the null hypothesis for the Mock2 catalogue (for numerical details see Table 2). Because of the strong connection between group mass and the velocity dispersion (see Eq. 4) the Mock1 catalogue shows (qualitatively) similar behavior in both Figs. 1 and 2. There is also a significant difference between HG82 and UZC-SSRS2 catalogue when the abundance of massive groups is studied. However, the most striking difference is between mock catalogues and HG82 when massive groups and clusters are considered. These differences can be explained with the small volume and the low number of large groups in HG82. In total, HG82 has only two groups with more than 30 members while the UZC-SSRS2 catalogue has 14 groups. The Mock1 (Mock2) catalogue has 45 (276) groups that have more than 30 members. It is obvious that the most massive groups are the ones that have highest velocity dispersion and that are the most expanded ones, meaning simply the ones having most members. The quartile values of 'observable' mass of the Mock1 (Mock2) catalogue groups are 0.7/3.5/7.9 × 10 13 M ⊙ (1.4/3.1/15.8 × 10 12 M ⊙ ). The values of Mock2 are close to the observed values (see Table 4 in Paper I for numerical details) even if there is an excess in the abundance of massive groups in comparison to observations. The good agreement in quartile values suggests that simulated groups are similar in a statistical sense, as the greatest differences in Fig. 2 are observed at very low densities. Despite the differences discussed above, our mock catalogues are comparable to observational catalogues in a statistical sense (see Table 2). This conclusion is supported by the K−S tests, by the good agreement on quartile values and by the fact that the properties of simulated groups are, in general, within 2σ error limits. We do not compare other properties of groups in this paper, as the focus of the paper is to study and explain the observed redshift asymmetries. However, before explaining these asymmetries we briefly compare our results to the findings of other authors. Eke et al. (2004Eke et al. ( , 2006 have previously applied the same method to galaxy groups identified in the 2dFGR Survey. These groups are typically much further away than our groups, as the median redshift of their groups is 0.11. Despite this difference we make an attempt to compare their results to ours. Eke et al. (2004) found the median velocity dispersion of 227 km s −1 for groups with at least three members. This is rather close to our Mock1 catalogue value (∼ 213 km s −1 ) implying that our mock groups are similar to their groups in a statistical sense. The median velocity dispersion of the Mock2 catalogue is only ∼ 131 km s −1 , and differs significantly from the value of Eke et al. (2004). This result is somewhat expected, as our mock groups are all found at very close distances. The median value of dynamical mass in our Mock1 catalogue is close to the value of Eke et al. (2004); even though their definition of dynamical mass is different to ours. Eke et al. (2004) found that as much as ∼ 40 per cent of groups can contain interlopers. This fraction is close to the fraction of groups we find to be gravitationally unbound. Eke et al. (2006) found that dynamical group masses give higher abundances to the mass function. It is possible that the reason for this discrepancy lies in the large fraction of unbound groups. If the (dynamical) mass of a group is calculated by adopting the virial theorem, one has to assume that the group is a bound structure. However, if one applies the virial theorem to a group that is gravitationally unbound, the dynamical mass of the group can be overestimated significantly. Even if the virial theorem is not used while calculating the dynamical mass, one can easily overestimate the mass, as unbound groups can have significantly higher velocity dispersion and size, leading to the higher abundance noticed by Eke et al. (2006). As groups observed by Eke et al. (2004Eke et al. ( , 2006 are typically much further away, these groups are not expected to show significant redshift asymmetry. In our groups the distance ratio of the far side of the group relative to the front side of the group can be rather large, and this causes the interesting effects that are discussed next. Redshift asymmetries in groups In this subsection we use our mock catalogues to study and explain the observed redshift asymmetries. However, before that, we discuss the suitability of our mock catalogues and quantify the methods of calculating the redshift asymmetries. The Mock1 catalogue has been generated with the same parameters as the original HG82 catalogue, from which S84 found positive redshift excess. Because of this it is most suitable for this study and for the comparison of redshift excesses found in S84. Mock1 is also favoured because a redshift excess is most noticeable for nearby groups for which there is a larger distance ratio between the background and the foreground galaxies. Mock2 is used to provide better statistics and for comparison. Note, however, that Mock2 probes groups deeper in redshift space than Mock1, and therefore the total effect of nearby groups is not as strong. It is also noteworthy that in this subsection we study small groups, therefore we limit the number of group members to less than 11 (i.e. 2 < n ≤ 10). Both mock catalogues have been generated from the Millennium Simulation. Even though the Millennium run provides reliable statistics its mass resolution is only ∼ 10 9 M ⊙ . This complicates the matter of identifying low mass groups with at least three members inside. The low mass resolution might have a large effect on studies of the abundance or space density of individual objects. An even greater effect would be noticed if one was interested in subhalo properties or abundances. However, as our purpose is to study the relative location and distribution of galaxies inside groups, we do not consider the low mass resolution as a significant problem. Further, Section 3.1 showed that the abundance of low mass groups in the Millennium Simulation is comparable to observable catalogues. This ensures that the mass resolution is good enough for a statistical study like ours, especially as in a statistical sense missing dwarf galaxies could reside anywhere inside the dark matter halo. We calculate redshift asymmetries both on a group and on a galaxy level. On a group level we use groups from five different observation points and calculate the sum of groups with redshift/blueshift excess in each point. The redshift or the blueshift excess percentage shown is the mean value of the excesses from the five observation points when ties have been removed. The error limits are standard deviation errors between five observation points. When the redshift asymmetries are studied on a member galaxy level, we quantify the redshift excess as in Byrd & Valtonen (1985): Z = N R − N B N R + N B ,(6) where N B is the number of galaxies having a redshift lower than the apparently brightest group member and N R is the number of galaxies having a redshift higher than the apparently brightest group member. Table 3 shows a general comparison of the mock catalogues and samples from S84 when the asymmetries have been calculated on a group level. Mock catalogues show a (weak) positive redshift excess when spiral-dominated (i.e. the central galaxy is a spiral galaxy) groups are studied. However, the excess is not significant for elliptical (E or S0) dominated groups. These groups actually show a small blueshift excess, similar to S84. By conventional theory, this suggests that these groups could be mostly gravitationally bound. However, we find that ∼ 35.0 ± 10.0 (27.4 ± 1.3) per cent of E/S0-dominated groups in Mock1 (Mock2) are gravitationally unbound. This suggests that Note: Sample refers either to S84, Mock1 or Mock2 catalogues, n G is the number of groups, n +∆z is the number of groups with positive redshift excess, n −∆z is the number of groups with negative redshift excess, n ±∆z is the number of groups with no excess, +∆z is the percentage of groups with positive redshift excess when ties have been removed, −∆z is the percentage of groups with negative redshift excess when ties have been removed and ±∆z is the percentage of groups having equal number of negative and positive redshifts. Errors for mock catalogues are standard deviation errors between five different observation points. the absence of redshift asymmetry does not alone guarantee that these groups are gravitationally bound systems. Table 4 further divides the mock catalogues to subsamples on the basis of whether the groups are gravitationally bound or not. The unbound groups are important, since about one half of all groups in both of our mock catalogues belong to this category. A further division in Table 4 is made on the basis of whether the dominant galaxy is correctly identified, i.e. whether the brightest, in apparent magnitudes group member is also the most massive galaxy in the group. A large and statistically highly significant redshift excess appears only in those subsamples where the groups are gravitationally unbound, and in addition, their dominant galaxies have been misidentified. These comprise approximately one quarter of all groups. The excess appears both among elliptical and spiral dominated groups but is stronger among the spiral dominated groups. Table 4 also shows that groups that are gravitationally bound show larger error limits for redshift asymmetries than groups which are unbound, suggesting that there are large differences in the fraction of bound groups between our five observation points. This result shows that projection effects can play a significant role when the FoF percolation algorithm is applied, and that it can produce groups which are spurious because of these projection effects. It also shows that the choice of free parameters can affect the results, as the Mock2 catalogue does not show larger errors for bound groups in comparison to unbound ones. When redshift asymmetries are studied at a member galaxy level, the results stay similar. If we consider only spiraldominated groups which are unbound and whose centre is wrongly identified, we find a redshift excess of Z = 0.22 and Z = 0.10 for the Mock1 and Mock2 catalogues, respectively. When only bound groups from Mock1 (Mock2) are considered, the redshift excess is only 0.07 (0.04). The former values are similar to observed ones, as S84 found a redshift excess of Z = 0.21 for spiral-dominated (i.e. the apparently brightest galaxy is a spiral galaxy) groups. Byrd & Valtonen (1985) and Valtonen & Byrd (1986) derived values of the redshift excess from an analytical model that ranges from Z = 0.1 to Z = 0.5 depending on the parameters adopted, while the most probable value was 0.2. The good agreement between observations, analytical models and simulations (unbound and wrongly identified groups) suggests that most of the membership of HG82's spiral dominated groups is unbound and that the centre has been misidentified for these groups. Simulations further show that to obtain as high a redshift excess as observed, we have to select groups that are gravitationally unbound and whose centre has been misidentified. If we concentrate on E/S0-dominant groups at a member galaxy level, we see similar results as in Tables 3 and 4. Now Mock1 (Mock2) shows a blueshift excess of Z = −0.05 (Z = −0.01) for gravitationally unbound groups, whose central galaxies are correctly identified. This is a slightly weaker blueshift excess than S84 found (−0.13), but comparable. These values are in agreement with the analytical calculations of Byrd & Valtonen (1985) and Valtonen & Byrd (1986) that showed the blueshift excess of correctly identified, but unbound E-dominated groups ranging from Z = −0.13 to Z = 0.03 depending on the values of free parameters. Here the agreement between observations, analytical models and simulations is reasonable. This points to the possibility that E/S0 galaxies do mark the position of the group centre correctly. Table 4 shows that E/S0 galaxies mark the position of the centre correctly more often than spiral galaxies. If we calculate the percentage of groups whose dominant member has been correctly identified it is clear that ∼ 73.9 per cent of E/S0-dominated groups are correctly identified while as much as ∼ 60.0 per cent of spiral-dominant groups are incorrectly identified. The incorrect identification of the central galaxy is important as it may lead to a redshift excess, since it is more likely that the apparently brightest galaxy is in the front part of the cluster than in the back part of it. The galaxies in the front part of the group appear brighter, and in the back part of the group fainter than what their absolute magnitudes would lead us to expect, i.e. compared with the situation when they all are at the same distance from us. This greater apparent brightness of the group members in the front makes it also more likely that the apparently brightest member is picked from the front volume of the group. Statistically this applies to groups of all sizes. In a gravitationally bound group this would not in itself cause a redshift excess, but if the whole group or a substantial part of it is in Hubble flow, then the galaxies in front are typically blueshifted while the galaxies in the back are redshifted relative to the apparently brightest galaxy. To further test the cause of redshift asymmetries we use absolute B-band magnitudes rather than apparent ones to identify the dominant group member and calculate the redshift asymmetries in this case. The use of absolute magnitudes leads to a small blueshift excess for all groups. The blueshift excess is highest for unbound, spiral-dominated and misidentified groups, being ∼ 56 ± 1.5 per cent (Z ∼ −0.06). For all gravitationally bound groups we find a small blueshift excess of ∼ 52.5 ± 2.5 Note: Sample is the subsample of the mock catalogue, n G is the number of groups, n +∆z is the number of groups with positive redshift excess, n −∆z is the number of groups with negative redshift excess, n ±∆z is the number of groups with no excess, +∆z is the percentage of groups with positive redshift excess when ties have been removed, −∆z is the percentage of groups with negative redshift excess when ties have been removed and ±∆z is the percentage of groups having equal number of negative and positive redshifts. Errors for mock catalogues are standard deviation errors between five different observation points. (Z ∼ −0.02). This small and hardly significant blueshift excess is expected and is due to the fact that magnitude limited observations will miss faint members from the back part of the group with higher probability than from the front part. Additionally, if the group is expanding (unbound) it can further lead to a case where the group is missing redshifted members. A small blueshift excess is also present if we consider only groups that are correctly identified. Although the redshift excess disappears when using absolute B-band magnitudes, it does not lead to a significantly smaller number of misidentifications as the fraction remains roughly the same as in the case of apparent magnitudes. The percentage of misidentifications for all galaxies in the case of apparent magnitudes for Mock2 is ∼ 52.6 ± 1.7% while it is ∼ 51.8 ± 1.8% in the case of absolute magnitudes. This result is due to the fact that both B-band magnitudes are a poor indicator of the most massive galaxy as they tend to favour spiral galaxies over elliptical ones. Summary and conclusions In this paper we have compared cosmological N−body simulations to observations. We have studied fractions of unbound groups, redshift asymmetries and their connection. We have found an explanation for the positive redshift excess found by many authors from different observational group catalogues. Our mock catalogues of groups of galaxies are in reasonable agreement (in general, within 2σ) with observational catalogues when dynamical properties of groups are studied. Both mock catalogues show a significant fraction of gravitationally unbound groups, independent of the choices of free parameter values in the Friends-of-Friends algorithm. Even though the density enhancement of the Mock2 catalogue is more than three times higher than the density enhancement of Mock1, we found that both mock catalogues contain roughly 50% gravitationally unbound groups. Even though the density enhancement does not have a significant effect on the fraction of gravitationally bound groups, the values of free parameters of the FoF algorithm have an effect on group properties. Our results show that the value of V 0 has an effect on the abundance of high velocity groups. This is an expected result as a higher V 0 value gives the percolation algorithm more room in redshift space leading to a higher number of high velocity dispersion groups. Mock catalogues produce similar redshift asymmetries as found in observations. The Mock1 catalogue produces higher redshift asymmetries than Mock2 in all cases, because the redshift asymmetries are most noticeable for nearby groups for which there is a larger distance ratio between the background and the foreground galaxies. The redshift excess is similar to observations for groups that are gravitationally unbound and in which, at the same time, the apparently brightest galaxy is not the most massive galaxy. The misidentification of the group centre is important as it can lead to a redshift excess, since it is more likely that the apparently brightest galaxy is in the front part of the cluster than in the back part of it. The galaxies in the front part of the group appear brighter, and in the back part of the group fainter than what their absolute magnitudes would lead us to expect, i.e. compared with the situation when they all are at the same distance from us. The use of absolute B-band magnitudes does not lead to a redshift excess; however, a small blueshift excess is present. This is due to the fact that magnitude limited observations miss faint group members from the back rather than from the front part of the group. Despite the lack of redshift excess, the fraction of groups in which the dominant group member has been incorrectly identified remains as we observe roughly the same number of misidentified groups (∼ 52 per cent) as in the case of apparent magnitudes when all groups from Mock2 are considered. Thus, absolute and apparent B-band magnitude is a poor indicator of the dominant member in a group. Our results also show that the E/S0 galaxies tend to mark the group centre correctly, as ∼ 75 per cent of E/S0-dominated groups have been correctly identified. These groups do not show significant redshift excess in any case. Gravitationally bound groups do not show any significant redshift excess. This is in agreement with conventional theory, where it is expected that distribution of redshift differentials should be evenly distributed. The subsample of the Mock1 catalogue that includes only gravitationally bound groups has an equal number of galaxies relative to the brightest member within statistical errors. We conclude that when the group centre is not correctly identified, it may cause the major part of the observed redshift excess. If the group is also gravitationally unbound, the level of the redshift excess becomes as high as in S84. Thus the explanation of Byrd & Valtonen (1985) and Valtonen & Byrd (1986) for the origin of the redshift excess is verified. It further means that there is no need to introduce any "anomalous" redshift mechanism to explain the redshift excess of Arp (1970). This paper shows that the Friends-of-Friends percolation algorithm picks out the expanding parts of the groups, in addition to the gravitationally bound group cores. Thus the group catalogues constructed in this way cannot be used as if the groups are purely bound systems. For example, the use of the virial theorem to estimate group masses easily leads to wrong answers. As about 50 per cent of groups in our mock catalogues are bound, in principle one could apply the virial theorem only to this subclass of groups, but then it is difficult to identify this subclass in observations. The redshift excess in a sample of groups would tell us readily that there must be many unbound groups in the sample. However, the absence of redshift excess alone does not guarantee that these groups are gravitationally bound systems. To overcome the difficulty of finding gravitationally bound groups, one can, for example, use stellar mass rather than luminosity for identifying the central galaxy. Even then dark matter and the lack of knowledge of relative distances inside observed groups complicates matters. If the grouping algorithm concentrates on finding satellite galaxies that belong to the same dark matter halo (see Yang et al. 2007), it could return groupings that are mainly gravitationally bound. Detection of extended X-ray radiation can also indicate gravitationally bound groups. Mock2) catalogue groups are 131.0/213.4/347.4 km s −1 (80.5/131.0/213.4 km s −1 ). The values of the Mock2 catalogue groups are closer to the observed ones than the values of mock catalogue groups of Paper I. The large values of Mock1 groups can be explained with larger number of interlopers due to the lower density enhancement and the relatively large value of V 0 . Fig. 1 . 1The cumulative number density of velocity dispersion σ v for galaxy groups. Simulation data are averaged over the ensemble of 5 observation points. The error bars are 1σ errors and are only shown for Mock2 for clarity. The error bars for other data have similar or slightly greater size due to poorer statistics. Fig. 2 . 2Group abundance by 'observable' mass of the groups. Simulation data are averaged over the ensemble of 5 observation points. The error bars are 1σ errors and are only shown for Mock2 for clarity. The error bars for other data have similar or slightly greater size due to poorer statistics. Table 1 . 1Values of free parameters used for creating mock and observed group catalogues.Authors Sample m lim D 0 V 0 δρ ρ α M * φ * This paper Mock1 13.2 0.64 400 19.8 −1.02 −19.06 0.0277 This paper Mock2 14.0 0.37 200 68.0 −1.15 −19.84 0.0172 Huchra & Geller (1982) HG82 13.2 0.64 400 20 −1.02 −19.06 0.0277 Ramella et al. (2002) UZC-SSRS2 15.5 0.25 350 80 −1.1 (−1.2) −19.1 (−19.73) 0.04 (0.013) Eke et al. (2004) 2dFGRS 19.45 0.13 143 − − − − Goto (2005) SDSS-DR2 17.77 1.5 1000 − − − − Merchán & Zandivarez (2005) SDSS-DR3 17.77 − 200 80 −1.05 −20.44 − Berlind et al. (2006) SDSS-DR3 17.5 0.14 75 − − − − Tago et al. (2006) 2dFGRS 19.45 0.25 200 − −1.21 −19.66 − Tago et al. (2008) SDSS-DR5 17.77 0.25 250 − − − − Columns: Table 2 . 2The significance levels of the K−S tests.Sample Property HG82 UZC-SSRS2 HG82 vs. UZC-SSRS2 Mock1 σ v 0.013 2.7 × 10 −4 0.08 Mock1 M G 4.2 × 10 −4 7.1 × 10 −7 0.05 Mock2 σ v 4.8 × 10 −4 0.015 0.08 Mock2 M G 0.042 0.462 0.05 Note: Significance levels of the K−S test for the null hypothesis that observations and the simulations are alike and are drawn from the same parent population (HG82 and UZC-SSRS2 columns). Significance level of the K−S test for the null hypothesis that the HG82 and the UZC- SSRS2 group catalogue are alike and are drawn from the same parent population (HG82 vs. UZC-SSRS2 column). 2σ error bars. However, there are clear differences visible be- tween all catalogues in both Figures. These differences are dis- cussed below in greater detail. We use a statistical Kolmogorov- Smirnov (K−S) test to prove or disprove the null hypothesis, H null , that the two distributions are alike and are drawn from the same population distribution function. Results of the K−S tests are presented as significance levels (value of the Q function) for the null hypothesis and are listen in Table 2. Both Figs. Table 3 . 3A general comparison of the mock catalogues and samples from S84.Sample n G n +∆z n −∆z n ±∆z +∆z (%) −∆z (%) ±∆z (%) S84, all groups (n ≤ 10) 85 42 32 11 57.0 43.0 12.9 S84, E/S0 dominant groups removed 60 33 18 9 65.0 35.0 15.0 S84, E/S0 dominant groups 21 9 10 2 47.0 53.0 9.5 Mock1, all groups (n ≤ 10) 1384 622 505 257 55.2 ± 5.1 44.8 ± 5.1 18.6 ± 1.9 Mock1, spiral (T≥ 0.92) dominant groups 1153 528 408 217 56.4 ± 5.6 43.6 ± 5.6 18.8 ± 2.4 Mock1, E/S0 (T< 0.92) dominant groups 231 94 97 40 49.2 ± 6.9 50.8 ± 6.9 17.3 ± 4.2 Mock2, all groups (n ≤ 10) 12100 5087 4658 2355 52.2 ± 1.3 47.8 ± 1.3 19.5 ± 0.6 Mock2, spiral (T≥ 0.92) dominant groups 9086 3854 3466 1748 52.5 ± 1.8 47.3 ± 1.6 19.2 ± 0.8 Mock2, E/S0 (T< 0.92) dominant groups 3013 1179 1227 607 49.0 ± 1.8 51.0 ± 1.8 20.1 ± 2.0 Table 4 . 4A detailed view of the redshift asymmetries in mock catalogues.Sample n G n +∆z n −∆z n ±∆z +∆z (%) −∆z (%) ±∆z (%) Mock1, gravitationally bound groups all groups (n ≤ 10) 675 282 259 134 50.5 ± 10.1 49.5 ± 10.1 19.9 ± 2.5 spiral (T≥ 0.92) dominant groups 537 231 199 107 52.4 ± 10.6 47.6 ± 10.6 19.9 ± 2.7 E/S0 (T< 0.92) dominant groups 138 51 60 27 46.0 ± 16.0 54.1 ± 16.0 19.6 ± 4.4 Mock1, gravitationally unbound groups all groups (n ≤ 10) 709 340 246 123 58.3 ± 2.9 41.7 ± 2.9 17.3 ± 2.2 spiral (T≥ 0.92) dominant groups 616 297 209 110 59.0 ± 3.2 41.0 ± 3.2 17.9 ± 2.8 E/S0 (T< 0.92) dominant groups 93 43 37 13 54.1 ± 7.1 45.9 ± 7.1 14.0 ± 8.5 Mock1, bound, wrongly identified all groups (n ≤ 10) 289 142 110 37 56.3 ± 12.9 43.7 ± 12.9 12.8 ± 4.2 spiral (T≥ 0.92) dominant groups 257 129 95 33 57.6 ± 12.8 42.4 ± 12.8 12.8 ± 3.7 E/S0 (T< 0.92) dominant groups 32 13 15 4 46.4 ± 18.2 53.6 ± 18.2 12.5 ± 12.0 Mock1, unbound, wrongly identified all groups (n ≤ 10) 338 193 96 49 66.8 ± 3.8 33.2 ± 3.8 14.5 ± 4.6 spiral (T≥ 0.92) dominant groups 297 171 83 43 67.6 ± 3.5 32.4 ± 3.5 14.5 ± 4.8 E/S0 (T< 0.92) dominant groups 41 22 13 6 62.9 ± 9.7 37.1 ± 9.7 14.6 ± 9.9 Mock2, gravitationally bound groups all groups (n ≤ 10) 5715 2327 2214 1174 51.3 ± 1.7 48.7 ± 1.7 20.5 ± 0.7 spiral (T≥ 0.92) dominant groups 3649 1515 1386 748 52.3 ± 2.9 47.7 ± 2.9 20.5 ± 1.2 E/S0 (T< 0.92) dominant groups 2066 801 839 426 48.8 ± 2.2 51.2 ± 2.2 20.6 ± 2.5 Mock2, gravitationally unbound groups all groups (n ≤ 10) 6385 2760 2444 1181 52.9 ± 1.6 47.1 ± 1.6 18.5 ± 0.9 spiral (T≥ 0.92) dominant groups 5437 2357 2080 1000 53.0 ± 1.7 47.0 ± 1.7 18.4 ± 1.1 E/S0 (T< 0.92) dominant groups 947 402 364 181 52.4 ± 2.8 47.6 ± 2.8 19.1 ± 1.7 Mock2, bound, wrongly identified all groups (n ≤ 10) 2618 1193 1028 397 53.7 ± 2.2 46.3 ± 2.2 15.2 ± 0.7 spiral (T≥ 0.92) dominant groups 2151 980 840 331 53.8 ± 2.7 46.2 ± 2.7 15.4 ± 1.0 E/S0 (T< 0.92) dominant groups 467 213 188 66 53.1 ± 1.7 46.9 ± 1.7 14.1 ± 3.4 Mock2, unbound, wrongly identified all groups (n ≤ 10) 3591 1661 1379 551 54.6 ± 1.9 45.4 ± 1.9 15.3 ± 1.3 spiral (T≥ 0.92) dominant groups 3291 1535 1260 496 54.8 ± 2.0 45.2 ± 2.0 15.1 ± 1.4 E/S0 (T< 0.92) dominant groups 300 126 119 55 51.4 ± 5.6 48.6 ± 5.6 18.3 ± 4.1 S.-M.Niemi and M. Valtonen: The origin of redshift asymmetries: How ΛCDM explains anomalous redshift S.-M.Niemi and M. Valtonen: The origin of redshift asymmetries: How ΛCDM explains anomalous redshift Acknowledgements. SMN acknowledges the funding by Finland's Academy of Sciences and Letters and the Nordic Optical Telescope (NOT) Scientific Association (NOTSA). SMN would like to thank Dr. Gerard Lemson for invaluable help with the Millennium Simulation database. SMN would also like to acknowledge the support and help provided by Dr. Pekka Heinämäki and Dr. Pasi Nurmi. We thank the anonymous referee for detailed reading of the manuscript and comments that helped us to improve the original manuscirpt. The Millennium Simulation databases used in this paper and the web application providing online access were constructed as part of the activities of the German Astrophysical Virtual Observatory. . H Arp, Nature. 2251033Arp, H. 1970, Nature, 225, 1033 . H Arp, ApJ. 25654Arp, H. 1982, ApJ, 256, 54 . 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Parallel maximal matching algorithms (adapted for switching) appear to have stricken the best such tradeoff. On one hand, they provide the following Quality of Service (QoS) guarantees: Using maximal matchings as crossbar schedules results in at least 50% switch throughput and order-optimal (i.e., independent of the switch size N ) average delay bounds for various traffic arrival processes. On the other hand, using N processors (one per port), their per-port computational complexity can be as low as O(log 2 N ) (more precisely O(log N ) iterations that each has O(log N ) computational complexity) for an N × N switch.In this work, we propose QPS-r, a parallel iterative switching algorithm that has the lowest possible computational complexity: O(1) per port. Yet, the matchings that QPS-r computes have the same quality as maximal matchings in the following sense: Using such matchings as crossbar schedules results in exactly the same aforementioned provable throughput and delay guarantees as using maximal matchings, as we show using Lyapunov stability analysis. Although QPS-r builds upon an existing add-on technique called Queue-Proportional Sampling (QPS), we are the first to discover and prove this nice property of such matchings. We also demonstrate that QPS-3 (running 3 iterations) has comparable empirical throughput and delay performances as iSLIP (running log 2 N iterations), a refined and optimized representative maximal matching algorithm adapted for switching.

1905.05392

4e76759eac9259fa9b3c8e7f8e5e49027da0ec7d

QPS-r: A Cost-Effective Crossbar Scheduling Algorithm and Its Stability and Delay Analysis 21 Aug 2019 Long Gong School of Computer Science School of Industrial & Systems Engineering Georgia Institute of Technology Georgia Institute of Technology Jun Xu School of Computer Science School of Industrial & Systems Engineering Georgia Institute of Technology Georgia Institute of Technology Liang Liu School of Computer Science School of Industrial & Systems Engineering Georgia Institute of Technology Georgia Institute of Technology Siva Theja Maguluri School of Computer Science School of Industrial & Systems Engineering Georgia Institute of Technology Georgia Institute of Technology QPS-r: A Cost-Effective Crossbar Scheduling Algorithm and Its Stability and Delay Analysis 21 Aug 2019Index Terms-Crossbar schedulingqueue-proportional sam- plingLyapunov stability analysisQPS-rQoS Parallel maximal matching algorithms (adapted for switching) appear to have stricken the best such tradeoff. On one hand, they provide the following Quality of Service (QoS) guarantees: Using maximal matchings as crossbar schedules results in at least 50% switch throughput and order-optimal (i.e., independent of the switch size N ) average delay bounds for various traffic arrival processes. On the other hand, using N processors (one per port), their per-port computational complexity can be as low as O(log 2 N ) (more precisely O(log N ) iterations that each has O(log N ) computational complexity) for an N × N switch.In this work, we propose QPS-r, a parallel iterative switching algorithm that has the lowest possible computational complexity: O(1) per port. Yet, the matchings that QPS-r computes have the same quality as maximal matchings in the following sense: Using such matchings as crossbar schedules results in exactly the same aforementioned provable throughput and delay guarantees as using maximal matchings, as we show using Lyapunov stability analysis. Although QPS-r builds upon an existing add-on technique called Queue-Proportional Sampling (QPS), we are the first to discover and prove this nice property of such matchings. We also demonstrate that QPS-3 (running 3 iterations) has comparable empirical throughput and delay performances as iSLIP (running log 2 N iterations), a refined and optimized representative maximal matching algorithm adapted for switching. I. INTRODUCTION The volume of network traffic across the Internet and in data-centers continues to grow relentlessly, thanks to existing and emerging data-intensive applications, such as big data analytics, cloud computing, and video streaming. At the same time, the number of network-connected devices is exploding, fueled by the wide adoption of smart phones and the emergence of the Internet of things. To transport and "direct" this massive amount of traffic to their respective destinations, switches and routers capable of connecting a large number of ports (called high-radix [1,2]) and operating at very high line rates are badly needed. Many present day switching systems in Internet routers and data-center switches employ an input-queued crossbar to interconnect their input ports and output ports (e.g., Cisco Nexus 5000 Series [3], Arista 7500 Switch [4], and Juniper QFX 10000 Switches [5]). Though it was commonly believed that such a (monolithic) crossbar is difficult to scale beyond 64 (input/output) ports, recent advances in switching hardware technologies (e.g., [1,6,7]) have made high-radix crossbars not only technologically feasible but also economically and environmentally (i.e., more energy-efficient) favorable, as compared to low-radix crossbars. In an N × N input-queued crossbar switch, each input port can be connected to only one output port and vice versa in each switching cycle or time slot. Hence, in every time slot, the switch needs to compute a one-to-one matching between input and output ports (i.e., the crossbar schedule). A major research challenge of designing high-link-rate high-radix switches (e.g., 128 ports or beyond of 40 Gbps each) is to develop algorithms that can compute "high quality" matchingsi.e., those that result in high switch throughput and low queueing delays for packets -in a few nanoseconds. For example, with a cell 1 size of 64 bytes (the minimum cell size used in Arista 7500 Switch [4]), a switch supporting 40 Gbps per-port rates has to compute a matching every 12.8 nanoseconds. Clearly, a suitable matching algorithm has to have very low (ideally O(1)) computational complexity, yet output "fairly good" matching decisions most of time. A. The Family of Maximal Matchings A family of parallel iterative algorithms for computing maximal matchings (to be precisely defined in §II-B) are arguably the best candidates for crossbar scheduling in highlink-rate high-radix switches, because they have reasonably low computational complexities, yet can provide fairly good QoS guarantees. More specifically, using maximal matchings as crossbar schedules results in at least 50% switch throughput in theory (and usually much higher throughput in practice), as shown in [10]. In addition, it results in low packet delays that also have excellent scaling behaviors such as orderoptimal (i.e., independent of switch size N ) under various traffic arriving processes when the offered load is less than 50% (i.e., within the provable stability region), as shown in [11,12]. In comparison, matchings of higher qualities such as maximum matching and maximum weighted matching are much more expensive to compute, as will be elaborated in §II-B. Hence, it is fair to say that, maximal matching algorithms overall deliver the biggest "bang" (performance) for the "buck" (computational complexity). Unfortunately, parallel maximal matching algorithms are still not "dirt cheap" computationally. More specifically, all existing parallel/distributed algorithms that compute maximal matchings on general N × N bipartite graphs (i.e., without additional constraints or conditions such as the graph being degree-bounded [13] and/or already edge-colored [14]) require a minimum of O(log N ) iterations (rounds of message exchanges). This minimum is attained by the classical algorithm of Israel and Itai [15]; the PIM algorithm [16] is a slight adaptation of this classical algorithm to the switching context, and iSLIP [17] further improves upon PIM by reducing its periteration per-port computational complexity to O(log N ) via de-randomizing a computationally expensive (O(N ) complexity to be exact) operation in PIM. Although parallel iterative maximal matching algorithms and their variants are found in real-world products (e.g., Cisco Nexus 5548P switch uses an enhanced iSLIP algorithm [18]), it is hard to scale them to a large number of switch ports. For example, recent experiments in [6,19] demonstrated the feasibility of using iSLIP (or its variants) for 128 × 128 or larger crossbar switches, but at the cost of cutting corners on the matching computation (e.g., running a single iteration instead of log 2 N iterations), which results in lower-quality crossbar schedules and poorer throughput and delay performances. B. QPS-r: Bigger Bang for the Buck In this work, we propose QPS-r, a parallel iterative algorithm that has the lowest possible computational complexity: O(1) per port. More specifically, QPS-r requires only r (a small constant independent of N ) iterations to compute a matching, and the computational complexity of each iteration is only O(1); here QPS stands for Queue-Proportional Sampling, an add-on technique proposed in [20] that we will describe shortly. Yet, even the matchings that QPS-1 (running only a single iteration) computes have the same quality as maximal matchings (running log 2 N iterations) in the following sense: Using such matchings as crossbar schedules results in exactly the same aforementioned provable throughput and delay guarantees as using maximal matchings, as we will show using Lyapunov stability analysis. Note that QPS-r performs as well as maximal matching algorithms not just in theory: We will show that QPS-3 (running 3 iterations) has comparable empirical throughput and delay performances as iSLIP (running log 2 N iterations) under various workloads. QPS-r has another advantage over parallel iterative maximal matching algorithms such as iSLIP and PIM: Its per-port communication complexity is also O(1), much smaller than that of maximal matching algorithms such as iSLIP. In each iteration of QPS-r, each input port sends a request to only a single output port. In comparison, in each iteration of PIM or iSLIP, each input port has to send requests to all output ports to which the corresponding VOQs are nonempty, which incurs O(N ) communication complexity per port. Although QPS-r builds on the QPS data structure and algorithm proposed in [20], our work on QPS-r is very different in three important aspects. First, in [20], QPS was used only as an add-on to other crossbar scheduling algorithms such as SERENA [21] and iSLIP [17] by generating a starter matching for other switching algorithms to further refine, whereas in this work, QPS-r is used only as a stand-alone algorithm. Second, we are the first to discover and prove that (QPS-r)generated matchings and maximal matchings provide exactly the same aforementioned QoS guarantees, whereas in [20], no such mathematical similarity or connection was mentioned. Third, the establishment of this mathematical similarity is an important theoretical contribution in itself, because maximal matchings have long been established as a cost-effective family both in switching [16,17] and in wireless networking [11,12], and with this connection we have considerably enlarged this family. Although we show that QPS-r has exactly the same throughput and delay bounds as that of maximal matchings established in [10][11][12], our proofs are different for the following reason. A departure inequality (see Property 1), satisfied by all maximal matching algorithms was used in the stability analysis of [10] and the delay analysis of [11,12]. This inequality, however, is not satisfied by QPS-r in general. However, QPS-r satisfies this departure inequality in expectation, which is a weaker guarantee and we show that this is enough to obtain the throughput and delay bounds in our proofs. The rest of this paper is organized as follows. In §II, we provide some background on the input-queued crossbar switches. In §III, we first review QPS, and then describe QPSr. Then in §IV, we derive the throughput and the queue length (and delay) bounds of QPS-r, followed by the performance evaluation in §V. In §VI, we survey related work before concluding this paper in §VII. II. BACKGROUND ON CROSSBAR SCHEDULING In this section, we provide a brief introduction to the crossbar scheduling (switching) problem, and describe and compare the aforementioned three different types of matchings. Throughout this paper we adopt the aforementioned standard assumption [8, Chapter 2, Page 21] that all the incoming variable-size packets are first segmented into fixedsize packets (also referred to as cells), and then reassembled at their respective output ports before leaving the switch. Each fixed-size cell takes one time slot to switch. We also assume that all input links/ports and output links/ports operate at the same normalized line rate of 1, and so do all wires and crosspoints inside the crossbar. A. Input-Queued Crossbar Switch In an N × N input-queued crossbar switch, each input port has N Virtual Output Queues (VOQs) [22]. The j th VOQ at input port i serves as a buffer for packets going from input port i to output port j. The use of VOQs solves the Headof-Line (HOL) blocking issue [23], which severely limits the throughput of the switch system. An N × N input-queued crossbar can be modeled as a weighted bipartite graph, of which the two disjoint vertex sets are the N input ports and the N output ports respectively. In this bipartite graph, there is an edge between input port i and output port j, if and only if the j th VOQ at input port i, the corresponding VOQ, is nonempty. The weight of this edge is defined as the length of (i.e., the number of packets buffered at) this VOQ. A set of such edges constitutes a valid crossbar schedule, or a matching, if any two of them do not share a common vertex. The weight of a matching is the total weight of all the edges belonging to it (i.e., the total length of all corresponding VOQs). Each such matching M can be represented as an N × N sub-permutation matrix (a 0-1 matrix that contains at most one entry of "1" in each row and in each column) S = (s ij ) as follows: s ij = 1 if and only if the edge between input port i and output port j is contained in M (i.e., input port i is matched to output port j in M ). To avoid any confusion, only S (not M ) is used to denote a matching in the sequel, and it can be both a set (of edges) and a matrix. B. Maximal Matching As mentioned in §I, three types of matchings play important roles in crossbar scheduling problems: (I) maximal matchings, (II) maximum matchings, and (III) maximum weighted matchings. A matching S is called a maximal matching, if it is no longer a matching, when any edge not in S is added to it. A matching with the largest possible number of edges is called a maximum matching or maximum cardinality matching. Neither maximal matchings nor maximum matchings take into account the weights of edges, whereas maximum weighted matchings do. A maximum weighted matching is one that has the largest total weight among all matchings. By definition, any maximum matching or maximum weighted matching is also a maximal matching, but neither converse is generally true. As mentioned earlier, the family of maximal matchings has long been recognized as a cost-effective family for crossbar scheduling. Compared to maximal matching, maximum weighted matching (MWM) (i.e., the well-known MaxWeight scheduler [24] in the context of crossbar scheduling) is much less cost effective. Although MWM provides stronger QoS guarantees such as 100% switch throughput [25,26] and O(N ) average packet delay [27] in theory (and usually even better empirical delay performance in practice as shown in [25]), the state of the art serial MWM algorithm (suitable for switching) has a prohibitively high computational complexity of O(N 2.5 log W ) [28], where W is the maximum possible weight (length) of an edge (VOQ). By the same measure, maximum matching is not a great deal either: It is only slightly cheaper to compute than MWM, yet using maximum matchings as crossbar schedules generally cannot guarantee 100% throughput [29]. Compared to maximal matching algorithms, QPS-r provides the same provable QoS guarantees at a much lower computational complexity. More specifically, in a single iteration (i.e., with r = 1), QPS-r computes a matching that is generally not maximal, yet using such matchings as crossbar schedules can result in the same provable throughput guarantee (at least 50%) and delay bounds as using maximal matchings, as we will show in §IV. QPS-r can make do with less (iterations) because the queue-proportional sampling operation implicitly makes use of the edge weight (VOQ length) information, which maximal matching algorithms do not. One major contribution of this work is to discover the family of (QPS-r)-generated matchings that is even more cost-effective. III. THE QPS-r ALGORITHM The QPS-r algorithm simply runs r iterations of QPS (Queue-Proportional Sampling) [20] to arrive at a matching, so its computational complexity per port is exactly r times those of QPS. Since r is a small constant, it is O(1), same as that of QPS. In the following two subsections, we describe QPS and QPS-r respectively in more details. A. Queue-Proportional Sampling (QPS) QPS was used in [20] as an "add-on" to augment other switching algorithms as follows. It generates a starter matching, which is then populated (i.e., adding more edges to it) and refined, by other switching algorithms such as iSLIP [17] and SERENA [30], into a final matching. To generate such a starter matching, QPS needs to run only one iteration, which consists of two phases, namely, a proposing phase and an accepting phase. We briefly describe them in this section for this paper to be self-contained. 1) The Proposing Phase: In this phase, each input port proposes to exactly one output port -decided by the QPS strategy -unless it has no packet to transmit. Here we will only describe the operations at input port 1; that at any other input port is identical. Like in [20], we denote by m 1 , m 2 , · · · , m N the respective queue lengths of the N VOQs at input port 1, and by m their total (i.e., m N k=1 m k ). Input port 1 simply samples an output port j with probability mj m , i.e., proportional to m j , the length of the corresponding VOQ (hence the name QPS); it then proposes to output port j, with the value m j that will be used in the next phase. The computational complexity of this QPS operation, carried out using a simple data structure proposed in [20], is O(1) per (input) port. 2) The Accepting Phase: We describe only the action of output port 1 in the accepting phase; that of any other output port is identical. The action of output port 1 depends on the number of proposals it receives. If it receives exactly one proposal from an input port, it will accept the proposal and match with the input port. However, if it receives proposals from multiple input ports, it will accept the proposal accompanied with the largest VOQ length (called the "longest VOQ first" accepting strategy), with ties broken uniformly at random. The computational complexity of this accepting strategy is O(1) on average and can be made O(1) even in the worst case [20]. B. The QPS-r Scheme The QPS-r scheme simply runs r QPS iterations. In each iteration, each input port that is not matched yet, first proposes to an output port according to the QPS proposing strategy; each output port that is not matched yet, accepts a proposal (if it has received any) according the "longest VOQ first" accepting strategy. Hence, if an input port has to propose multiple times (once in each iteration), due to all its proposals (except perhaps the last) being rejected, the identities of the output ports it "samples" (i.e., proposes to) during these iterations are samples with replacement, which more precisely are i.i.d. random variables with a queue-proportional distribution. At the first glance, sampling with replacement may appear to be an obviously suboptimal strategy for the following reason. There is a nonzero probability for an input port to propose to the same output port multiple times, but since the first (rejected) proposal implies this output port has already accepted "someone else" (a proposal from another input port), all subsequent proposals to this output port will surely go to waste. For this reason, sampling without replacement (i.e., avoiding all output ports proposed to before) may sound like an obviously better strategy. However, it is really not, since compared to sampling with replacement, it has a much higher computational complexity of O(log N ), but improves the throughput and delay performances only slightly according to our simulation studies. IV. THROUGHPUT AND DELAY ANALYSIS In this section, we show that QPS-1 (i.e., running a single QPS iteration) delivers exactly the same provable throughput and delay guarantees as maximal matching algorithms. When r > 1, QPS-r in general has better throughput and delay performances than QPS-1, as more input and output ports can be matched up during subsequent iterations, although we are not able to derive better bounds. A. Preliminaries In this section, we introduce the notation and assumptions that will later be used in our derivations. We define three N × N matrices Q(t), A(t), and D(t). Let Q(t) q ij (t) be the queue length matrix where each q ij (t) is the length of the j th VOQ at input port i during time slot t. With a slight abuse of notation, we refer to this VOQ as q ij (without the t term). We define Q i * (t) and Q * j (t) as the sum of the i th row and the sum of the j th column respectively of Q(t), i.e., Q i * (t) j q ij (t) and Q * j (t) i q ij (t). With a similar abuse of notation, we define Q i * as the VOQ set {q i1 , q i2 , · · · , q iN } (i.e., those on the i th row), and Q * j as {q 1j , q 2j , · · · , q N j } (i.e., those on the j th column). Now we introduce a concept that lies at the heart of our derivations: neighborhood. For each VOQ q ij , we define its neighborhood as Q i * Q * j , the set of VOQs on the i th row or the j th column. We denote this neighborhood as Q † ij , since it has the shape of a cross. Figure 1 illustrates Q † ij , where the row and column in the shadow are the VOQ sets Q i * and Q * j respectively. Q † ij can be viewed as the interference set of VOQs for VOQ q ij [11,12], as no other VOQ in Q † ij can be active (i.e., transmit packets) simultaneously with q ij . We q 11 q 12 · · · q 1j · · · q 1N q 21 q 22 · · · q 2j · · · q 2N . . . . . . . . . . . . . . . . . . q i1 q i2 · · · q ij · · · q iN . . . . . . . . . . . . . . . . . . q N 1 q N 2 · · · q N j · · · q N N                                                 · · · q iN q i1 q i2 · · · . . . q N j q 1j q 2j . . . q ij Fig. 1. Illustration of neighborhood of q ij , i.e., Q † ij . define Q † ij (t) as the total length of all VOQs in (the set) Q † ij at time slot t, that is Q † ij (t) Q i * (t) − q ij (t) + Q * j (t).(1) Here we need to subtract the term q ij (t) so that it is not double-counted (in both Q i * (t) and Q * j (t)). Let A(t) = a ij (t) be the traffic arrival matrix where a ij (t) is the number of packets arriving at the input port i destined for output port j during time slot t. For ease of exposition, we assume that, for each 1 ≤ i, j ≤ N , {a ij (t)} ∞ t=0 is a sequence of i.i.d. random variables, the second moment of their common distribution (= E a 2 ij (0) ) is finite, and this sequence is independent of other sequences (for a different i and/or j). Our analysis, however, holds for more general arrival processes (e.g., Markovian arrivals) that were considered in [11,12], as we will elaborate shortly. Let D(t) = d ij (t) be the departure matrix for time slot t output by the crossbar scheduling algorithm. Similar to S, D(t) is a 0-1 matrix in which d ij (t) = 1 if and only if a packet departs from q ij during time slot t. For any i, j, the queue length process q ij (t) evolves as follows: q ij (t + 1) = q ij (t) − d ij (t) + a ij (t).(2) Let Λ = λ ij be the (normalized) traffic rate matrix (associated with A(t)) where λ ij is normalized (to the percentage of the line rate of an input/output link) mean arrival rate of packets to VOQ q ij . With a ij (t) being an i.i.d. process, we have λ ij = E a ij (0) . We define ρ Λ as the maximum load factor imposed on any input or output port by Λ, ρ Λ max max 1≤i≤N { j λ ij }, max 1≤j≤N { i λ ij }(3) A switching algorithm is said to achieve 100% throughput or be throughput-optimal if the (packet) queues are stable whenever ρ Λ < 1. As mentioned before, we will prove in this section that, same as the maximal matching algorithms, QPS-1 is stable under any traffic arrival process A(t) whose rate matrix Λ satisfies ρ Λ < 1/2 (i.e., can provably attain at least 50% throughput, or half of the maximum). We also derive the average delay bound for QPS-1, which we show is orderoptimal (i.e., independent of switch size N ) when the arrival process A(t) further satisfies that for any i, j, a ij (0) has finite variance. In the sequel, we drop the subscript term from ρ Λ and simply denote it as ρ. Similar to Q † ij (t), we define A † ij (t) as the total number of packet arrivals to all VOQs in the neighborhood set Q † ij : A † ij (t) A i * (t) − a ij (t) + A * j (t),(4) where A i * (t) and A * j (t) are similarly defined as Q i * (t) and Q * j (t) respectively. D † ij (t), D i * (t), and D * j (t) are similarly defined, so is Λ † ij (t). We now state some simple facts concerning D(t), A(t), and Λ as follows. Fact 1. Given any crossbar scheduling algorithm, for any i, j, we have, D i * (t) ≤ 1 (at most one packet can depart from input port i during time slot t), D * j (t) ≤ 1, and D † ij (t) ≤ 2. Fact 2. Given any i.i.d. arrival process A(t) and its rate matrix is Λ whose maximum load factor is defined in (3), for any i, j, we have E[A † ij (t)] = Λ † ij ≤ 2ρ. The following fact is slightly less obvious. Fact 3. Given any crossbar scheduling algorithm, for any i, j, we have d ij (t)D † ij (t) = d ij (t).(5) Fact 3 holds because, as mentioned earlier, no other VOQ in Q † ij (see Figure 1) can be active simultaneously with q ij . More precisely, if d ij (t) = 1 (i.e., VOQ q ij is active during time slot t) then D † ij (t) D i * (t)−d ij (t)+D * j (t) = 1−1+1 = 1; otherwise d ij (t)D † ij (t) = 0 · D † ij (t) = 0 = d ij (t). B. Why QPS-1 Is Just as Good? The provable throughput and delay bounds of maximal matching algorithms were derived from a "departure inequality" (to be stated and proved next) that all maximal matchings satisfy. This inequality, however, is not in general satisfied by matchings generated by QPS-1. Rather, QPS-1 satisfies a much weaker form of departure inequality, which we discover is fortunately barely strong enough for proving the same throughput and delay bounds. Property 1 (Departure Inequality, stated as Lemma 1 in [11,12]). If during a time slot t, the crossbar schedule is a maximal matching, then each departure process D † ij (t) satisfies the following inequality q ij (t)D † ij (t) ≥ q ij (t),(6) Proof: We reproduce the proof of Property 1 with a slightly different approach for this paper to be self-contained. Suppose the contrary is true, i.e., q ij (t)D † ij (t) < q ij (t). This can only happen when q ij (t) > 0 and D † ij (t) = 0. However, D † ij (t) = 0 implies that no nonempty VOQ (edge) in the neighborhood Q † ij (see Figure 1) is a part of the matching. Then this matching cannot be maximal (a contradiction) since it can be enlarged by the addition of the nonempty VOQ (edge) q ij . Clearly, the departure inequality (6) above implies the following much weaker form of it: i,j E q ij (t)D † ij (t) ≥ i,j E q ij (t) .(7) In the rest of this section, we prove the following lemma: Lemma 1. The matching generated by QPS-1, during any time slot t, satisfies the much weaker "departure inequality" (7). Before we prove Lemma 1, we introduce an important definition and state four facts about QPS-1 that will be used later in the proof. In the following, we will run into several innocuous possible 0 0 situations that all result from queueproportional sampling, and we consider all of them to be 0. We define α ij (t) as the probability of the event that the proposal from input port i to output port j is accepted during the accepting phase, conditioned upon the event that input port i did propose to output port j during the proposing phase. With this definition, we have the first fact E d ij (t) | Q(t) = q ij (t) Q i * (t) · α ij (t),(8) since both sides (note d ij (t) is a 0-1 r.v.) are the probability that i proposes to j and this proposal is accepted. Applying the " j operator" to both sides, we obtain the second fact E D i * (t) | Q(t) = j q ij (t) Q i * (t) · α ij (t).(9) The third fact is that, for any output port j, E D * j (t) | Q(t) = 1 − i 1 − q ij (t) Q i * (t) .(10) In this equation, the LHS is the conditional probability (D * j (t) is also a 0-1 r.v.) that at least one proposal is received and accepted by output port j, and the second term on the RHS of (10) is the probability that no input port proposes to output port j (so j receives no proposal). This equation holds since when j receives one or more proposals, it will accept one of them (the one with the longest VOQ). The fourth fact is that, for any i, j, α ij (t) ≥ k =i 1 − q kj (t) Q k * (t) .(11) This inequality holds because when input port i proposes to output port j, and no other input port does, j has no choice but to accept i ′ s proposal. C. Proof of Lemma 1 Now we are ready to prove Lemma 1. It suffices to show that for any i and j, we have i,j E q ij (t)D † ij (t) | Q(t) ≥ i,j q ij (t)(12) because with (12), we have i,j E q ij (t)D † ij (t) = E E i,j q ij (t)D † ij (t) | Q(t) ≥ E i,j q ij (t) = i,j E q ij (t) . By the definition of D † ij (t) D i * (t)− d ij (t)+ D * j (t), we have, i,j E q ij (t)D † ij (t) | Q(t) = i,j q ij (t)E D i * (t) | Q(t) − i,j q ij (t)E d ij (t) | Q(t) + i,j q ij (t)E D * j (t) | Q(t) .(13) Focusing on the first term on the RHS of (13) and using (9), we have, i,j q ij (t)E D i * (t) | Q(t) = i Q i * (t)E D i * (t) | Q(t) = i Q i * (t) j q ij (t) Q i * (t) · α ij (t) = i,j q ij (t)α ij (t). (14) Focusing the second term on the RHS of (13) and using (8), we have − i,j q ij (t)E d ij (t) | Q(t) = − i,j q ij (t)α ij (t) q ij (t) Q i * (t) .(15) Hence, the sum of the first two terms in (13) is equal to i,j q ij (t)α ij (t) 1− q ij (t) Q i * (t) ≥ i,j q ij (t) k =i 1 − q kj (t) Q k * (t) 1 − q ij (t) Q i * (t) (16) = i,j q ij (t) i 1− q ij (t) Q i * (t) = i,j q ij (t) 1 − E D * j (t) | Q(t) .(17) Note that (16) is due to (11) and (17) is due to (10). We now arrive at (12), when adding the third and last term in (13) to the RHS of (17). D. Throughput Analysis In this section we prove, through Lyapunov stability analysis, the following theorem (i.e., Theorem 1), which states that QPS-1 can attain at least 50% throughput. The proof will make use of the much weaker departure inequality (7). The same throughput bound was proved in [10], through fluid limit analysis, for maximal matching algorithms using the (stronger) departure inequality (6) which as stated earlier is not satisfied by matchings generated by QPS-1. Theorem 1. Whenever the maximum load factor ρ < 1/2, QPS-1 is stable in the following sense: The queueing process {Q(t)} ∞ t=0 is a positive recurrent Markov chain. Proof: {Q(t)} ∞ t=0 is clearly a Markov chain, since in (2), the term d ij (t) is a function of Q(t) and a ij (t) is a random variable independent of Q(t). We define the following Lyapunov function of Q(t): (1). This Lyapunov function was first introduced in [11] for the delay analysis of maximal matching algorithms for wireless networking. By the Foster-Lyapunov stability criterion [31, Proposition 2.1.1], to prove that {Q(t)} ∞ t=0 is positive recurrent, it suffices to show that, there exists a constant B > 0 such that whenever Q(t) 1 > B (because it is not hard to verify that the complement set of states {Q(t) : Q(t) 1 ≤ B} is finite and the drift is bounded whenever Q(t) belongs to this set), we have L Q(t) = i,j q ij (t)Q † ij (t), where Q † ij (t) is defined earlier inE L Q(t + 1) − L Q(t) | Q(t) ≤ −ǫ,(18) where ǫ > 0 is a constant. It is not hard to check (for more detailed derivations, please refer to [11]), L Q(t + 1) − L Q(t) =2 i,j q ij (t) A † ij (t) − D † ij (t) + i,j a ij (t) − d ij (t) A † ij (t) − D † ij (t) .(19) Hence the drift (LHS of (18)) can be written as E L Q(t + 1) − L Q(t) | Q(t) =E 2 i,j q ij (t) A † ij (t) − D † ij (t) | Q(t) + E i,j a ij (t)−d ij (t) A † ij (t)−D † ij (t) | Q(t) . (20) Now we claim the following two inequalities, which we will prove shortly. E 2 i,j q ij (t) A † ij (t)−D † ij (t) | Q(t) ≤ 2(2ρ−1) Q(t) 1 . (21) E i,j a ij (t)−d ij (t) A † ij (t)−D † ij (t) | Q(t) ≤ CN 2 . (22) With (21) and (22) substituted into (20), we have E L Q(t + 1) − L Q(t) | Q(t) ≤ 2(2ρ−1) Q(t) 1 + CN 2 . where C > 0 is a constant. Since ρ < 1/2, we have 2ρ − 1 < 0. Hence, there exist B, ǫ > 0 such that, whenever Q(t) 1 > B, E L Q(t + 1) − L Q(t) | Q(t) ≤ −ǫ. Now we proceed to prove (21). E 2 i,j q ij (t) A † ij (t) − D † ij (t) | Q(t) =2 i,j E q ij (t)A † ij (t) | Q(t) − i,j E q ij (t)D † ij (t) | Q(t) ≤2 2ρ i,j E q ij (t) | Q(t) − i,j E[q ij (t) | Q(t)](23)=2(2ρ − 1) Q(t) 1 .(24) In the above derivations, inequality (23) holds due to (12), A(t) being independent of Q(t) for any t, and Fact 2 that E[A † ij (t)] ≤ 2ρ. Now we proceed to prove (22), which upper-bounds the conditional expectation E a ij (t)−d ij (t) A † ij (t)−D † ij (t) | Q(t) . It suffices however to upper-bound the unconditional expectation E a ij (t)−d ij (t) A † ij (t)−D † ij (t) , which we will do in the following, since we can obtain the same upper bounds on E[D † ij (t)] and E[d ij (t)] (2 and 1 respectively) whether the expectations are conditional (on Q(t)) or not. Note the other two terms A † ij (t) and a ij (t) are independent of (the condition) Q(t). As for any i, j, a ij (t) is i.i.d., we have, E a ij (t) − d ij (t) A † ij (t) − D † ij (t) (25) =E[a ij (t)A † ij (t) − d ij (t)A † ij (t) − a ij (t)D † ij (t) + d ij (t)D † ij (t)] =E[a 2 ij (t)]−λ 2 ij +λ ij Λ † ij −E[d ij (t)]Λ † ij − λ ij E[D † ij (t)]+E[d ij (t)D † ij (t)] =E[a 2 ij (t)]−λ 2 ij +λ ij Λ † ij −E[d ij (t)]Λ † ij − λ ij E[D † ij (t)]+E[d ij (t)].(26) In arriving at (26), we have used (5). The RHS of (26) can be bounded by a constant C > 0 due to the following assumptions and facts: E[a 2 ij (t)] = E[a 2 ij (0)] < ∞ for any t, d ij (t) ≤ 1, D † ij (t) ≤ 2, λ ij ≤ ρ < 1/2, and Λ † ij ≤ 2ρ < 1. Therefore, we have (by applying i,j to both (25) and the RHS of (26)) i,j E a ij (t)−d ij (t) A † ij (t)−D † ij (t) ≤ CN 2 . Remarks. Now that we have proved that {Q(t)} ∞ t=0 is positive recurrent. Therefore, for any i, j, the long term departure rate lim T →∞ 1 T T −1 t=0 E[d ij (t)] = λ ij . Hence, we have, lim T →∞ 1 T T −1 t=0 E a ij (t)−d ij (t) A † ij (t)−D † ij (t) =σ 2 ij −λ ij Λ † ij +λ ij .(27) where σ 2 ij = E[a 2 ij (t)]−λ 2 ij is the variance of a ij (t), because LHS of (27) is the long term average of (25), and the long term average of (26) can be simplified as the RHS of (27). E. Delay Analysis In this section, we derive the queue length bound (readily convertible to the delay bound by Little's Law) of QPS-1 using the following moment bound theorem [31,Proposition 2.1.4]. Although in the interest of space, in this work we only show the delay analysis for the i.i.d. traffic arrivals, those for more general arrivals are almost identical. It can be shown that the delay analysis results for general Markovian arrivals derived in [11,12] for maximal matchings (using the stronger "departure inequality" (6)) hold also for QPS-1. Theorem 2. Suppose that {Y t } ∞ t=0 is a positive recurrent Markov chain with countable state space Y. Suppose V , f , and g are non-negative functions on Y such that, V (Y t+1 ) − V (Y t ) ≤ −f (Y t ) + g(Y t ), for all Y t ∈ Y (28) Then lim T →∞ 1 T T −1 t=0 E[f (Y t )] ≤ lim T →∞ 1 T T −1 t=0 E[g(Y t )]. Now we derive the queue length bound for QPS-1 when the maximum load factor ρ < 1/2. The queue length bound is formally stated in the following theorem. Theorem 3. Given a switching system under QPS-1, whenever the maximum load factor ρ < 1/2, the steady state mean queue length is bounded. More precisely, we have, E[ Q 1 ] ≤ 1 2(1 − 2ρ) i,j σ 2 ij −λ ij Λ † ij +λ ij .(29) Proof: We define V , Y t , f , and g terms in Theorem 2 in such a way that the LHS and the RHS of (28) become the LHS and the RHS of (19) respectively (e.g., define V as L, Y t as Q(t), and f (Y t ) as −2 i,j q ij (t) A † ij (t) − D † ij (t) ). Then, we have, − 2(2ρ − 1)E[ Q 1 ] ≤E[f (Ȳ )] (30) ≤E[g(Ȳ )](31)= i,j σ 2 ij −λ ij Λ † ij + λ ij .(32) In the above derivation, inequality (30) is due to (21) (whose LHS is −f (Y t )), inequality (31) is due to Theorem 2, and equality (32) is due to (27). Therefore, we have, in steady state, E[ Q 1 ] ≤ 1 2(1 − 2ρ) i,j σ 2 ij −λ ij Λ † ij +λ ij . This queue-length bound is identical to that derived in [11, 12, Section III.B] for maximal matchings under i.i.d. traffic arrivals. It is not hard to check (by applying Little's Law) that the average delay (experienced by packets) is bounded by a constant independent of N (i.e., order-optimal) for a given maximum load factor ρ < 1/2, if the variance σ 2 ij for any i, j is assumed to be finite. For the special case of Bernoulli i.i.d. arrival (when σ 2 ij = λ ij − λ 2 ij ), this bound (the RHS) can be further tightened to i,j λij 1−2ρ . This implies, by Little's Law, the following "clean" bound:ω ≤ 1 1−2ρ whereω is the expected delay averaged over all packets transmitting through the switch. V. EVALUATION In this section, we evaluate, through simulations, the performance of QPS-r under various load conditions and traffic patterns. We compare its performance with that of iSLIP [17], a refined and optimized representative parallel maximal matching algorithm (adapted for switching). The performance of the MWM (Maximum Weighted Matching) is also included in the comparison as a benchmark. Our simulations show conclusively that QPS-1 performs very well inside the provable stability region (more precisely, with no more than 50% offered load), and that QPS-3 has comparable throughput and delay performances as iSLIP, which has much higher computational and communication complexities. A. Simulation Setup In our simulations, we first fix the number of input/output ports, N to 64. Later, in section V-C we investigate how the mean delay performances of these algorithms scale with respect to N . To measure throughput and delay accurately, we assume each VOQ has an infinite buffer size and hence there is no packet drop at any input port. Each simulation run is guided by the following stopping rule [32,33]: The number of time slots simulated is the larger between 500N 2 and that is needed for the difference between the estimated and the actual average delays to be within 0.01 time slots with probability at least 0.98. We assume in our simulations that each traffic arrival matrix A(t) is Bernoulli i.i.d. with its traffic rate matrix Λ being equal to the product of the offered load ρ and a traffic pattern matrix (defined next). Similar Bernoulli arrivals were studied in [17,20,30]. Note that only synthetic traffic (instead of that derived from packet traces) is used in our simulations because, to the best of our knowledge, there is no meaningful way to combine packet traces into switch-wide traffic workloads. The following four standard types of normalized (with each row or column sum equal to 1) traffic patterns are used: (I) Uniform: packets arriving at any input port go to each output port with probability 1 N . (II) Quasi-diagonal: packets arriving at input port i go to output port j = i with probability 1 2 and go to any other output port with probability 1 2(N −1) . (III) Log-diagonal: packets arriving at input port i go to output port j = i with probability 2 (N −1) 2 N −1 and go to any other output port j with probability equal 1 2 of the probability of output port j−1 (note: output port 0 equals output port N ). (IV) Diagonal: packets arriving at input port i go to output port j = i with probability 2 3 , or go to output port (i mod N )+1 with probability 1 3 . These traffic patterns are listed in order of how skewed the volumes of traffic arrivals to different output ports are: from uniform being the least skewed, to diagonal being the most skewed. B. QPS-r Throughput and Delay Performances We first compare the throughput and delay performances of QPS-1 (1 iteration), QPS-3 (3 iterations), iSLIP (log 2 64 = 6 iterations), and MWM (length of VOQ as the weight measure). Figure 2 shows their mean delays (in number of time slots) under the aforementioned four traffic patterns respectively. Each subfigure shows how the mean delay (on a log scale along the y-axis) varies with the offered load ρ (along the x-axis). We make three observations from Figure 2. First, Figure 2 clearly shows that, when the offered load is no larger than 0.5, QPS-1 has low average delays (i.e., more than just being stable) that are close to those of iSLIP and MWM, under all four traffic patterns. Second, the maximum sustainable throughputs (where the delays start to "go through the roof" in the subfigures) of QPS-1 are roughly 0.634, 0.645, 0.681, and 0.751 respectively, under the four traffic patterns respectively; they are all comfortably larger than the 50% provable lower bound. Third, the throughput and delay performances of QPS-3 and iSLIP are comparable: The former has slightly better delay performances than the latter under all four traffic patterns except the uniform. The exception is that we did not obtain the delay values for MWM (not a "main character" in our story) for N = 1, 024, as it proved to be prohibitively expensive computationally to do so. In all these plots, the offered load is 0.75, which is quite high compared to the maximum achievable throughputs of QPS-3 and iSLIP (shown in Figure 2) under these four traffic patterns. Figure 3 shows that the mean delays of QPS-3 are slightly lower (i.e., better) than those of iSLIP under all traffic patterns except the uniform. In addition, the mean delay curves of QPS-3 remain almost flat (i.e., constant) under logdiagonal and diagonal traffic patterns. Although they increase with N under uniform and quasi-diagonal traffic patterns, they eventually almost flatten out when N gets larger (say when N ≥ 128). These delay curves show that QPS-3, which runs only 3 iterations, deliver slightly better delay performances, under all non-uniform traffic patterns, than iSLIP (a refined and optimized parallel maximal matching algorithm adapted for switching), which runs log 2 N iterations with each iteration has O(log 2 N ) computational complexity. C. Scale with Port Numbers VI. RELATED WORK Scheduling in crossbar switches is a well-studied problem with a large amount of literature. So, in this section, we provide only a brief survey of prior work that is directly related to ours, focusing on those we have not described earlier. O(N )-complexity algorithms that attain 100% throughput. Several serial randomized algorithms, starting with TASS [34] and culminating in SERENA [30], have been proposed that have a total computational complexity of only O(N ) yet can provably attain 100% throughput; SERENA, the best among them, also delivers a good empirical delay performance. However, this O(N ) complexity is still too high for scheduling high-line-rate high-radix switches, and none of them has been successfully parallelized (i.e., converted to a parallel iterative algorithm) yet. Notice that O(N ) computational complexity is not the complexity barrier for attaining 100% throughput, sub-linear algorithms attaining 100% throughput do exist. However, those algorithms compromise delay performances and/or generality. For example, the O(log N ) algorithm proposed in [35] can provably attain 100% throughput under the assumption that the traffic rate matrix is known a prior. O(1)-complexity algorithms. In [36], a crossbar scheduling algorithm specialized for switching variable-size packets was proposed, that has O(1) total computational complexity (for the entire switch). Although this algorithm can provably attain 100% throughput, its delay performance is poor. For example, as shown in [20], its average delays, under the aforementioned four standard traffic matrices, are roughly 3 orders of magnitudes higher than those of SERENA [30] even under a moderate offered load of 0.6. A parallel iterative algorithm called RR/LQF (Round Robin combined with Longest Queue First), that has O(1) time complexity per iteration per port was recently proposed, in [37]. Even though N iterations of this algorithm have to be run (for each scheduling) for it to provably attain at least 50% throughput, running only 1 iteration leads to reasonably good empirical throughput and delay performance over round-robin-friendly workloads such as uniform and hot-spot. Batch scheduling algorithms. In all algorithms above, a matching decision is made in every time slot. An alternative type of algorithms [38][39][40] is frame-based, in which multiple (say K) consecutive time slots are grouped as a frame. These K matching decisions in a frame are batch-computed, which usually has lower time complexity than K independent matching computations. However, since K is usually quite large (e.g.,= O(log N )), and a packet arriving at the beginning of a frame has to wait till at least the beginning of the next frame to be switched, frame-based scheduling generally lead to higher queueing delays. The best known provable delay guarantee for this type of algorithms is O(log N ) average delays for an N × N switch [39]. However, this algorithm has a high computational complexity of O(N 1.5 log N ) per time slot. VII. CONCLUSION In this work, we propose QPS-r, a parallel iterative crossbar scheduling algorithm with O(1) computational complexity per port. We prove, through Lyapunov stability analysis, that it achieves the same QoS (throughput and delay) guarantees in theory, and demonstrate through simulations that it has comparable performances in practice as the family of maximal matching algorithms (adapted for switching); maximal matching algorithms are much more expensive computationally (at least O(log N ) iterations and a total of O(log 2 N ) perport computational complexity). These salient properties make QPS-r an excellent candidate algorithm that is fast enough computationally and can deliver acceptable throughput and delay performances for high-link-rate high-radix switches. Figure 3 3shows how the mean delays of QPS-3, iSLIP (running log 2 N iterations given any N ), and MWM scale with the number of input/output ports N , under the four different traffic patterns. With one exception, we have simulated the following different values of N : N = 8, 16, 32, 64, 128, 256, 512, 1, 024. 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Compressive sensing relies on the sparse prior imposed on the signal to solve the ill-posed recovery problem in an under-determined linear system. The objective function that enforces the sparse prior information should be both effective and easily optimizable. Motivated by the entropy concept from information theory, in this paper we propose the generalized Shannon entropy function and Rényi entropy function of the signal as the sparsity promoting objectives. Both entropy functions are nonconvex, and their local minimums only occur on the boundaries of the orthants in the Euclidean space. Compared to other popular objective functions such as the x 1, x p p , minimizing the proposed entropy functions not only promotes sparsity in the recovered signals, but also encourages the signal energy to be concentrated towards a few significant entries. The corresponding optimization problem can be converted into a series of reweighted l1 minimization problems and solved efficiently. Sparse signal recovery experiments on both the simulated and real data show the proposed entropy function minimization approaches are better than other popular approaches and achieve state-of-the-art performances.

10.1109/tsp.2018.2889951

1703.10556

55cd3877bc5956794150f586c8ab822ec411825f

Sparse Signal Recovery via Generalized Entropy Functions Minimization Student Member, IEEEShuai Huang Fellow, IEEETrac D Tran Sparse Signal Recovery via Generalized Entropy Functions Minimization 1Index Terms-Compressive sensingentropy function mini- mizationentropy minimizationimage recovery Compressive sensing relies on the sparse prior imposed on the signal to solve the ill-posed recovery problem in an under-determined linear system. The objective function that enforces the sparse prior information should be both effective and easily optimizable. Motivated by the entropy concept from information theory, in this paper we propose the generalized Shannon entropy function and Rényi entropy function of the signal as the sparsity promoting objectives. Both entropy functions are nonconvex, and their local minimums only occur on the boundaries of the orthants in the Euclidean space. Compared to other popular objective functions such as the x 1, x p p , minimizing the proposed entropy functions not only promotes sparsity in the recovered signals, but also encourages the signal energy to be concentrated towards a few significant entries. The corresponding optimization problem can be converted into a series of reweighted l1 minimization problems and solved efficiently. Sparse signal recovery experiments on both the simulated and real data show the proposed entropy function minimization approaches are better than other popular approaches and achieve state-of-the-art performances. I. INTRODUCTION Nowadays, there is an increasing amount of digital information constantly generated from every aspect of our life and data that we work with grow in both size and variety. Fortunately, most of the data have sparse structures. Compressive sensing [1]- [3] offers us an efficient framework to not only collect data but also to process and analyze them in a timely fashion. Various compressive sensing tasks eventually boil down to the sparse signal recovery problem in the following underdetermined linear system: y = Ax + w ,(1) where y ∈ R M is the linear measurement, A ∈ R M ×N is the sensing matrix, x ∈ R N is the sparse signal with mostly zero entries, and w ∈ R M is the measurement noise. In compressive sensing, we try to recover the sparse signal x given {y, A} with M N . In this case, the sensing matrix A contains more columns than rows, and there would be more than one solutions that satisfy the constrain y − Ax 2 2 ≤ , where ≥ 0 is an upper bound on the noise contribution. This makes the recovery of x an ill-posed problem. On the other hand, since the signal of interest itself is sparse, the most straight-forward way is to search for the solutions that also share this sparse property. Following the well known Occam's razor, we can use the l 0 norm as the criterion and choose the sparsest (simplest) one: P 0 (x) : min x x 0 subject to y − Ax 2 2 ≤ . (2) This rather naïve attempt is actually backed up by sound theories [4]- [8]. Under noiseless conditions, it can be shown that the sparsest solution is indeed the true signal when x is sufficiently sparse and A satisfies the corresponding restricted isometry property [4], [7]. P 0 (x) is a nonconvex NP-hard problem whose solutions requires an intractable combinatorial search [9]. In practice, two alternative approaches are usually employed to solve P 0 (x): 1) Greedy search under the constrain x 0 ≤ S. 2) Relaxation of the l 0 norm x 0 . S > 0 is an upper bound on the number of nonzero entries in x. The greedy search approach leads to various matching pursuit methods [10]- [13], while the relaxation approach leads to methods that minimize different objective functions to promote sparsity in the recovered solution [8], [14]- [23]. Here we focus on studying the "relaxation" approach that tries to solve the following unconstrained recovery problem: P g (x) : min x y − Ax 2 2 + λg(x) ,(3) where λ > 0 is the parameter that balances the trade-off between the data fidelity term y − Ax 2 2 and the sparsity regularizer g(x). The sparse prior information is enforced via the regularizer g(x), and a proper g(x) is crucial to the success of the sparse signal recovery task: it should favor sparse solutions and make sure the problem P g (x) can be solved efficiently in the mean time. In this paper we propose the generalized Shannon entropy function and Rényi entropy function of x as new sparsityregularization objectives, and show their effectiveness and advantages over other popular regularizers in promoting sparse solutions with both theoretical analyses and experimental evaluations. A. Prior Work Various sparsity regularizers have been proposed as the relaxation of the l 0 norm. Most popular among them are the convex l 1 norm and the nonconvex l p norm to the p-th power [14], [18], [20], [21], [24], [25]: Both x 1 and x p p are separable functions, and strict error bounds on the recovered solutions from l 1 -minimization and l p p -minimization problems in (3) can be established [20], [21]. Compared to l 1 -minimization, l p p -minimization has a tighter error bound and better sparse recovery performances. [24] uses the following "logarithm of energy" function as a measure of sparsity: • g 1 (x) = i log |x i | 2 = 2 i log |x i | Minimizing it is equivalent the l p p -minimization method when p → 0 [25]. [8] later proposed the reweighted l 1 -minimization algorithm as a way to enhance the sparsity, which is essentially the iterative minimization of the first-order approximation of a modified g 1 (x) function: i log(|x i | + ), > 0. Entropy-based functionals have also been widely used to promote sparsity [14], [15], [17], [18], [24]. The Shannon entropy functions considered all share the following form: • g 2 (x) = − ix i logx i ,x i > 0, ix i = 1. x i is constructed from the sparse signal x. In [15], [17], [18], [24], the case wherex i = |xi| 2 x 2 2 is studied. [17] also considers the cases wherex i = |x i | andx i = |xi| x 1 , and the corresponding Rényi entropy functions. Choosingx i = |x i | fails to promote sparsity in the signal x whose l 1 -norm is not 1, i.e. x 1 = 1. There are also some imprecisions in [17]'s analysis on the local minimums of the l 2 2 -normalized entropy functions, [17] states that the local minimums "occur just shy of the boundaries defined by the coordinate axes". In section II-C, we can prove that the local minimums actually occur exactly on the coordinate axes. The previously mentioned entropy functions are all nonconvex. [23] later proposes the following convex entropy function as an approximation to the l 1 -norm x 1 : • g 3 (x) = i (|x i | + 1 e ) log(|x i | + 1 e ) + 1 e . The entropy function g 3 (x) by [23] maintains the strictly convex property of x 1 and is continuously differentiable in R. However, g 3 (x) only produces concentrated but not truly sparse solutions. The convex l 1 -norm minimization problem can be efficiently solved by many available algorithms [26]- [32], such as interior-point method, FISTA, AMP, etc. However, it is often quite difficult to directly minimize the aforementioned nonconvex sparsity regularizers. In this case, we can iteratively minimize the approximation or upper bound of the regularizer using the reweighted l 1 or l 2 approach [8], [16], [25], [33]. B. Main Contribution Compared to previously adapted entropy functions [15], [17], [18], [24], our proposed Shannon entropy function h p (x) and Rényi entropy function h p,α (x) are more generalized. They are defined with respect to the probability distribution in (6) where p can choose any positive number. This gives us more freedom in constructing the proper entropy function for the sparse signal recovery task. As is evident from the experiments on both simulated and real data, a good choice of the p value enables us to fully exploit the sparsity-promoting property of the entropy functions and to achieve better performances over the state-of-the-art x 1 and x p p minimization approaches. Previous works [17], [18] focus on the study of the Schurconcavity of the entropy functions with respect to x where p = 1, 2, and believe that the Shannon entropy function produces truly sparse solutions only when p = 1. Here we can show that it's the concavity or Schur-concavity with respect to the distribution P(·) that really matters in the sparsity promotion analysis. In fact, ∀p > 0 and 1 = α > 0, we can prove that the local minimums of the proposed entropy functions only occur at the boundaries of the orthants in R N . Hence minimizing h p (x) or h p,α (x) in said orthant O will lead us to the solutions on its boundaries, i.e. sparser solutions. The Shannon entropy function with p = 1 is not the only case where truly sparse solutions can be obtained. Additionally, minimizing the proposed entropy functions promotes large-magnitude entries in the recovered signal and encourages the energy ofx to be concentrated towards a few significant entries. Using the proximal approximation [34], [35] of the data fidelity term and the first order approximations of the entropy functions, we can convert the nonconvex minimization problems into a series of classical reweighted l 1 problems, and solve them efficiently. II. SPARSITY-REGULARIZATION ENTROPY FUNCTION A. Introduction to Entropy We first introduce the entropy concepts in information theory [36], [37]. Both the Shannon entropy and Rényi entropy are defined with respect to the probability distribution P(V) of some random variable V. Here we give the following definitions in terms of discrete probability distribution 1 : • Shannon entropy 2 : H(V) = − |V| i=1 P(v i ) log P(v i ) .(4) H(V) is strictly concave with respect to the probability distribution P(V) = {P(v 1 ), · · · , P(v |V| )}. • Rényi entropy: H α (V) = 1 1 − α log   |V| i=1 P(v i ) α   ,(5) where α ≥ 0 and α = 1. When α ∈ (0, 1), H α (V) is strictly concave with respect to P(V) [38]; when α ∈ (1, ∞), H α (V) is strictly Schur concave with respect to P(V) [39]. We should make it clear that Shannon entropy H(V) is not a special case of the Rényi entropy, but the limiting value of the Rényi entropy H α (V) as α → 1 [40]. Hence we need to discuss them respectively in this paper. B. Entropy Function of the Sparse Signal Entropy measures the uncertainty about the random variable V with |V| = N . The lower the entropy is, the more predictable the variable V is, which corresponds to a skewed distribution P(V). The idea of a skewed distribution could translate naturally to the idea of a sparse probability vector P V = [P(v 1 ), · · · , P(v N )] T in the sense that only a few probability values of P V ∈ R N are significant. In other words, the entropy can be used as a measure of how sparse the probability vector P V is. This observation motivates us to adapt the concept of entropy as a sparsity-measure for the general signal x and to use it as a regularizer in the sparse signal recovery task. As we have mentioned before, the entropy is defined with respect to a probability distribution P(·). Here we can construct the following discrete probability distribution out of the signal x ∈ R N : x → |x 1 | p x p p , |x 2 | p x p p , · · · , |x N | p x p p ,(6) where p > 0. The adaptation from the classical entropy to the entropy function is then pretty straightforward. Specifically, the following two types of entropy functions are proposed: 1) Shannon entropy function: h p (x) = − N i=1 |x i | p x p p log |x i | p x p p ,(7) where p > 0. h p (x) is the "Shannon entropy function" of x, it should not be confused with the "Shannon entropy" of x in (4): H(x) = − x P(x) log P(x) dx. 2) Rényi entropy function: h p,α (x) = 1 1 − α log   |V| i=1 |x i | p x p p α   ,(8) where p > 0, α > 0 and α = 1. Again, this should not be confused with the Rényi entropy of x in (5). Both h p (x) and h p,α (x) are nonconvex functions, their local minimums occur at the boundary of each orthant in the Euclidean space R N , i.e. the axes. Take the Shannon entropy function for example, the 2-dimensional level plots of h p (x) with p = {0.5, 1, 2} are shown in Fig. 1. We can see that the local minimums occur at the two axises in all three cases. In order to promote sparsity in the recovered solutions, we would like to minimize the entropy functions. The sparse signal recovery problems in (3) based on the Shannon entropy function (SEF) minimization and the Rényi entropy function (REF) minimization then become: P hp (x) : min x y − Ax 2 2 + λh p (x) (9) P hp,α (x) : min x y − Ax 2 2 + λh p,α (x) .(10) C. Sparsity Promotion Analysis We next show that h p (x) and h p,α (x) can be used as sparsity regularizers in the following sense: minimizing them in an orthant O of the Euclidean space R N leads us to solutions on the boundary of said orthant, i.e. sparser solutions. K Noiseless recovery: In this case we are minimizing h p (x) or h p,α (x) subject to the constrain y = Ax. We first show that there is a one to one mapping in each orthant between x = [x 1 , · · · , x N ] T andẍ = [ẍ 1 , · · · ,ẍ N ] T , whereẍ i = sign(x i ) · |xi| p x p p . This will be done in two steps: Lemma 1 and Lemma 2. Lemma 1. If x is the solution to y = Ax, y = 0, then there is a one to one mapping in each orthant between x and x = x x p . Proof. We just need to prove x ←→x: • It is easy to verify that x →x. • Suppose there are two solutions of y = Ax: x (1) , x (2) in the same orthant, and they are both mapped tox. We then have: which tells us x (1) x (1) p =x = x (2) x (2) p(11)y x (1) p = Ax (1) x (1) p = Ax = Ax (2) x (2) p = y x (2) p ,(12)y x (1) p = y x (2) p . Since y = 0, we have x (1) p = x (2) p . Using (11), we get x (1) = x (2) . Hence x ←x. Lemma 2. There is a one to one mapping in each orthant betweenx andẍ Proof. We just need to provex ←→ẍ: • We can rewriteẍ in terms ofx:ẍ = sign(x)·|x| p . Hencẽ x →ẍ. • Suppose there are two pointsx (1) ,x (2) in the same orthant mapped to the sameẍ. We then have: sign(x (1) ) · |x (1) | p =ẍ = sign(x (2) ) · |x (2) | p ,(13) which tells us |x (1) | = |x (2) |. Since sign(x (1) ) = sign(x (2) ), we getx (1) =x (2) . Hencex ←ẍ. Combining Lemma 1 and Lemma 2, we have x ←→ẍ, as is shown in Fig. 2. Let X = {x 1 , x 2 , · · · } be the solutions of y = Ax in one of the orthants O. Specifically, X = X 1 ∪ X 2 and X 1 ∩X 2 = ∅, where X 1 contains solutions on the boundary of the orthant O and X 2 contains the rest solutions that are not on the boundary. The solution x is then mapped toẍ one by one, producing the corresponding mapped setsẌ 1 ,Ẍ 2 . We can verify that the solutions in X 1 are sparser than those in X 2 , and we have the following Lemma 3: Lemma 3. For every solution x ∈ X 2 , there is a solution x * ∈ X 1 on the boundary of the orthant O such that h p (x * ) < h p (x) and h p,α (x * ) < h p,α (x). Proof. By definition we have: For the SEF h p (x), we first study the local minimums on the plane ẍ 1 = 1. g(ẍ) is strictly concave with respect tö x, and the local minimums of g(ẍ) are on the boundary of the orthant O. Hence for everyẍ ∈Ẍ 2 , there is aẍ * ∈Ẍ 1 such that g(ẍ * ) < g(ẍ). h p (x) = g(ẍ) = − N i=1 |ẍ i | log |ẍ i | (14) h p,α (x) = g α (ẍ) = 1 1 − α log N i=1 |ẍ i | α .(15) For the REF h p,α (x), when α ∈ (0, 1), g α (ẍ) is strictly concave with respect toẍ, the local minimums of g α (ẍ) are on the boundary of the orthant O. When α ∈ (1, ∞), g α (ẍ) is strictly Schur concave [39], since the boundary of the orthant O majorizes theẍ inside O, the local minimums of g α (ẍ) are also on the boundary of O. Hence for everyẍ ∈Ẍ 2 , there also exists aẍ * ∈Ẍ 1 such that g α (ẍ * ) < g α (ẍ) for α ∈ (0, 1) ∪ (1, ∞). There is a one to one mapping in O between x andẍ: x ←→ẍ. Since h p (x) = g(ẍ) and h p,α (x) = g α (ẍ), for every x ∈ X 2 , there is a x * ∈ X 1 such that h p (x * ) < h p (x) and h p,α (x * ) < h p,α (x). From Lemma 3 we can see that minimizing h p (x) or h p,α (x) in the orthant O will lead us to the sparser solutions in X 1 . K Noisy recovery: We can show similarly that minimizing h p (x) or h p,α (x) subject to the constrain y − Ax 2 2 ≤ in an orthant O of the Euclidean space ∈ R N also produces sparse solutions. First, we have the following Lemma 4: Lemma 4. Let X = {x 1 , x 2 , · · · } are the nonzero solutions satisfying the constrain y − Ax 2 2 ≤ , y = 0 such that: ∀x i = x j , x i = τ x j for some τ > 0. Pick any x i ∈ X , there is a one to one mapping in each orthant between the set X andx i = xi xi p . Proof. We need to prove X ←→x i • Suppose that there are two sets X (1) , X (2) in the same orthant being mapped to the samex i . Let x 1 ∈ X (1) and x 2 ∈ X (2) , we have: • ∀x j ∈ X \x i , we havex i = xi xi p = τ xj τ xj p = xj xj p = x j . It is easy to verify that x j →x j =x i . Hence X → x i .x 1 x 1 p =x 1 =x i =x 2 = x 2 x 2 p .(16) We then have x 1 = x1 p x2 p x 2 , which means that x 1 , x 2 belongs to the same set, i.e. X (1) = X (2) . Hence X ←x i . We can see that the solutions in X all have the same entropy function value. Combining Lemma 4 and Lemma 2, we have X ←→ẍ i for anyẍ i ∈ X , as is illustrated in Fig. 3. Let X = {x 1 , x 2 , · · · } denote the nonzero solutions that satisfy y − Ax 2 2 ≤ in one of the orthants O. Specifically, X = X 1 ∪ X 2 and X 1 ∩ X 2 = ∅, where X 1 contains solutions on the boundary of the orthant O and X 2 contains the rest solutions that are not on the boundary. Lemma 3 also applies in the noisy case. We can see that minimizing h p (x) or h p,α (x) in the orthant O also leads to sparser solutions in X 1 in the noisy case. Proof. Without loss of generality, suppose x o ∈ R N is a local minimum inside some orthant O, as is shown in Fig. 4. There exists a small neighborhood ρ( x o ) surrounding x o such that ∀x ∈ ρ(x o ), h p (x) ≥ h p (x o ) or h p,α (x) ≥ h p,α (x o ). As is done in Lemma 1 and 4, we project x o along with its neighborhood ρ(x o ) tox o and ρ(x o ) on the sphere x p p = 1 inside the same orthant O: x o ←→x o and ρ(x o ) ←→ ρ(x o ) .(17) x o and ρ(x o ) are further projected toẍ o and ρ(ẍ o ) respectively on the sphere ẍ 1 = 1 inside the same orthant O. x o ←→ẍ o and ρ(x o ) ←→ ρ(ẍ o ) .(18) Consequently So what happens if we minimize the "true" entropy of x? Take the Shannon entropy in (4) for example, it is defined with respect to some probability distribution P(·). Here we can assume the signal x follows the pre-specified distribution P(x|θ) parameterized by θ. For the sparse signal recovery task, we rely on the posterior distribution P(x|y, θ) to perform the MAP estimation or MMSE estimation of the signalx. , ∀ẍ ∈ ρ(ẍ o ), we have h p (ẍ) ≥ h p (ẍ o ) or h p,α (ẍ) ≥ h p,α (ẍ o ).ẍ o is In practice the entries x are further assumed to be independently distributed given y, and the following Shannon entropy will be minimized: θ = arg min θ H(x|y, θ) = arg min θ − P(x|y, θ) log P(x|y, θ) dx = arg min θ − i P(x i |y, θ) log P(x i |y, θ) dx i .(19) P(x i |y, θ) can be computed using the approximate message passing algorithms (AMP) [31], [41]- [43]. In this case (19) essentially serves as a parameter estimation step in the PE-GAMP algorithm [43]. D. Energy Concentration Analysis Apart from promoting sparsity in the recovered signal, minimizing the entropy functions h p (x) and h p (x) also encourages the energy of x to be concentrated towards a few significant entries. This energy concentration behavior can be best illustrated from an optimization point of view. Take the SEF h p (x) for example, we use gradient descent to minimize it and the solution x is updated as follows: where η > 0 is some suitable step size, and ∇ |x| h p (x) is the derivative with respect to the magnitude |x|: x = x − η · sign(x) · ∇ |x| h p (x) ,(20)∂h p (x) ∂|x i | = − p|x i | (p−1) log |x i | p x p p + p|x i | (p−1) l |x l | p log |x l | p x 2p p .(21) It's easy to verify the following remark: Remark. Let ν = exp l |x l | p log |x l | p p x p p , ν > 0, we then have: ∂h p (x) ∂|x i |    < 0 = 0 > 0 if |x i | > ν if |x i | = ν if |x i | < ν .(22) For the entries with relatively large magnitudes |x i | > ν, we can see that the update in (20) makes their magnitudes larger. Conversely, if the entries have relatively small magnitudes |x i | < ν, the update makes their magnitudes even smaller. In this way minimizing h p (x) promotes large-magnitude entries in the recovered signal, hence energy concentration. For the REF h p,α (x), the derivative with respect to the magnitude |x| is: ∂h p,α (x) ∂|x i | = 1 1 − α × 1 N l=1 |x l | x p pα × pα x p+pα p × |x i | pα−1 x p p − |x i | p−1 x pα pα .(23) Similarly we have the following remark: Remark. Let ν = exp 1 pα−p log x pα pα x p p , ν > 0, we then have: ∂h p,α (x) ∂|x i |    < 0 = 0 > 0 if |x i | > ν if |x i | = ν if |x i | < ν .(24) We can see that minimizing h p,α (x) also encourages energy concentration in the recovered signal. In III. ENTROPY FUNCTION MINIMIZATION In this section we propose the algorithms to perform the sparse signal recovery tasks in (9,10). Specifically, the proximal regularization [34], [35] of the data fidelity term f (x) = y − Ax 2 2 and the first order approximations of the entropy functions h p (x), h p,α (x) are minimized in alternation iteratively until convergence. The proposed entropy functions h p (x), h p,α (x) are nonconvex, a good initialization is needed to ensure good performance. Here we will use the solution from l 1 normminimization as the initialization to our proposed algorithm. The sparsity-promotion analysis in section II-C shows that we are able to obtain sparse solutions by minimizing the entropy functions. In order to solve the problems in (9,10), the following two steps are repeated in alternation until convergence. 1) In the first step, the data fidelity term f (x) = y −Ax 2 2 is approximated: For the (t + 1)-th iteration to solve the problems P hp (x) and P hp,α (x), we use its quadratic approximation, a.k.a. proximal regularization [34], at the previous t-th iteration's solutionx (t) : f (x) = y − Ax 2 2 ≤ f (x (t) ) + x −x (t) , ∇f (x (t) ) + κ 2 x −x (t) 2 2 = o(x (t) ) + κ 2 x − x (t) − 1 κ ∇f (x (t) ) 2 2 ,(25) where o(x (t) ) is a relative constant depending on the previous solutionx (t) , ∇f (x (t) ) = 2(A T Ax (t) − A T y), κ is the Lipschitz constant of the gradient ∇f [44]. The smallest value κ can take is twice the largest eigenvalue of A T A to ensure that f (x) is bounded by the proximal regularization. it can be viewed as a suitable step size to ensure the upper bound on f (x) in (25). When κ is unknown or difficult to compute, we can use a backtracking strategy to find it. The problems in (9, 10) then becomes: P (1) hp (x) : min x κ 2 x − x (t) − 1 κ ∇f (x (t) ) 2 2 + λh p (x)(26)P (1) hp,α (x) : min x κ 2 x − x (t) − 1 κ ∇f (x (t) ) 2 2 + λh p,α (x) .(27) 2) In the second step, the problems P hp (x) and P (1) hp,α (x) are iteratively solved: In the "inner" (r + 1)-th iteration to solve P (1) hp (x) and P (1) hp,α (x), h p (x), h p,α (x) are approximated with their first order approximations with respect to |x (t+1,r) | from the previous r-th iteration: h p (x) ≈ |x| − |x (t+1,r) |, ∇h p (x (t+1,r) ) + h p (x (t+1,r) ) (28) h p,α (x) ≈ |x| − |x (t+1,r) |, ∇h p,α (x (t+1,r) ) + h p,α (x (t+1,r) ) ,(29) where ∇h p (x), ∇h p,α (x) are the first order derivatives with respect to |x i | given in (21,23). Since log 0 is −∞, when computing ∇ p (|x hp (x) and P (1) hp,α (x) then become: P (2) hp (x) : min x κ 2 x − x (t) − 1 κ ∇f (x (t) ) 2 2 + λ |x|, ∇h p (x (t+1,r) )(30)P (2) hp,α (x) : min x κ 2 x − x (t) − 1 κ ∇f (x (t) ) 2 2 + λ |x|, ∇h p,α (x (t+1,r) ) .(31)P (2) hp (x) and P (2) hp,α (x) are simple reweighted l 1 normminimization problems that can be converted to a series of independent one-dimensional problems. The solutionŝ x (t+1,r+1) i to the above problems can be obtained using the iterative shinkage thresholding algorithm (ISTA): x (t+1,r+1) i = Γ λ κ ∇hp(|x (t+1,r) i |) x (t) i − 1 κ ∇f (x (t) i ) (32) x (t+1,r+1) i = Γ λ κ ∇hp,α(|x (t+1,r) i |) x (t) i − 1 κ ∇f (x (t) i ) ,(33) where Γ τ (·) is the soft thresholding function, a.k.a. shrinkage operator, defined as follows: Γ τ (x) = 0 (|x| − τ ) · sign(x) if |x| ≤ τ if |x| > τ .(34) Conventional ISTA solves a convex problem and requires the threshold τ to be positive. However, the derivatives ∇h p (x (t+1,r) and ∇h p,α (x (t+1,r) in (30,31) could be negative. In Appendix A we can show that the optimal solution can still be obtained using the soft thresholding operator given in (34), yet with a different derivation process. The proposed entropy function minimization approach can be summarized in Algorithm 1. (25); 4: Initialize {x (t+1,r) , r = 0} withx (t) ; 5: for r = {0, 1, · · · } do 6: Computex (t) − 1 κ ∇f (x (t) ) in Compute ∇h p (x) or ∇h p,α (x) in (21, 23); 7: Obtainx (t+1,r+1) by solving P (2) hp (x) or P (2) hp,α (x) in (32, 33); 8: ifx (t+1,r+1) reaches convergence or the objective functions in (26,27) As is shown in Appendix B, Algorithm 1 produces a sequence {x (t) , t = 0, 1, · · · } that decreases objective functions in (9,10) monotonically. Since the data fidelity term f (x) = y − Ax 2 2 ≥ 0 and the entropy functions h p (x) ≥ 0, h p,α(x) ≥ 0 are all bounded from below, eventually Algorithm 1 is going to converge to some local minimums of the nonconvex objective functions. A proper initialization is thus need for best performance, here we use the solution from l 1 norm minimization to initializex (0) . ISTA usually converges slowly in practice. [29] proposes a fast iterative shrinkage thresholding algorithm (FISTA) to address this issue. Although FISTA is proposed for convex regularizers, we find that it could also speed up the convergence of the nonconvex regularizers such as x p p , the entropy functions, etc. by a great deal. Naturally, choosing a proper λ is the key to the success of sparse signal recovery. For noiseless signals, the solution can be obtained as λ → 0 [32]. Here we use solve a series of minimization problems characterized by a decreasing Solve the minimization problem characterized by λ k : P λ k hp (x) : min x y − Ax 2 2 + λ k h p (x) .(35) 4: Update λ k+1 = ρ · λ k . 5: if convergence is reached then break. 6: end for To ensure the best performance, ρ is chosen to be 0.9 ≤ ρ < 1. For noisy signals, the optimal λ depends on the noise level and is usually unknown. A fixed λ is tuned on some development set and used in practice. IV. EXPERIMENTAL RESULTS We compare the proposed Shannon entropy function (SEF) minimization and Rényi entropy function (REF) minimization approaches with the state-of-the-art l 1 norm (L1) minimization and l p norm to the p-th power (Lp) minimization approaches on simulated and real datasets. x 2 / x 2 < 10 −3 , the recovery is considered to be a success. The parameters are selected to obtain best performance for each method: for the SEF minimization approach, p = 1.1; for the REF minimization approach, p = 1.1, α = 1.1; for the Lp minimization approach, p = 0.5. FPC method [45] is used to approach the optimal λ = 0. Based on the 100 trials, we compute the success recovery rate for each combination of σ and ρ and plot the PTCs in Fig. 7. A. Simulated sparse signal recovery The PTC is the contour that corresponds to the 0.5 success rate in the domain (σ, ρ) ∈ (0, 1) 2 , it divides the domain into a "success" phase (lower right) and a "failure" phase (upper left). We can see that the proposed SEF minimization and REF minimization approaches generally perform equally well, and they both perform better than the L1 and Lp minimization approaches. We next try to recover the sparse signal x from a noisy measurement vector y. Specifically, we fix S = 100, N = 1000 and increase the number of measurement M . y ∈ R M is generated as follows: Noisy measurements: y = Ax + νw ,(36) where ν > 0 controls the amount of noise added to y, the entries of w are i.i.d Gaussian N (0, 1). We choose ν = 0.05, this creates a measurement y with signal to noises ratio (SNR) around 25 dB. We randomly generate 100 triples of {x, A, w}. The average SNRs of the recovered signalsx are shown in Fig. 8. We can see that the proposed SEF/REF minimization approaches and the Lp minimization approach perform better than the L1 minimization approach. When σ < 0.5), the SEF and REF minimization approaches outperform the Lp minimization approach. B. Real image recovery Real images are considered to be approximately sparse under some proper basis, such as the DCT basis, wavelet basis, etc. Here we compare the recovery performances of the aforementioned sparsity regularization approaches based on varying noiseless and noisy measurements of the 4 real images in Fig. 9: Barbara, Boat, Lena, Peppers. Specifically, in order to reveal the sparse coefficients x of the real images s, we use the sparsity averaging method by [46] to construct an over-complete wavelet basis by concatenating Db1-Db4 [47] as follows: V = 1 2 × [V Db1 V Db2 V Db3 V Db4 ] .(37) It is easy to verify that s = V x, and x = V T s. The sampling matrix U is constructed using the structurally random matrix approach by [48]: U = DF R ,(38) where R is a uniform random permutation matrix that scrambles the signal's sample locations globally while a diagonal matrix of Bernoulli random variables flips the signal's sample signs locally, F is an orthonormal DCT matrix that computes fast transforms, D is a sub-sampling matrix that randomly selects a subset of the rows of the matrix F R. The noiseless measurements y of the image s are obtained as follows: Noiseless measurements: y = DF RV x = U V x = U s .(39) The noisy measurements y are obtained as follows: Noisy measurements: y = U s + νw . The entries of the noise w are generated using i.i.d. Gaussian distribution N (0, 1), ν is chosen to be 0.02 so that the SNR of the measurement vector y is around 30 dB 3 . Take the SEF minimization for example, we have the following recovery problem: min s y − U s 2 2 + λ h p (V T s) .(41) Since the recovery problem is with respect to x, we need to modified Algorithm 1: we also use the proximal regularization of the data fidelity term y − U s 2 2 , the optimization problem in the (t + 1)-th iteration then becomes: min s κ 2 s − ŝ (t) − 1 κ · 2U T (Uŝ (t) − y) 2 2 + λ h p (V T s) ,(42) where κ = 2 for the chosen U in (38). In the (r + 1)-th iteration to minimize (42), let Q (t+1,r) be a diagonal matrix whose diagonal entries are the partial derivative of h p (V T s) with respect to V T s at the solutionŝ (t+1,r) , the optimization problem is as follows: min s s − ŝ (t) − U T (Uŝ (t) − y) 2 2 + λQ (t+1,r) V T s .(43) (43) can be efficiently solved using the alternating split bregman shrinkage algorithm by [49]. Since the real images are only approximately sparse, both the noiseless and noisy recovery experiments are done using a fixed λ. The parameters are tuned to obtain best performance for each approach. For the L1 minimization approach, λ = 0.1; for the SEF minimization approach, p = 1, λ = 5000; for the REF minimization approach, p = 0.9, α = 1.1, λ = 10000; for the Lp minimization approach, p = 0.8, λ = 0.01. The peak signal to noise ratios (PSNR) of the noiseless and noisy recovery experiments are shown in Fig. 10 and 11 respectively. We can see that the proposed SEF and REF entropy function minimization approaches perform equally well, and they give the best performances in terms of PSNR (dB). V. CONCLUSION AND FUTURE WORK In this paper we proposed the generalized Shannon entropy function h p (x) and Rényi entropy function h p,α (x) as the sparsity regularizers for the sparse signal recovery task. Regardless of the values of p, α, the local minimums of the entropy functions occur on the boundaries of the orthants in R N and minimizing them promotes sparsity in the recovered signals. Both h p (x) and h p,α (x) are noncovnex function, the corresponding minimization problems (9,10) can be solved by iteratively minimizing the their first order approximations until convergence. Compared to the popular l 1 norm x 1 and the l p p norm to the p-th power x p p , minimizing the entropy functions encourages the energy of the recovered signal to be concentrated towards a few significant entries. Sparse signal recovery experiments on both the simulated and real data show the proposed entropy function regularizations perform better than the popular x 1 and x p p regularizations. This motivates us to explore theoretical guarantees of the advantage over other approaches in the future by establishing error bounds on the recovered signalx and providing sufficient conditions under which the successful recovery is warranted. Additionally, we would also like to apply the entropy function minimizations to other Compressive Sensing applications such as the SRC [50], [51], RPCA [52]- [55], dictionary learning [56]- [58], etc. APPENDIX A GENERALIZED SOFT SHRINKAGE THRESHOLDING Take P (2) hp (x) in (30) for example, let τ following problem for x i : (t+1,r) i = λ κ ∇h p (|x (t+1,r) i |),x (t) i =x (t) i − 1 κ ∇f (x (t) i ), we have the x i (t) < τ i (t+1, r) x i −3 −2 −1 0 1 2 3 −2 0 2 4 6 8 q (a) τ i (t+1, r) ≤ x i (t) <0 x i −3 −2 −1 0 1 2 3 −1 0 1 2 3 4 q (b) 0 ≤ x i (t) <− τ i (t+1, r) x i −3 −2 −1 0 1 2 3 −1 0 1 2 3 4 q (c) x i (t) ≥ − τ i (t+1, r) x i −3 −2 −min xi 1 2 x i −x (t) i 2 2 + τ (t+1,r) i |x i | .(44) When τ (t+1,r) i ≥ 0, (44) is a convex problem. Its solution is given by applying the shrinkage operator given in (34) oñ x (t) i with the threshold τ (t+1,r) i . When τ (t+1,r) i < 0, (44) is a not necessarily a convex problem. Luckily this is a simple one dimensional problem, its global optimal solution can be still found as follows: 1) Whenx (t) i < τ (t+1,r) i , as is shown in Fig. 12(a): For x i ≥ 0, we have: min xi x i + τ (t+1,r) i −x (t) i 2 − τ (t+1,r) i −x (t) i 2 + (x (t) i ) 2 .(45) Since τ (t+1,r) i −x (t) i > 0, the x i that minimizes (45) is 0. For x i < 0, we have: min xi x i − τ (t+1,r) i −x (t) i 2 − τ (t+1,r) i +x (t) i 2 + (x (t) i ) 2 .(46) Since −τ i < 0, as is shown in Fig. 12(b): For x i ≥ 0, we have (45). Since τ i . It's easy to verify that the minimum of (46) is smaller than the minimum of (45). Hence the global minimum of (44) is obtained by τ i . It's easy to verify that the minimum of (46) is larger than the minimum of (45). Hence the global minimum of (44) is obtained by −τ APPENDIX B MONOTONIC OBJECTIVE FUNCTION MINIMIZATION Take Shannon entropy minimization (SEF) problem P hp (x) in (9) for example, we can have the following: f (x (t+1) ) + h p (x (t+1) ) (a) ≤ f (x (t) ) + x (t+1) −x (t) , ∇f (x (t) ) + κ 2 x (t+1) −x (t) 2 2 + h p (x (t+1) ) (b) ≤ f (x (t) ) + x (t) −x (t) , ∇f (x (t) ) + κ 2 x (t) −x (t) 2 2 + h p (x (t) ) = f (x (t) ) + h p (x (t) ) . The first inequality "(a)" in (47) is obtained using (25). The second inequality "(b)" in (47) holds sincex (t+1) is a solution of P hp (x) in (26). In this case, the objective function f (x) + h p (x) monotonically decreases on the sequence {x (t) , t = 0, 1, · · · }. Similar remark can also be made for the Rényi entropy function minimization problem P hp,α (x) in (10). This work is supported by the National Science Foundation under grants NSF-CCF-1117545, NSF-CCF-1422995 and NSF-EECS-1443936.The authors are with the Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD, 21218 USA (email: [email protected]; [email protected]). Fig. 1 : 1Shannon entropy function h p (x) in the 2-dimensional space: (a) p = 0.5; (b) p = 1; (c) p = 2. Fig. 2 : 2The one-to-one mapping: x ←→x ←→ẍ when p = 0.5. Fig. 3 : 3The one-to-one mapping: X ←→x ←→ẍ when p = 0.5. Fig. 4 : 4The one-to-one mapping: x o ←→x o ←→ẍ o when p = 0.5. K Local minimum: Here we formally introduce the following lemma 5 about the local minimums of the proposed entropy functions: Lemma 5. The local minimums of the entropy functions h p (x) and h p,α (x) only occur at the boundaries of each orthant in R N . also a corresponding local minimumẍ o on the sphere ẍ 1 = 1 inside the orthant O.However, in the following we can show that such aẍ o does not exist on the sphere ẍ 1 = 1 inside the orthant O:1) SEF h p (ẍ) with p > 0: It is strictly concave with respect toẍ, there are no local minimums on the sphereẍ 1 = 1 inside the orthant O, i.e.ẍ o does not exist. 2) REF h p,α (ẍ) with p > 0, α > 0 and α = 1: When α ∈ (0, 1), h p,α (ẍ)is strictly concave with respect toẍ. When α ∈ (1, ∞), h p,α (ẍ) is strictly Schur concave with respect toẍ.ẍ o also does not exist in this case. We can thus see that there are no local minimums inside each orthant. Furthermore, the local minimums of h p (ẍ) and h p,α (ẍ) occur at the boundaries of the sphere ẍ 1 = 1.Hence the local minimums of h p (x) and h p,α (x) only occur at the boundaries of each orthant.O Discussion: In this section we have showed that minimizing the entropy functions(7,8) leads to sparser solutions. Fig. 5 : 5Popular sparsity regularizers in the 2-dimensional space: (a) x 0.5 0.5 ; (b) x 1 . Fig. 5 , 5the 2-dimensional level plots of x 1 and x 0.5 0.5 are shown. Following similar derivation process, we can see that the popular x 1 and x p p with 0 < p < 1 don't have the energy concentration properties. Ignoring the relative constant terms in(28,29) that depend onx (r) , the problems P Algorithm 1 1Sparse signal recovery via entropy function minimization Require: {y, A}, λ, κ, {p, α} 1: Initializex (0) with the solution from l 1 norm minimization; 2: for t = {0, 1, · · · } do 3: Fig. 6 : 6The phase transition curve divide the (ρ, σ) plane into the success phase and the failure phase: (a) l 1 norm; (b) l p norm to the p-th power; (c) The entropy functions h p (x) and h p,α (s).λ sequence. Take the Shannon entropy function h p (x) for example, we have the following:1: Start with a relatively large λ 0 . 2: for k = {0, 1, · · · } do 3: Fig. 7 : 7For the noiseless sparse signal recovery experiments, we fix N = 1000 and vary the sampling ratio σ = M N ∈ [0.05, 0.1, 0.15, · · · , 0.95] and the sparsity ratio ρ = S M ∈ [0.05, 0.1, 0.15, · · · , 0.95], where S is the sparsity of the signal, i.e. the number of nonzero coefficients. For each combination of σ and ρ, we randomly generate 100 pairs of {x, A}: A is a M × N random Gaussian matrix with normalized and centralized rows; the nonzero entries of the sparse signal x ∈ R N are i.i.d. generated according to the Gaussian distribution N (0, 1).Given the measurement vector y = Ax and the sensing matrix A, we try to recover the signal x. The phase transition curves (PTC) of different sparsity regularization approaches in the noiseless case. Fig. 8 : 8The signal-to-noise-ratio (SNR) of the recovered signalx using different sparsity regularization approaches in the noisy case. Fig. 9 : 9The real images used in the recovery experiments: (a) Barbara; (b) Boat; (c) Lena; (d) Peppers. Fig. 10 :Fig. 11 : 1011The peak-signal-to-noise-ratio (PSNR) of the recovered images from "noiseless" measurements using different sparsity regularizationapproaches. (a) Barbara; (b) Boat; (c) Lena; (d) Peppers. The peak-signal-to-noise-ratio (PSNR) of the recovered images from "noisy" measurements using different sparsity regularization approaches. (a) Barbara; (b) Boat; (c) Lena; (d) Peppers. Fig. A12 : A12Generalized soft shrinkage threshoding when the threshold τ < 0. ) is continuous at the point x i = 0. Hence the global minimum of (44) is obtained by x i = τ the x i that minimizes(46) is τ is shown inFig. 12(c): For x i ≥ 0, we have(45). Since τ the x i that minimizes (46) is 0. (44) is continuous at the point x i = 0. 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J Mairal, F Bach, J Ponce, G Sapiro, Proceedings of the 26th Annual International Conference on Machine Learning. the 26th Annual International Conference on Machine LearningJ. Mairal, F. Bach, J. Ponce, and G. Sapiro, "Online dictionary learning for sparse coding," in Proceedings of the 26th Annual International Conference on Machine Learning, 2009, pp. 689-696.

We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations for which the approximate velocity field is pointwise divergence-free. The method proposed here builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889-A913]. We show that with simple modifications of the function spaces in the method of Labeur and Wells that it is possible to formulate a simple method with pointwise divergence-free velocity fields, and which is both momentum conserving and energy stable. Theoretical results are verified by two-and three-dimensional numerical examples and for different orders of polynomial approximation.

10.1007/s10915-018-0671-4

1704.07569

59f6c47e802cf8e02a4ef616beddbad364e7d936

A hybridizable discontinuous Galerkin method for the Navier-Stokes equations with pointwise divergence-free velocity field Sander Rhebergen Department of Applied Mathematics University of Waterloo N2L 3G1WaterlooCanada Garth N Wells Department of Engineering University of Cambridge Trumpington StreetCB2 1PXCambridgeUnited Kingdom A hybridizable discontinuous Galerkin method for the Navier-Stokes equations with pointwise divergence-free velocity field Navier-Stokes equationshybridizeddiscontinuous Galerkinfinite element methodssolenoidal We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations for which the approximate velocity field is pointwise divergence-free. The method proposed here builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889-A913]. We show that with simple modifications of the function spaces in the method of Labeur and Wells that it is possible to formulate a simple method with pointwise divergence-free velocity fields, and which is both momentum conserving and energy stable. Theoretical results are verified by two-and three-dimensional numerical examples and for different orders of polynomial approximation. Introduction Numerous finite element methods for the incompressible Navier-Stokes equations result in approximate velocity fields that are not pointwise divergence-free. This lack of pointwise satisfaction of the continuity equation typically leads to violation of conservation laws beyond just mass conservation, such as conservation of energy. A key issue is that, in the absence of a pointwise solenoidal velocity field, the conservative and advective format of the Navier-Stokes equations are not equivalent. The review paper by John et al. [1] presents cases for the Stokes limit where the lack of pointwise enforcement of the continuity equation can lead to large errors. Elements that are stable (in sense of the infsup condition), but do not enforce the continuity equation pointwise, such as the Taylor-Hood, Crouzeix-Raviart, and MINI elements, can suffer from large errors in the pressure which in turn can pollute the velocity approximation. The concept of 'pressure-robustness' to explain the aforementioned issues is discussed by John et al. [1]. A second issue is when a computed velocity field that is not pointwise divergence-free is used as the advective velocity in a transport solver. The lack of pointwise incompressibility can lead to spurious results and can compromise stability of the transport equation. Discontinuous Galerkin (DG) finite element methods provide a natural framework for handling the advective term in the Navier-Stokes equations, and have been studied extensively in this context, e.g. [2,3,4,5,6,7]. A difficulty in the construction of DG methods for the Navier-Stokes equations is that it is not possible to have both an energy-stable and locally momentum conserving method unless the approximate velocity is exactly divergence-free [3, p. 1068]. To overcome this problem, a post-processing operator was introduced by Cockburn et al. [3]. The operator, which is a slight modification of the Brezzi-Douglas-Marini interpolation operator (see e.g. [8]), when applied to the DG approximate velocity field provides a post-processed approximate velocity that is pointwise divergence-free. Key to this operator is that it can be applied element-wise and is therefore inexpensive to apply. A second issue with DG methods, and a common criticism, is that the number of degrees-of-freedom on a given mesh is considerably larger than for a conforming method. This is especially the case in three dimensions. An approach to generating pointwise divergence-free velocity fields is to use a H(div)-conforming velocity field, in which the normal component of the velocity is continuous across facets, together with a discontinuous pressure field from an appropriate space. Such a velocity space can be constructed by using a H(div)-conforming finite element space, or by enforcing the desired continuity via hybridization [8]. However, construction of H(div)-conforming methods is not straightforward as the tangential components of the viscous stress on cell facets must be appropriately handled. Moreover, for advection dominated flows it is not immediately clear how the advective terms can be appropriately stabilised. Examples of hybridization for the Stokes equations can be found in [9,10,11], and in Navier-Stokes equations [12]. A synthesis of discontinuous Galerkin and hybridized methods has lead to the development of hybridizable Discontinuous Galerkin (HDG) finite element methods [13,14]. These methods were introduced with the purpose of reducing the computational cost of DG methods on a given mesh. This is achieved as follows. Approximate variables and their numerical fluxes across element boundaries are expressed in terms of approximate traces of the variables on facets. By coupling the degrees of freedom on an element only with degrees of freedom of approximate trace variables, element degrees of freedom can be eliminated in favour of facet degrees of freedom only. The result is that the HDG global system of algebraic equations is significantly smaller than those obtained using DG. It has been shown that, after post-processing, solutions obtained by HDG methods may show super-convergence results for elliptic problems (for polyno-mial approximations of order k, the order of accuracy is order k + 2). This property has been exploited also in the context of the Navier-Stokes equations by, e.g., [15,16]. Although their velocity field is not automatically pointwise divergence-free, they introduce a post-processing that not only results in an approximate velocity field that is exactly divergence-free and H(div)-conforming, but also a velocity field that super-converges for low Reynolds number flows. Super-convergence is, however, lost when the flow is convection dominated. In this paper we use the HDG approach to construct a simple discretization of the Navier-Stokes equations in which the approximate velocity field is H(div)conforming and pointwise divergence-free. To achieve this, we first note that unlike many other HDG methods for incompressible flows [15,17,18,12,19,16,20], the HDG methods of Labeur and Wells [21] and Rhebergen and Cockburn [22] contain also facet unknowns for the pressure. The element pressure unknowns play the role of cell-wise Lagrange multiplier to enforce the continuity, whereas the facet pressure unknowns play the role of Lagrange multipliers enforcing continuity of the normal component of the velocity across cell boundaries. It was shown already in [21] that if the polynomial approximation of the element pressure on simplices is one order lower than the polynomial approximation of the element velocity, that the approximate velocity field is exactly divergence-free element-wise. However, they were unable to obtain H(div)-conforming function spaces, because their facet pressure approximation spaces were not rich enough to enforce normal continuity of the velocity across element boundaries exactly. A consequence was that while the method satisfied the continuity equation exactly, it could not satisfy both momentum conservation and energy stability simultaneously. We show in sections 3.2 and 3.3 that if this facet pressure space is chosen rich enough we obtain approximate velocity fields that are H(div)conforming and exactly divergence-free on the whole domain. A consequence is that the HDG method introduced here exactly conserves mass, momentum and is energy stable. The remainder of this paper is organized as follows. Section 2 briefly introduces the Navier-Stokes problem, which is followed by the main result of this paper in section 3; a momentum conserving and energy stable HDG method for the Navier-Stokes equations with pointwise solenoidal velocity field. Numerical results are presented in section 4 and conclusions are drawn in section 5. Incompressible Navier-Stokes problem Let Ω ⊂ R d be a polygonal (d = 2) or polyhedral (d = 3) domain with boundary outward unit normal n, and let the time interval of interest be given by I = (0, t N ]. Given the kinematic viscosity ν ∈ R + and forcing term f : Ω × I → R d , the Navier-Stokes equations for the velocity field u : Ω × I → R d and kinematic pressure field p : Ω × I → R are given by ∂ t u + ∇ · σ = f in Ω × I, (1a) ∇ · u = 0 in Ω × I,(1b) where σ is the momentum flux: σ := σ a + σ d with σ a := u ⊗ u and σ d := pI − ν∇u,(2) and I is the identity tensor and (a ⊗ b) ij = a i b j . We partition the boundary of Ω such that ∂Ω = Γ D ∪ Γ N and Γ D ∩ Γ N = ∅. Given h : Γ N × I → R d and a solenoidal initial velocity field u 0 : Ω → R d , we prescribe the following boundary and initial conditions: u = 0 on Γ D × I, (3a) σ · n − max (u · n, 0) u = h on Γ N × I, (3b) u(x, 0) = u 0 (x) in Ω.(3c) On inflow parts of Γ N (u · n < 0) we impose the total momentum flux, i.e., σ · n = h. On outflow parts of Γ N (u · n ≥ 0), only the diffusive part of the momentum flux is prescribed, i.e., σ d · n = h. Equation (1a) is the conservative form of the Navier-Stokes equation. With satisfaction of the incompressibility constraint, eq. (1b), the momentum equation (1a) can be equivalently expressed as: ∂ t u + (1 − χ)u · ∇u + χ∇ · σ a + ∇ · σ d = f,(4) where χ ∈ [0, 1]. For numerous finite element methods, the approximate velocity field is not pointwise or locally (in a weak sense) solenoidal. In such cases, it can be shown that momentum is conserved if χ = 1, while energy stability can be proven if χ = 1/2. For stabilised finite element methods in which the continuity equation is not satisfied locally, manipulations of the advective term can be applied to achieve momentum conservation [23]. The hybridizable discontinuous Galerkin method of Labeur and Wells [21] is based on a weak formulation of eq. (4) with χ = 1/2. It thereby gives up momentum conservation in favour of energy stability. We will show how the method of Labeur and Wells [21] can be formulated to satisfy the continuity equation pointwise. As a consequence, on the discrete level eq. (4) is unchanging for all values of χ. We will show that this leads to a method that is both energy stable and momentum conserving. A hybridizable discontinuous Galerkin method We present a hybridizable discontinuous Galerkin method for the Navier-Stokes problem for which the approximate velocity field is pointwise divergencefree. Preliminaries Let T := {K} be a triangulation of the domain Ω into non-overlapping simplex cells K. The boundary of a cell is denoted by ∂K and the outward unit normal vector on ∂K by n. Two adjacent cells K + and K − share an interior facet F := ∂K + ∩ ∂K − . A facet of ∂K that lies on the boundary of the domain ∂Ω is called a boundary facet. The sets of interior and boundary facets are denoted by F I and F B , respectively. The set of all facets is denoted by F := F I ∪ F B . Semi-discrete formulation Consider the following finite element spaces: V h := v h ∈ L 2 (T ) d , v h ∈ P k (K) d ∀K ∈ T ,(5a)V h := v h ∈ L 2 (F) d ,v h ∈ P k (F ) d ∀F ∈ F,v h = 0 on Γ D ,(5b)Q h := q h ∈ L 2 (T ), q h ∈ P k−1 (K) ∀K ∈ T ,(5c)Q h := q h ∈ L 2 (F),q h ∈ P k (F ) ∀F ∈ F ,(5d) where P l (D) denotes the space of polynomials of degree l > 0 on a domain D. Note that the spaces V h and Q h are defined on the whole domain T , whereas the spacesV h andQ h are defined only on facets of the triangulation. The spaces V h and Q h are discontinuous across cell boundaries, hence the trace of a function a ∈ V h may be double-valued on cell boundaries. At an interior facet, F , we denote the traces of a ∈ V h by a + and a − . We introduce the jump operator a := a + · n + + a − · n − , where n ± the outward unit normal on ∂K ± . We now state the weak formulation of the proposed method: given a forcing term f ∈ L 2 (Ω) d , boundary condition h ∈ L 2 (Γ N ) d and viscosity ν, find u h ,ū h , p h ,p h ∈ V h ×V h × Q h ×Q h such that: 0 = K K u h · ∇q h dx − K ∂K u h · n q h ds ∀q h ∈ Q h ,(6a)0 = K ∂K u h · nq h ds − ∂Ωū h · nq h ds ∀q h ∈Q h ,(6b) and Ω f · v h dx = Ω ∂ t u h · v h dx − K K σ h : ∇v h dx + K ∂Kσ h : (v h ⊗ n) ds + K ∂K ν (ū h − u h ) ⊗ n : ∇v h ds ∀v h ∈ V h , (6c) Γ N h·v h ds = K ∂Kσ h : (v h ⊗ n) ds− Γ N (1 − λ) (ū h · n)ū h ·v h ds ∀v h ∈V h ,(6d) whereσ h :=σ a,h +σ d,h is the 'numerical flux' on cell facets. The advective part of the numerical flux is given by: σ a,h := σ a + (ū h − u h ) ⊗ λu h ,(7) where λ is an indicator function that takes on a value of unity on inflow cell boundaries (where u h · n < 0) and a value of zero on outflow cell facets (where u h · n ≥ 0). This definition of the numerical flux provides upwinding of the advective component of the flux. The diffusive part of the numerical flux is defined asσ d,h :=p h I − ν∇u h − να h K (ū h − u h ) ⊗ n,(8) where α > 0 is a penalty parameter as is typical of Nitsche and interior penalty methods. It is proven in [24,25] that α needs to be sufficiently large to ensure stability. A key feature of this formulation, and what distinguishes it from standard discontinuous Galerkin methods, is that functions on cells (functions in V h and Q h ) are not coupled across facets directly via the numerical flux. Rather, fields on neighbouring cells are coupled via the facet functionsū h andp h . The fields u h and p h can therefore be eliminated locally via static condensation, resulting in a global system of equations in terms of the facet functions only. This substantially reduces the size of the global systems compared to a standard discontinuous Galerkin method on the same mesh. The weak formulation presented here is the weak formulation of Labeur and Wells [21] with conservative form of the advection term (χ = 1 in eq. (4)). The key difference is that we have been more prescriptive on the relationships between the finite element spaces in eq. (5), and this leads to some appealing properties, as we will prove. In particular, the spaces in eq. (5) are such that: for u h ∈ P k (K) d , ∇·u h ∈ P k−1 (K) and u h ·n ∈ P k (F ); and forū h ∈ P k (F ) d , u h · n ∈ P k (F ). Furthermore, the function spaces have been chosen such that the resulting method is inf-sup stable, see [25]. The resulting weak formulation can be shown to be equivalent to a weak formulation in which the approximate velocity field lies in the Brezzi-Douglas-Marini (BDM) finite element space. Hybridization of other H(div) conforming finite element spaces, see e.g. [8], are also possible. Proposition 1 (mass conservation). If u h ∈ V h andū h ∈V h satisfy eq. (6), with V h andV h defined in eq. (5), then ∇ · u h = 0 ∀x ∈ K, ∀K ∈ T ,(9) and u h = 0 ∀x ∈ F, ∀F ∈ F I , (10a) u h · n =ū h · n ∀x ∈ F, ∀F ∈ F B . (10b) Proof. Applying integration-by-parts to eq. (6a): 0 = K q h ∇ · u h dx ∀q h ∈ P k−1 (K), ∀K ∈ T .(11) Since q h , ∇ · u h ∈ P k−1 (K), pointwise satisfaction of the continuity equation, eq. (9), follows. It follows from eq. (6b) that: 0 = F ∈F I F u h q h ds + F ∈F B F (u h −ū h ) · nq h ds ∀q h ∈Q h .(12) Sinceq h , u h · n,ū h · n ∈ P k (F ), eq. (10) follows. Proposition 1 is a stronger statement of mass conservation than in Labeur and Wells [21,Proposition 4.2], in which mass conservation was proved locally (cell-wise) in an integral sense only. Under certain conditions, implementations in [21] satisfy eq. (9), but not eq. (10). We will show that this difference is critical for the new formulation in this work as it allows simultaneous satisfaction of momentum conservation and energy stability. We next show momentum conservation for the semi-discrete weak formulation in terms of the numerical flux. Proposition 2 (momentum conservation). Let u h ,ū h , p h ,p h ∈ V h ×V h × Q h × Q h satisfy eq. (6). Then, d dt K u h dx = K f dx − ∂Kσ h n ds ∀K ∈ T .(13)Furthermore, if Γ D = ∅, d dt Ω u h dx = Ω f dx − ∂Ω (1 − λ)(ū h · n)ū h ds − ∂Ω h ds.(14) Proof. In eq. (6c), set v h = e j on K, where e j is a canonical unit basis vector, and set v h = 0 on T \K in eq. (6c): d dt K u h · e j dx + ∂K (σ h · n) · e j ds = K f · e j dx,(15) which proves eq. (13). Equation (14) follows immediately by setting v h = e j in eq. (6c),v h = −e j in eq. (6d) and summing the two results. We next prove that the method is also globally energy stable. Proposition 3 (global energy stability). If u h ,ū h , p h ,p h ∈ V h ×V h × Q h ×Q h satisfy eq. (6) , for homogeneous boundary conditions, f = 0 and for a suitably large α: d dt K K |u h | 2 dx ≤ 0.(16) Proof. Setting q h = −p h ,q h = −p h , v h = u h andv h = −ū h in eqs. (6a) to (6d) and inserting the expressions for the numerical fluxes (eqs. (2), (7) and (8)), and summing: K 1 2 K ∂ t |u h | 2 dx + K 1 2 ∂K (u h · n)|u h | 2 ds − K 1 2 ∂K (u h · n)|ū h | 2 ds + K 1 2 ∂K |u h · n||u h −ū h | 2 ds + K K ν|∇u h | 2 dx + K ∂K να h K |ū h − u h | 2 ds + 2 K ∂K ν (∇u h · n) · (ū h − u h ) ds + Γ N (1 − λ)(ū h · n)|ū h | 2 ds − K K (u h ⊗ u h ) : ∇u h dx = 0,(17) where we have used that λu h ·n = u h · n − |u h · n| /2, and applied integrationby-parts to the pressure gradient terms. Sinceū h is single-valued on facets, the normal component of u h is continuous across facets andū h · n = u h · n on the domain boundary (see proposition 1), the third integral on the left-hand side of eq. (17) can be simplified: − K 1 2 ∂K (u h · n)|ū h | 2 ds = − 1 2 Γ N (ū h · n)|ū h | 2 ds.(18) We consider now the last term on the left-hand side of eq. (17). On each cell K it holds that −u h ⊗ u h : ∇u h = (∇ · u h )(u h · u h )/2 − ∇ · ((u h ⊗ u h ) · u h )/2 = −∇ · ((u h ⊗ u h ) · u h )/2, since ∇ · u h = 0 (by proposition 1). It follows that − K K (u h ⊗ u h ) : ∇u h dx = − 1 2 K ∂K (u h · n)|u h | 2 ds.(19) Combining eqs. (17) to (19), 1 2 K K ∂ t |u h | 2 dx = − 1 2 K ∂K |u h · n||u h −ū h | 2 ds − K K ν|∇u h | 2 dx − K ∂K να h K |ū h − u h | 2 ds − 2 K ∂K ν (∇u h n) · (ū h − u h ) ds − 1 2 Γ N |ū h · n||ū h | 2 ds,(20) where we have used that Γ N (1−λ)(ū h ·n)|ū h | 2 ds− 1 2 Γ N (ū h · n)|ū h | 2 ds = 1 2 Γ N |ū h · n||ū h | 2 ds.(21) It can be proven that there exists an α > 0, independent of h K , such that K ∂K να h K |ū h − u h | 2 ds ≥ 2 K ∂K ν (∇u h · n) · (ū h − u h ) ds ,(22) (see [24,Lemma 5.2] and [25,Lemma 2]). Therefore, the right-hand side of eq. (20) is non-positive, proving eq. (16). The keys to be able to prove global energy stability for this conservative form of the Navier-Stokes equations are: (a) the pointwise solenoidal velocity field; and (b) continuity of the normal component of the velocity field across facets. The latter point was not fulfilled by the method in [21]. A fully-discrete weak formulation We now consider a fully-discrete formulation. We partition the time interval I into an ordered series of time levels 0 = t 0 < t 1 < · · · < t N . The difference between each time level is denoted by ∆t n = t n+1 − t n . To discretize in time, we consider the θ-method and denote midpoint values of a function y by y n+θ := (1 − θ)y n + θy n+1 . Following Labeur and Wells [21], the convective velocity will be evaluated at the current time t n , thereby linearizing the problem, i.e.: The time-discrete counterpart of eq. (6) is: given u n h ,ū n h , p n h ,p n h ∈ V h ×V h × Q h ×Q h at time t n , the forcing term f n+θ ∈ L 2 (Ω) d , the boundary condition h n+θ ∈ L 2 (Γ N ) d , and the viscosity ν, find u n+1 σ n+θ h = σ n+θ a,h + σ n+θ d,h where σ n+θ a,h = u n+θ h ⊗ u n h ,(23)h ,ū n+1 h , p n+1 h ,p n+1 h ∈ V h ×V h × Q h ×Q h such that mass conservation, 0 = K K u n+1 h · ∇q h dx − K ∂K u n+1 h · n q h ds,(25a)0 = K ∂K u n+1 h · nq h ds − ∂Ωū n+1 h · nq h ds,(25b) and momentum conservation, Ω f n+θ · v h dx = Ω u n+1 h − u n h ∆t n · v h dx − K K σ n+θ h : ∇v h dx + K ∂Kσ n+θ h : v h ⊗ n ds + K ∂K ν ū n+θ h − u n+θ h ⊗ n : ∇v h ds, (25c) Γ N h n+θ ·v h ds = K ∂Kσ n+θ h :v h ⊗ n ds − Γ N (1 − λ) (ū n h · n)ū n+θ h ·v h ds, (25d) are satisfied for all v h ,v h , q h ,q h ∈ V h ×V h × Q h ×Q h . Here λ is evaluated using the known velocity field at time t n . In section 3.2 we proved that the semi-discrete formulation eq. (6) is momentum conserving, energy stable and exactly mass conserving when using the function spaces given by eq. (5). We show next that the fully-discrete formulation given by eq. (25) inherits these properties. Proposition 4 (fully-discrete mass conservation). If u n+1 h ∈ V h andū n+1 h ∈V h satisfy eq. (25), then ∇ · u n+1 h = 0 ∀x ∈ K, ∀K ∈ T ,(26) and u n+1 h = 0 ∀x ∈ F, ∀F ∈ F I , (27a) u n+1 h · n =ū n+1 h · n ∀x ∈ F, ∀F ∈ F B . (27b) Proof. The proof is similar to that of proposition 1 and therefore omitted. Proposition 5 (fully-discrete momentum conservation). If u n h ,ū n h , p n h ,p n h ∈ V h ×V h × Q h ×Q h and u n+1 h ,ū n+1 h , p n+1 h ,p n+1 h ∈ V h ×V h × Q h ×Q h satisfy eq. (25), then K u n+1 h − u n h ∆t n dx = K f n+θ dx − ∂Kσ n+θ h n ds ∀K ∈ T . (28) Furthermore, if Γ D = ∅, K K u n+1 h − u n h ∆t n dx = K K f n+θ dx − ∂Ω (1 − λ)(ū n h · n)ū n+θ h ds − ∂Ω h n+θ ds. (29) Proof. The proof is similar to that of proposition 2 and therefore omitted. Proposition 6 (fully-discrete energy stability). If u n h ,ū n h , p n h ,p n h ∈ V h ×V h × Q h ×Q h and u n+1 h ,ū n+1 h , p n+1 h ,p n+1 h ∈ V h ×V h × Q h ×Q h satisfy eq. (25) , then with homogeneous boundary conditions, no forcing terms, for suitably large α, and θ ≥ 1/2, K K u n+1 h 2 dx ≤ K K |u n h | 2 dx. (30) Proof. Setting q h = −θp n+θ h ,q h = −θp n+θ h , v h = u n+θ h andv h = −ū n+θ h , in eqs. (25a) to (25d), adding the results, using the expressions for the diffusive fluxes, given by eqs. (2) and (8), partial integration of the pressure gradient terms and using that ∇ · u n h = 0 by proposition 4, we obtain, using the same steps as in the proof of proposition 3, Ω u n+1 h − u n h ∆t n · u n+θ h dx + K 1 2 ∂K |u n h · n| u n+θ h −ū n+θ h 2 ds + K K ν ∇u n+θ h 2 dx + K ∂K να h K ū n+θ h − u n+θ h 2 ds + 2 K ∂K ν ∇u n+θ h · n ū n+θ h − u n+θ h ds + 1 2 Γ N |ū n h · n| ū n+θ h 2 ds = 0. (31) The first term on the left-hand side of eq. (31) can be reformulated as Ω u n+1 h − u n h ∆t n · u n+θ h dx = θ − 1 2 Ω u n+1 h − u n h 2 ∆t n dx + 1 2 Ω u n+1 h 2 ∆t n dx − 1 2 Ω u n h 2 ∆t n dx. (32) Inserting this expression into eq. (31): 1 2 Ω u n+1 h 2 ∆t n dx − 1 2 Ω u n h 2 ∆t n dx = − θ − 1 2 Ω u n+1 h − u n h 2 ∆t n dx − K 1 2 ∂K |u n h · n| u n+θ h −ū n+θ h 2 ds − 2 K ∂K ν ∇u n+θ h · n ū n+θ h − u n+θ h ds − K ∂K να h K ū n+θ h − u n+θ h 2 ds − 1 2 Γ N |ū n h · n| ū n+θ h 2 ds − K K ν ∇u n+θ h 2 dx. (33) As in proposition 3, there exists an α > 0, independent of h K , such that the right hand side of eq. (33) is non-positive. The result follows. Numerical examples We now demonstrate the performance of the method for a selection of numerical examples, paying close attention to mass and momentum conservation, In the implementation we apply cell-wise static condensation such that only the degrees-of-freedom associated with the facet spaces appear in the global system. Compared to standard discontinuous Galerkin methods, this significantly reduces the size of the global system. It is possible to reduce the size of the global system even further by taking into account that velocity is continuous in the normal direction and therefore we could remove the normal components fromV h , see [12]. In the current paper, however, we only apply static condensation. The examples have been implemented using the NGSolve finite element library [26]. All examples use unstructured simplicial meshes. Kovasznay flow We consider the steady, two-dimensional analytical solution of the Navier-Stokes equations from Kovasznay [27] on a domain Ω = (−0.5, 1) × (−0.5, 1.5). For a Reynolds number Re, let the viscosity be given by ν = 1/Re. The solution to the Kovasznay problem is: u x = 1 − e λx1 cos(2πx 2 ),(34a)u y = λ 2π e λx1 sin(2πx 2 ),(34b)p = 1 2 1 − e 2λx1 + C,(34c) where C is an arbitrary constant, and where λ = Re 2 − Re 2 4 + 4π 2 1/2(35) We choose C such that the mean pressure on Ω is zero. The Kovasznay flow solution in eq. (34) is used to set Dirichlet boundary conditions for the velocity on ∂Ω. The L 2 -error and rates of convergence are presented in table 1 for Re = 40 using a series of refined meshes. Optimal rates of convergence are observed for both the velocity field (order k + 1) and pressure field (order k). The divergence of the approximate velocity field is of machine precision in all cases. Position-dependent Coriolis force We now consider the test case from [28,Section 3.2]. In particular, we consider on the unit square (0, 1) × (0, 1) the steady Navier-Stokes equations augmented with a position-dependent Coriolis force: ∇ · σ + 2C × u = 0 and ∇ · u = 0, where we set 2C × u = −2x 2 (−u 2 , u 1 ) and ν = 1. On boundaries we set u = (1, 0). The exact solution to this problem is given by p = x 2 2 − 1/3 and u = (1, 0). ν = 0.001, k = 2 ν = 1, k = 2 Cells u h − u ∇ · u h p h − p rate u h − u ∇ · u h p h − p It was shown in [28] that the Scott-Vogelius finite element, in which the velocity is approximated in divergence-free function spaces, is able to produce the exact velocity field while the velocity computed using a Taylor-Hood finite element method is polluted by the pressure error, in part due to the approximate velocity field not being exactly divergence-free. Furthermore, it is shown in [28] that as ν → 0, the velocity error increases for the Taylor-Hood finite element method. In table 2 we show the results obtained using the HDG method presented in section 3 for k = 2. Table 2 shows the computed error in the L 2 norm for the velocity, pressure and divergence errors. Errors in the velocity and velocity divergence are of machine precision, regardless of ν. The HDG method therefore obtains the same quality of solution as produced using the Scott-Vogelius finite element in [28]. We do not consider the k = 3 case because for this discretization the pressure is approximated by quadratic polynomials and so the pressure error is also of machine precision. Transient higher-order potential flow In this test, taken from [29, Section 6.6], we solve the time dependent Navier-Stokes equations eq. (1) on the domain Ω = [−1, 1] 2 . This test case studies the time-dependent exact velocity u(t) = min(t, 1)∇χ where χ is a smooth harmonic potential given by χ = x 3 1 x 2 − x 3 2 x 1 . The pressure gradient then satisfies ∇p = −∇|u| 2 /2 − ∂ t min(t, 1)∇χ . We impose the exact velocity solution as Dirichlet boundary condition on all of ∂Ω. For the simulations we used a grid with 2048 cells, set the time step equal to ∆t = 0.01 and compute the solution on the time interval [0, 2]. Figure 1 shows the velocity and pressure errors as a function of time. We used both k = 2 and k = 3, and consider ν = 1/500 and ν = 1/2000. We observe that the error in pressure and velocity is more or less the same regardless of ν. Over the computational time interval, using k = 2 or k = 3 on a mesh with 2048 cells, for either ν = 1/500 and ν = 1/2000, the L 2 -norm of the divergence reaches 1.4 × 10 −10 in one point but is otherwise always of the order 10 −11 . The momentum balance, in absolute value, never exceeds 3.4 × 10 −12 . Two-dimensional flow past a circular obstacle In this test case we consider flow past a circular obstacle (see e.g. [12,30]). The domain is a rectangular channel, [0, 2.2] × [0, 0.41], with a circular obstacle of radius r = 0.05 centred at (0.2, 0.2). On the inflow boundary (x 1 = 0) we prescribe the x 1 -component of the velocity to be u 1 = 6x 2 (0.41−x 2 )/0.41 2 . The x 2 -component of the velocity is prescribed as u 2 = 0. Homogeneous Dirichlet boundary conditions are applied on the walls (x 2 = 0 and x 2 = 0.41), and on the obstacle. On the outflow boundary (x 1 = 2.2) we prescribe σ d · n = 0. The viscosity is set as ν = 10 −3 . We choose k = 3 and set ∆t = 5 × 10 −5 so that the spatial discretization error dominates the temporal discretization error. For the initial condition, we impose the steady Stokes solution of this problem. The mesh of the domain has 6784 cells and we consider the time interval [0, 5]. At each time step we compute the drag and lift coefficients, which are defined as C D = − 1 r Γc (σ d · n) · e 1 ds, C L = − 1 r Γc (σ d · n) · e 2 ds,(36) where e 1 and e 2 are unit vectors in the x 1 and x 2 directions, respectively, and Γ C is the surface of the circular object. We compute a maximum drag coefficient of C D = 3.23232 and minimum drag coefficient of C D = 3.16583. The maximum and minimum lift coefficients we compute are, respectively, C L = 0.98251 and C L = −1.02246. These are comparable to those found in literature [12,30]. The velocity magnitude at t = 5 is shown fig. 2. Three-dimensional flow past a cylinder In this test case we consider three-dimensional flow past a cylinder (see e.g. [12,30] 4 . The x 2 -and x 3 -components of the velocity are prescribed as u 2 = 0 and u 3 = 0. We impose homogeneous Dirichlet boundary conditions on the walls (x 2 = 0, x 2 = 0.41, x 3 = 0 and x 3 = 0.41) and on the cylinder. On the outflow boundary (x 1 = 2.5) we prescribe σ d · n = 0. The viscosity is set as ν = 10 −3 . We choose k = 3 and set ∆t = 5 × 10 −4 so that the spatial discretization error dominates the temporal discretization error. The initial condition is the Stokes solution to this problem. The mesh has 4091 cells and we compute on the time interval [0,8]. At each time step we compute the drag and lift coefficients, defined by eq. (36), where r = 0.41r cyl and Γ C is the surface of the cylinder. We compute maximum drag and lift coefficients of C D = 2.98815 and C L = 0.00348, respectively. Compared to Schäfer et al. [30], in which the maximum drag and lift coefficients lie in the intervals C D ∈ [3.2000, 3.3000] and C L ∈ [0.0020, 0.0040], we slightly under-predict the drag coefficient, but the lift coefficient lies within the same interval. Figure 3 shows the velocity magnitude at t = 4. Conclusions We have introduced a formulation of a hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations that computes velocity fields that are pointwise divergence-free. The construction of solenoidal velocity fields does not require post-processing or the use of finite dimensional spaces of divergence-free functions. The pointwise satisfaction of the continuity equation and the continuity of the normal component of the velocity field across cell facets allows us to prove that the method conserves momentum locally (cell-wise) and is energy stable. This is in contrast with the closely related method in Labeur and Wells [21] which can satisfy local momentum conservation or global energy stability, but not both simultaneously. The analysis that we present is supported by a range of numerical examples in two and three dimensions. and energy stability. For all stationary examples considered, exact solutions are known. For the stationary examples we use a fixed-point iteration with stopping criterion |e i+1 p − e i p |/(e i+1 p + e i p ) ≤ TOL, where e i p is the pressure error in the L 2 norm at the ith iterate, and TOL is a given tolerance that we set to 10 −4 . All unsteady examples use θ = 1. In all examples we set the penalty parameter to be α = 6k 2 . Figure 1 : 1Velocity and pressure errors for the transient higher-order potential flow test case. Approximations were obtained using k = 2 and k = 3 on a mesh with 2048 cells. Figure 2 : 2Two-dimensional flow past a cylinder test case: velocity magnitude past a twodimensional circular object in a channel at t = 5. Approximations were obtained using k = 3 on a mesh with 6784 cells. ) with a time dependent inflow velocity. The domain is a cuboid shaped channel [0, 2.5] × [0, 0.41] × [0, 0.41] with a cylinder of radius r cyl = 0.05 around the x 3 -axis centred at (x 1 , x 2 ) = (0.5, 0.2). On the inflow boundary (x 1 = 0) we prescribe the x 1 -component of the velocity to be u 1 = 36 sin(πt/8)x 2 x 3 (0.41− x 2 )(0.41 − x 3 )/0.41 Figure 3 : 3Three-dimensional flow past a cylinder test: slice through a 3D channel showing the 3D velocity magnitude past a cylinder in a channel at t = 4. Approximations were obtained using k = 3 on a mesh with 4091 cells. Table 1 : 1Computed velocity, pressure and divergence errors in the L 2 norm for the HDG method applied to the Kovasznay problem.k = 2 Cells u h − u rate p h − p rate ∇ · u h 64 1.8e-2 - 1.6e-2 - 3.8e-14 256 2.2e-3 3.0 4.0e-3 2.0 6.7e-14 1024 2.8e-4 3.0 9.8e-4 2.0 1.3e-13 4096 3.5e-5 3.0 2.4e-4 2.0 2.5e-13 k = 3 Cells u h − u rate p h − p rate ∇ · u h 64 1.4e-3 - 2.0e-3 - 1.9e-13 256 9.4e-5 3.9 2.0e-4 3.3 6.1e-13 1024 5.8e-6 4.0 2.3e-5 3.1 7.8e-13 4096 3.6e-7 4.0 2.8e-6 3.1 1.6e-12 Table 2 : 2Computed errors in the L 2 norm for the HDG method with the position-dependent Coriolis forcing term with different viscosity values. AcknowledgementsSR gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada through the Discovery Grant program (RGPIN-05606-2015) and the Discovery Accelerator Supplement (RGPAS-478018-2015).References On the divergence constraint in mixed finite element methods for incompressible flows. 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Hirschel (Ed.), Flow Simulation with High-Performance Computers II, 547-566, 1996.

In metaphysics, there are a number of distinct but related questions about the existence of "further facts"-facts that are contingent relative to the physical structure of the universe. These include further facts about qualia, personal identity, and time. In this article I provide a sequence of examples involving computer simulations, ranging from one in which the protagonist can clearly conclude such further facts exist to one that describes our own condition. This raises the question of where along the sequence (if at all) the protagonist stops being able to soundly conclude that further facts exist.Keywords: metaphysics, philosophy of mind, epistemology.Case A. Fonda has just attended an inspiring department colloquium. On her way out, she absentmindedly takes a wrong turn and wanders into a computer lab. She approaches one of the computers. * This paper will appear in Erkenntnis.

10.1007/s10670-018-9979-6

1802.01161

e1aa6c8a28231d22898498ba77545c7accf99357

A Puzzle about Further Facts * 4 Feb 2018 Vincent Conitzer Duke University A Puzzle about Further Facts * 4 Feb 2018 In metaphysics, there are a number of distinct but related questions about the existence of "further facts"-facts that are contingent relative to the physical structure of the universe. These include further facts about qualia, personal identity, and time. In this article I provide a sequence of examples involving computer simulations, ranging from one in which the protagonist can clearly conclude such further facts exist to one that describes our own condition. This raises the question of where along the sequence (if at all) the protagonist stops being able to soundly conclude that further facts exist.Keywords: metaphysics, philosophy of mind, epistemology.Case A. Fonda has just attended an inspiring department colloquium. On her way out, she absentmindedly takes a wrong turn and wanders into a computer lab. She approaches one of the computers. * This paper will appear in Erkenntnis. the literature on metaphysics and philosophy of mind. The first example of a further fact in the case is related to qualia and the possibility of inverted spectra. 2 It especially relates to strong versions of the inverted spectrum scenario where qualia do not supervene on the physical, i.e., where two microphysically identical twins nevertheless have inverted spectra. Stated otherwise, a variant of the question for which the case above is arguably especially relevant is: did the laws of our universe (and its initial conditions 3 ) necessitate that when I see red things they phenomenally appear the way they do, or would they have allowed for them to appear the way blue things do now? Whether the phenomenal nature of color perception is contingent (holding fixed the physical laws of our universe and its initial conditions) has been the subject of much philosophical debate. Fonda's question is clearly related, though it would seem that in her case she is entirely right to conclude that the way things in the simulated universe appear to her is contingent, even holding fixed the physical laws and initial conditions of the simulated universe. The analogy between Fonda's questions in Case A and the standard ones from the metaphysics and philosophy of mind literature breaks down at some points, and we will explore this in what follows. Before we do so, let us consider the other example, Fonda's question of why it is Alpha, and not some other agent, whose perspective is displayed. This is related to questions in metaphysics about personal identity and the self. Most closely, it is related to the question of whether "I could have been someone else," and the closely related question of whether it is contingent that this perspective is the present one. 4 Another type of further fact that could be included in the discussion is that of further facts about time. Fonda might ask herself why this point in the simulation's timeline is being displayed to her right now, as opposed to another point in simulated time. Also, she might ask herself why the 3 simulation runs at the rate that it does, as opposed to (say) twice as fast. 5 Here, it may not be clear that things could have been different. For example, perhaps the simulation simply started running when the program was first executed and it is running at the fastest possible rate on the hardware provided. On the other hand, the entire simulation may have been precomputed from beginning to end, so that a type of block universe is already stored in computer memory and Fonda is just watching a replay of some part of it. In this case, there must indeed be some further code governing which temporal part is replayed and at which rate the replay runs. While I believe that the cases presented here may indeed provide some helpful insights for the metaphysics of time, things are clearer for the other types of further facts. Hence, I will avoid discussion of time in what follows. We are now ready to introduce the next case. Case B. This case proceeds similarly to Case A, though with an important difference. Instead of plain monitors, the computer lab now has sophisticated virtual reality (VR) headsets. Fonda puts on one of these that shows her the perspective of Alpha. The VR system is so remarkably good that Fonda spends a long time using it and becomes completely engrossed-so engrossed that she completely forgets the world outside the simulation, her own identity in it, and, we may suppose, even basic facts such as what the color of grass is, or even the existence of such a thing as grass at all (assuming there is no such thing in the simulated universe). All that is left to her is the simulation, displayed from Alpha's perspective-so that presumably she feels rather identified with Alpha. Again, she quickly learns the laws of physics in this universe. Then she has the following thought. There must be further facts to this universe, namely the ones concerning my own perspective in it. 4 The (simulated) sky could have appeared to me in the color in which the (simulated) ground appears to me now, without the fundamental laws of the universe changing. Moreover, there must be further facts regarding my identity-why is it this perspective that appears (to me 6 ) and not that of some other agent? Finally, Case C returns to day-to-day life. Case C. In this version, Fonda does not walk into any computer lab; she just walks outside and experiences the world as we normally do. She knows the laws of physics well from her undergraduate studies and nothing in the world seems mysterious to her (unresolved questions in physics aside). Then, she has a thought just like the one in Case B, but now about our own familiar universe. Why does the sky appear to me the way it does? Why is it this perspective that appears (to me)? There must be further facts beyond the laws of physics and any initial conditions. In each case, Fonda reaches the conclusion that there are further facts to the universe at hand. In which of these cases is her conclusion justified? It is worth emphasizing that the question is not whether there are actually further facts, but rather whether the reasoning that leads her to conclude this is sound. In the same way as it is possible to give a wrong proof for a (true) theorem, in principle Fonda's belief in further facts can fail to be justified even if there are in fact further facts in these cases. Now, there are four possibilities: 1. Her reasoning is not sound in Case A. 5 2. Her reasoning is sound in Case A, but not in Case B. 3. Her reasoning is sound in Case B, but not in Case C. 4. Her reasoning is sound in Case C. It is straightforward to check that these four options are exhaustive in the sense that at least one of them must hold. 7 One may of course choose Option 4, having been convinced by the sequence of cases (or already believing prior to picking up this paper) that we are justified in concluding that there are further facts in our own world. I have little to say that is new about advantages and disadvantages of such a view, so the remainder of the paper is devoted to the following question. If we believe that Option 4 is false, then which of the first three options is most plausible? In what follows, I argue that Option 2 is the most appealing of the three, though attempts to decisively establish it as correct lead us to variants of known arguments about qualia and personal identity. The exercise does cast a new light on these arguments, in particular clarifying some of their epistemological aspects. It also demonstrates commonalities among various types of putative further facts that I believe have not been sufficiently appreciated in the literature. Option 1: Fonda's reasoning in Case A is not sound This, to me, seems the least appealing option of the three, so I will not spend much space on it. Fonda's reasoning seems entirely sound to me: if the simulation is displaying on the monitor, there must in fact be some code that governs this display. It would certainly be possible to write code for the simulation without any instructions to display anything on the screen, but then the simulation 6 would just run silently 8 on the machine without any output. This is in fact a mistake programmers make on occasion: they write the code (for, say, computing 2 n as a function of n) correctly but forget to write instructions to display the result to (say) the screen. Such a mistake is typically easily corrected by adding a line to the code. 9 One could argue for Option 1 by arguing that in fact, there are no further facts in Case A. Such an argument might proceed as follows. We should distinguish between two claims. One is that the qualia associated with seeing an object supervene (only) on facts about the physical properties of that object. Let us call these the "narrow" physical facts. The other is that they supervene on facts about the physical properties of the object being viewed, those of the observer viewing the object, and those of anything else mediating the viewing. Let us call these, collectively, the "broad" physical facts. The former claim is untenable, for example because the qualia are different when the observer is color blind or the air between the object and the observer is hazy. It is the latter claim that is of interest. And in Case A, the broad physical facts include facts about the code governing the display, the monitor itself, Fonda's eyes and brain, etc. Hence, there is no reason to think that there are any further facts in Case A. However, this argument relies on misunderstanding the sense in which the phrase "further fact" is being used here, which is quite modest. The point is that Fonda's experience, or even just what appears on the monitor, is not fully determined by the physical-in the sense of the simulated physics-facts of the simulated universe. These include facts about the simulated objects, Alpha, and anything mediating the viewing within the simulated universe. They do not include physical (in the common sense) facts about the monitor, Fonda's body, and the space between them. They also do not include facts about the additional code governing the display. Again, the relevant physics is the physics of the simulation, not the physics of the broader world. 10 It is clear that there are further facts in Case A in the modest sense of being contingent relative to just the facts about the simulated physics, and this modest sense is the one of interest in this paper. Why this interpretation is the one of interest is made clear by considering the analogous move of using an immodest interpretation in Case C. This move would result in arguments such as the following. Even if experiences were had by souls outside the world through some process mediating between brains in the world and souls outside it, then we should simply consider a broader physics that includes the souls and the mediating process. By doing so the experiences again supervene on the broader physical facts, so there is still no evidence for further facts. Clearly this argument is unsatisfactory; in arguing against further facts, we mean to argue against the existence of things such as extraworldly souls, not to accommodate them through a technical maneuver. While all this may seem rather obvious, it is important to keep straight, especially in Case B, where the relevant physics is still the physics of the simulation-which, in that case, is the only physics of which Fonda is aware. I will now skip to Option 3 before returning to Option 2. 8 Option 3: Fonda's reasoning is sound in Case B but not in Case C While this option seems more appealing to me than Option 1, it still seems difficult to argue for it. Key to this difficulty, of course, is that by assumption, Fonda has forgotten everything about the outside world in Case B. If she retains some memory of the outside world, the case will reduce to one that is not substantively different from Case A. Is Case B substantively different from Case C? Of course: in Case B the universe under consideration is a simulation in a larger universe. But are the two cases substantively different in terms of Fonda's epistemic situation? This is what seems difficult to argue. Whether we have reason to believe that we are not a brain in a vat or (in) a computer simulation is a topic that has been explored at length in the literature. Bostrom (2003) has argued that there is a large probability that we are in fact in a computer simulation, under some assumptions including that posthuman civilizations are likely to be reached and likely to run a large number of such simulations. In contrast, Markosian (2014) has argued that all the evidence speaks in favor of the external world being real; evidence in favor of being a brain in a vat would be exemplified by a major glitch in the simulated environment. For our purposes, it is not necessary to resolve this debate. What matters is not whether Fonda is justified in believing that the world around her is real (in the sense of not being a simulation) in either Case B or C, but rather whether there is a substantive difference between these two cases in terms of her epistemic situation. Let us simply assume that there are no glitches in Case B. Furthermore, at least in principle, the simulation in Case B could provide Fonda with a very rich experience. As Case B has been described so far, Fonda is not able to take actions in it; she is just observing. This indeed constitutes a difference between what Fonda observes in Cases B and C. It is not immediately clear to me whether and how this particular difference is relevant to the soundness of her argument for further facts. In any case, we can easily modify the example to give her some control over Alpha's actions (with perhaps other human beings who similarly wandered into labs controlling the other agents in the simulation). Perhaps more interestingly, she may wonder about the place of her own thoughts in the simulated universe. In our own world, we have reasons to believe that our thoughts are generated by our brains. If a similar account does not seem reasonable in the simulated universe-for example, because there does not seem to be any physical structure in it capable of generating these thoughts-she may conclude that such a structure must exist somewhere outside of her observable universe, and from there it is a short step to conclude the existence of further facts. Then again, we could modify the example so that there appear to be brains inside the simulated agents; we could even go so far as to imagine that, unbeknownst to Fonda, her brain is being scanned while she is standing in the lab, and what goes on in it is then reflected in Alpha's simulated brain. 11 (A similar idea is described by Chalmers (2005).) Overall, it seems difficult to draw a sharp distinction between Fonda's epistemic situation in Cases B and C that cannot be addressed with a simple modification of the cases. 12 Of course, this is precisely the point of Case B, to make Fonda's epistemic situation in it essentially identical to that in Case C; and if we can in fact succeed at this, then Option 3 fails. This leaves us with Option 2. Option 2: Fonda's reasoning is sound in Case A but not in Case B This appears to me the most attractive of the three options. A first attempt at an argument proceeds as follows. In Case A, Fonda recognizes (say) that the color of the sky in the simulation is the same as the color of grass in the outside world (i.e., green), whereas it could just as well have been displayed as the color of the sky in the outside world (i.e., blue). That is, the correspondence between colors in the simulation (as displayed on the screen) and colors in the outside world clearly could have been different, without the code that governs the laws of physics and the initial conditions in the simulation being any different. Thus there is clearly a "further fact" present, consisting in the additional code that governs how a perspective is displayed on the monitor. In Case B, however, Fonda cannot recognize the existence of any such correspondence, because she no longer remembers the outside world. Hence-so the argument goes-she cannot conclude further facts exist. Now, this argument does not seem entirely satisfactory to me. Even in Case B, it seems entirely possible for Fonda to imagine a scenario where the color of the sky (in the simulation, though she does not know it is a simulation) would be the color that the grass is now (in the simulation), and vice versa. This is the familiar inverted spectrum scenario, except in this case, by virtue of all this taking place in a virtual reality system, it is clearly true that the spectrum could be inverted; all this would require is some changes to the code governing the display. Nevertheless, it is still possible that Fonda's belief that the spectrum could have been inverted is not justified. Having forgotten about the outside world, she certainly does not know about the simple mechanism-changing a few lines of code in the outside world-by which the spectrum could indeed be inverted. But few would argue that awareness of a specific mechanism by which the spectrum could be inverted is necessary to justify belief in the possibility of an inverted spectrum (though it is clearly sufficient). Nevertheless, it seems that the physicalist, arguing that Fonda's belief is not justified, has arguments available in Case B that are unavailable in Case A. The physicalist can argue that Fonda cannot be sure that experiential properties corresponding to her seeing the simulated sky are not, at bottom, physical properties. (Again, here, "physical" refers to the physics of the environment, which we happen to know is simulated but she does not.) Even in our own case (Case C), fleshing out such an argument and addressing immediate counterarguments requires substantial work; see, for example, Hawthorne (2002). But I do not see that any additional obstacles to such an argument are introduced when moving from Case C to Case B. In contrast, in Case A such an argument becomes untenable. In that case, Fonda clearly knows that the experiential properties corresponding to her seeing the simulated sky are not, at bottom, properties of the simulated physics; she knows that the code governing the display, the monitor itself, her eyes and brain, etc., are also involved. So, perhaps this all reduces to standard arguments about inverted spectra. Perhaps Fonda cannot reasonably reject the possibility that the way colors appear to her necessarily emerges from the laws of her universe. If so, it at least suggests that the debate on inverted spectra has been on the right track. But it also provides a clearer lens on these arguments. 13 This is because unlike in the standard inverted spectrum scenario, in this case it is clearly true that the spectrum could have been inverted. This, I believe, reduces the intuitive appeal of the argument that which quale appears must supervene upon properties of the physical world (and that therefore a strong type of inverted spectrum is not possible). It makes it clear that if this argument is to succeed, it should be fundamentally epistemological in nature: it should argue just that we cannot know that there is no such supervenience. At least, this is so if our epistemic situation is sufficiently like that of Fonda in Case B, and it appears that it is, as discussed in the previous section. It is useful to note that even slightly nudging Case B towards Case A-for example, allowing Fonda to remember only that she is in a simulation, but effectively nothing else about the outside world, including even whether her color experiences there were anything like the ones she is experiencing now-would again allow her to soundly conclude that an inverted spectrum in her simulated world is a genuine possibility. This is why it is important to be strict about Fonda not remembering anything in Case B. Next, let us consider further facts about personal identity and the self. Again, in Case A, Fonda can soundly conclude that there are further facts about how the perspectives of agents in the simulation are assigned to monitors in the outside world. Even if every agent's perspective is displayed on some monitor, clearly the correspondence-on which particular monitor each agent's perspective is displayed-could have been different, without the code that governs the laws of physics and the initial conditions of the simulation being any different. Thus there is clearly a further fact present, consisting in the additional code that governs on which (if any) monitors each perspective is displayed. Again, however, in Case B, Fonda cannot recognize the existence of any such correspondence, because she no longer remembers the outside world. So one might argue that in Case B she is not justified in believing in the existence of further facts about personal identity and the self. How satisfying is this argument? Could she nevertheless, in Case B, imagine the perspective of an agent other than Alpha appearing to her? Certainly it is true that a different agent's perspective could be made to appear to her; all this would require is a change to the code governing which 13 perspective is displayed by the VR system. But would she be justified in believing that a different perspective could have appeared to her? Again, it seems that the moment we allow her to remember even just the mere fact that she is in a simulation, even if she remembers nothing else about her identity in the outside world, she can indeed conclude that a different agent's perspective could have been made to appear to her. But we explicitly rule out such a memory in Case B. Hence, it is not clear that the notion of a different perspective appearing to her makes sense from Fonda's perspective. For all she knows, she is Alpha, and how could any perspective other than Alpha's appear to Alpha? It is interesting to note that this argument is not entirely analogous to the corresponding argument regarding color appearance given earlier. It may be plausible to Fonda that the way colors appear necessarily emerges from the laws of the universe in which she finds herself. But it seems implausible that somehow Alpha's perspective, to the exclusion of any other, necessarily emerges as the "present" one from these laws, given that Alpha is just one agent among many similar ones as far as these laws go. 14 Instead, the argument here relies on the possibility of her complete identification with Alpha. Again, perhaps this all reduces to standard arguments about personal identity and the self. If so, then again, this on the one hand suggests that the debate has been on the right track, while on the other hand also casting a clearer lens on it. This is because unlike in standard scenarios in the literature on personal identity and the self, here it is clearly true that Fonda "could have been someone else"-i.e., she could have had a different agent's perspective in the simulation displayed to her on the VR system. This highlights, again, that the problem fundamentally has an important epistemological component. 14 Conclusion How can we avoid concluding that further facts exist in Case C, which corresponds to our own world? It seems that the approach most likely to succeed is to argue that, while in Case A the conclusion of further facts is justified, it is not in Case B. Moreover, the most natural way to do so is to counter the argument in favor of further facts in Case B in a way that is similar to how arguments in favor of further facts in our own world (Case C) are often countered. But it appears that these counterarguments lose at least some of their immediate intuitive appeal when moving from Case C to Case B. This is because, by construction, there are in fact further facts in Case B, making it difficult to point out where exactly the argument that further facts exist goes wrong. Moreover, this argument is essentially the same as in Case A, where presumably we do believe the argument is correct. Therefore, the counterargument needs to rely entirely on Fonda's epistemic limitations in Case B. Again, I believe that this is the most natural way to avoid the conclusion that further facts exist. But I also believe that the counterargument is in need of further fleshing out. In any case, if one agrees that Fonda's situation in Case B is epistemically sufficiently like ours (Case C), so that any arguments available to us against further facts should be available to Fonda in Case B as well, we obtain a nontrivial conclusion. This is that we cannot know with certainty that qualia supervene on the physical facts, because after all, (say) an inverted spectrum is genuinely possible in Case B. 15 At most, one can argue that we have no good reason to believe that they fail to supervene on the physical facts, and hence are not justified in concluding that there are further facts. One may take other routes. I do not see how one could reasonably hold that Fonda's argument in Case A is flawed. On the other hand, perhaps one could successfully argue that there is a relevant difference in Fonda's epistemic situation between Cases B and C. Of course, Case B is intended to be epistemically as similar to Case C as possible, and for any remaining potentially relevant epistemic difference between the two cases, it seems we can modify Case B appropriately to make the difference go away. One might perhaps argue that Case B describes a scenario that is inherently impossible-or at least that it would become so after sufficiently many of these modifications. But I see no convincing reason to think so. 16 Debates about further facts are ancient and clearly I have not settled them. I do believe that the three cases presented here, besides putting a modern spin on these questions, help to disentangle some of the different aspects relevant to these debates. They also allow us to treat different types of further facts in a more uniform manner. Acknowledgments I am thankful to anonymous referees who provided especially thorough and helpful comments, which significantly improved the paper. Notes tion from the main issues, I will omit a detailed review of, and comparisons to, the specific scenarios and arguments in this literature. I hope that the reader familiar with this literature can easily fill in the blanks. 3 For ease of exposition, I will assume that the laws of each universe are such that the initial conditions completely determine the physical structure of the universe. However, this is not essential to the arguments in this paper. 4 Whether one believes that these two questions get at the same issue will depend on one's interpretation of them, perhaps especially of the former. The former question has featured prominently in the literature on whether imagination provides a reliable guide to possibility. Namely, if I can imagine myself being (say) Napoleon, then should we not conclude that I could have been Napoleon? But it is difficult to make sense of this conclusion. One way to do so is to interpret "I" as referring to a Cartesian self, and "Napoleon" as referring to an empirical self. See Williams (1973) for discussion of these points, or Ninan (2016) for a more recent discussion and further references. Of course, most contemporary philosophers will eschew such an interpretation. The latter question avoids Cartesian dualism by focusing on the presence of the experience, rather than on which entity has the experience. This approach is closely related to the theory of "egocentric presentism" proposed by Hare (2007Hare ( , 2009) (see also Hare (2010) and the closely related subjectivist theory laid out by Merlo (2016)), which is a subtle form of solipsism according to which only one human being's perspective is "present." Other recent work on these questions includes that by Johnston (2011) (e.g., the section "Am I Now Contingently Johnston?", pages 151-154) and Hellie (2013), who discusses the "vertiginous question" of why Hellie is the human being whose experiences appear "live." Valberg (2007, page 62), in support of similar ideas, discusses in detail the example of himself having a dream in which he occupies the perspective of someone other than himself, even though he-meaning, Valberg-is one of the characters in the dream. (Ninan (2016) 6 It should be pointed out that the referent of "me" is not clear here. Since by assumption Fonda no longer remembers her life outside the simulation and she feels identified with Alpha, one could argue that for the thought to make sense from her perspective, "me" should refer to Alpha, in which case we end up with the familiar uninteresting question of why Alpha's perspective (rather than Beta's) should appear to Alpha. On the other hand, she could take "me" to refer to some abstract observer, one that is difficult for her to identify, rather than a clearly identifiable agent in the (simulated) universe. If she were to do so, it would make more sense for the referent to be Fonda, i.e., the human being wearing the VR headset, even though she is no longer aware of the existence of such a human being. And then, the question does have a nontrivial answer that involves the code governing the simulation and its display on the VR system. As yet another alternative, we can just leave out "to me" altogether-that is why this phrase is in parentheses. Again, in this particular context, this seems to be a sensible question with a nontrivial answer. (See also Endnote 13 on phenomenal concepts.) 7 Letting +X denote that her reasoning is sound in case X and −X that it is not, there are 2 3 = 8 combinations; 8 By using the word "silently" I do not intend to take any stance on whether and in what sense there might be such a thing as Alpha's own experience; this is irrelevant to the arguments presented here, which concern Fonda's experience. See also Endnote 10. 9 It is not required here that the code for running the simulation and the code governing the display are neatly separated. Even if they are intermingled in horribly messy ways, somewhere in the code there must be commands of roughly the following form: display (x,y,z) indicating that on the screen the pixel at coordinates x and y is to be given color z. Removing all (and only) these commands will result in nothing being displayed, even though the simulation of the physics is running. Alternatively, replacing z by z + 1 in every such command will result in all the displayed colors changing a bit. Is it conceivable that the code was written (and, perhaps, for some reason had to be written) in a strange programming language that would prevent changing the colors? Or that the monitor for some reason (say) cannot display large amounts of red at the same time, necessitating red to be used for a particular simulated wavelength? Perhaps, but it is easy to argue that Fonda has strong reason to believe that a sensible programming language was used and that the monitor does not have strange constraints. In any case, here, in Case A, we can simply sidestep these concerns by specifying that Fonda knows the programming language and the type of monitor used. 10 But then, could it similarly be the case that the relevant experience is Alpha's, not Fonda's? One can argue that Alpha's experience is fully determined by the simulated physics, so that no further facts are needed to explain Alpha's experience. However, this will not resolve the puzzle considered in this paper. It is not clear under what conditions there is something it is like to be Alpha-that is, not to be someone to whom Alpha's perspective is being displayed, but to really be Alpha-but in any case this is irrelevant to the issues we investigate here. In Cases A and B as I have specified them, the conclusions about further facts are reached by Fonda, not Alpha, on the basis of Fonda's experience, not Alpha's. So Fonda's experience is the relevant one. But, one might ask, is Fonda in Case C perhaps more similar to Alpha in Case B than to Fonda in Case B? At least for some aspects of these three entities, this is surely true. Does this mean that there is a gap between Cases B and C, and that to close the gap we should modify Case B to have Alpha, not Fonda, conclude that there are further facts? No. For our purposes, it is not important that Fonda in Case B and Fonda in Case C are similar in every aspect. All that matters is that their epistemic situations are similar across these cases. I will discuss this in more detail in the section on Option 3. See also Endnote 6 on what the referent of "me" is. 11 Neuroscience aside, in our own world we also observe, to some extent, how children learn to think. If there is nothing analogous in the simulation, again this may raise suspicions about there being an "outside" world where the ability to think rationally is acquired. Again, though, it does not seem difficult to modify the case appropriately, for example with Fonda and perhaps others having been in the virtual reality system since childhood. contributed causally to her current thoughts about further facts, even though she is not currently aware of this. (Examples in which one is not aware of exactly how one has come to believe something are common in the literature about internalist vs. externalist views of epistemic justification; see, e.g., Goldman (2009).) If so, externalists, and even some internalists, may consider Fonda's belief in further facts justified in Case B but not in Case C. However, we can simply specify that no such causal link exists in Case B-say, Fonda's brain has rewired itself from scratch after entering the simulation. Given this additional detail, it seems few externalists would hold that Fonda's belief in further facts is justified in Case B but not in Case C. For example, if we consider reliable process theory (see, e.g., Goldman (1979) for a classic version), it is not clear in what sense the process leading to Fonda's belief should be more reliable in Case B than in Case C. 13 Of course, I do not claim that these examples have significant implications for every argument in the literature. An exhaustive analysis of where they can provide insight is far beyond the scope of this paper, but, for instance, it is instructive to reconsider phenomenal concepts (for a survey article, see Balog (2009)), and especially their role in Chalmers' work (see, e.g., Chalmers (1996), in light of them. Even for this, a thorough analysis is beyond the scope of this paper, but here is a sketch of how part of such an analysis might proceed. Phenomenal concepts are taken to pick out phenomenal qualities. However, in the context of examples with simulations (Cases A and B), we can define analogous concepts that simply pick out the color displayed on the screen (as opposed to the phenomenal color quality experienced by Fonda). Even in Case B Fonda could herself form such a concept by means of imagination ("the color displayed on the screen assuming I am in a simulation"). Standard arguments, including ones about the possibility of inverted spectra or zombies, can then be applied to these concepts instead of the phenomenal ones, and may become less controversial (since, e.g., we know we can invert the colors on a screen). Then again, they may have less bite. If all that such an argument allows Fonda to conclude in Case B is that an inverted spectrum-in the limited sense of the colors on a screen being inverted-is possible if she is in a simulation, then it is not clear how this by itself could justify an unconditional belief in the possibility of an inverted spectrum in the original phenomenal sense. Fundamentally, the challenge for this approach seems to be that the new concepts are not infallible in the way that phenomenal concepts are widely held to be, and, relatedly, that they do not refer to something that can justify beliefs through acquaintance in the way that phenomenal qualities are widely held to be able to. (See also 20 Endnote 6 on what "me" could refer to for Fonda in Case B.) 14 To flesh this out, we may specify that all the agents in the simulation-Alpha, Beta, Gamma, . . . -are objectively (from a standpoint within the simulation) extremely similar. Then, given that the (simulated) physical laws treat similar agents similarly, these laws could not by themselves determine which agent's perspective becomes the present (displayed) one. In contrast, the phenomenal properties of seeing blue are inherently different from those of seeing red, so it does not seem possible to make a similar argument for the case of such qualia. Conceivably to Fonda in Case B, the phenomenal properties of seeing blue are somehow inherently linked to the physical properties of certain wavelengths in her environment. 15 Unlike for standard conceivability arguments (see, e.g., Chalmers (2010b)), where a key issue is whether the jump from conceivability to possibility can be made, it does not seem that possibility is at issue for Cases A and B. Our current state of technology already enables at least Case A. As for Case B, I am not aware of anyone having ever become so completely lost in a VR simulation, presumably at least in part due to remaining limitations of the technology. But this technology is advancing rapidly with no apparent fundamental obstacles in its path. Hence, it is hard to see what could keep Case B from being possible even in the near future. Moreover, inverting the spectrum or changing the identity of the displayed agent is clearly possible in Cases A and B. Unless an epistemic line can be drawn between Cases B and C, this seems to imply a strong type of epistemic possibility in our own case as well. 16 See also Endnote 15. 5 Again, the literature addressing apparently related questions in the philosophy of time is vast, and I hope that the reader familiar with this literature can easily draw the connections. For relatively recent references, see, e.g., Balashov (2005); Zimmerman (2005); Olson (2009), and Skow (2011). − A−B−C,−A−B+C,−A+B−C, and −A+B+C are covered (at least) under 1, +A−B−C and +A−B+C under 2, +A+B−C under 3, and +A+B+C under 4. Alternatively, it is not hard to see that (for example) the negation of the first three possibilities implies the fourth. Due to the existence of the additional code, these further facts are in fact ontologically further facts, as opposed to merely epistemologically further facts. For more on the distinction, see, e.g., Chalmers (2010a).2 See, e.g.,Shoemaker (1982) andBlock (1990). The literature on inverted spectrum (and closely related) scenarios, their possibility, and their implications is, of course, vast. To keep the length of this paper reasonable and avoid distrac- One may ask whether this presupposes an internalist view of epistemic justification. Might an externalist not argue that Fonda's belief in further facts is justified in Case B but not in Case C, because whether her belief is justified hinges on aspects of the external world? For one, it may be that in Case B, Fonda's prior experiences outside the simulation Times of Our Lives: Negotiating the Presence of Experience. Yuri Balashov, American Philosophical Quarterly. 424Yuri Balashov. Times of Our Lives: Negotiating the Presence of Experience. American Philosophi- cal Quarterly, 42(4):295-309, October 2005. Phenomenal Concepts. Katalin Balog, Ansgar Beckermann, Brian P. McLaughlin, and Sven 21Katalin Balog. Phenomenal Concepts. In Ansgar Beckermann, Brian P. McLaughlin, and Sven 21 The Oxford Handbook of Philosophy of Mind. Walter, Oxford University PressWalter, editors, The Oxford Handbook of Philosophy of Mind, pages 292-312. Oxford University Press, 2009. . Ned Block. Inverted Earth. Philosophical Perspectives. 4Ned Block. Inverted Earth. Philosophical Perspectives, 4:53-79, 1990. Are You Living in a Computer Simulation?. Nick Bostrom, Philosophical Quarterly. 53211Nick Bostrom. Are You Living in a Computer Simulation? Philosophical Quarterly, 53(211): 243-255, 2003. The Conscious Mind. David J Chalmers, Search of a Fundamental Theory. Oxford University PressDavid J. Chalmers. The Conscious Mind: In Search of a Fundamental Theory. Oxford University Press, 1996. The Content and Epistemology of Phenomenal Belief. David J Chalmers, Consciousness: New Philosophical Perspectives. Quentin Smith and Aleksandar JokicOxford University PressDavid J. Chalmers. The Content and Epistemology of Phenomenal Belief. In Quentin Smith and Aleksandar Jokic, editors, Consciousness: New Philosophical Perspectives, pages 220-72. Ox- ford University Press, 2003. The Matrix as Metaphysics. David J Chalmers, Philosophers Explore The Matrix. Christopher GrauOxford University PressDavid J. Chalmers. The Matrix as Metaphysics. In Christopher Grau, editor, Philosophers Explore The Matrix, chapter 9, pages 132-176. Oxford University Press, 2005. The Character of Consciousness. David J Chalmers, Oxford University PressDavid J. Chalmers. The Character of Consciousness. Oxford University Press, 2010a. The Two-Dimensional Argument against Materialism. David J Chalmers, The Character of Consciousness, chapter 6. Oxford University PressDavid J. Chalmers. The Two-Dimensional Argument against Materialism. In The Character of Consciousness, chapter 6. Oxford University Press, 2010b. What Is Justified Belief?. Alvin I Goldman, George S. PappasD. Reidel Publishing CompanyAlvin I. Goldman. What Is Justified Belief? In George S. Pappas, editor, Justification and Knowl- edge, pages 1-25. D. Reidel Publishing Company, 1979. Internalism, Externalism, and the Architecture of Justification. Alvin I Goldman, Journal of Philosophy. 1066Alvin I. Goldman. Internalism, Externalism, and the Architecture of Justification. Journal of Phi- losophy, 106(6):309-338, 2009. Self-Bias, Time-Bias, and the Metaphysics of Self and Time. Caspar Hare, The Journal of Philosophy. 1047Caspar Hare. Self-Bias, Time-Bias, and the Metaphysics of Self and Time. The Journal of Philoso- phy, 104(7):350-373, July 2007. On Myself, And Other, Less Important Subjects. Caspar Hare, Princeton University PressCaspar Hare. On Myself, And Other, Less Important Subjects. Princeton University Press, September 2009. Realism About Tense and Perspective. Caspar Hare, Philosophy Compass. 59Caspar Hare. Realism About Tense and Perspective. Philosophy Compass, 5(9):760-769, September 2010. Advice for Physicalists. John Hawthorne, Philosophical Studies. 1091John Hawthorne. Advice for Physicalists. Philosophical Studies, 109(1):17-52, 2002. . Benj Hellie. Against Egalitarianism. 732AnalysisBenj Hellie. Against Egalitarianism. Analysis, 73(2):304-320, 2013. Surviving Death. Mark Johnston, Princeton University PressMark Johnston. Surviving Death. Princeton University Press, October 2011. Do You Know That You Are Not a Brain In a Vat?. Ned Markosian, Logos and Episteme. 52Ned Markosian. Do You Know That You Are Not a Brain In a Vat? Logos and Episteme, 5(2): 161-181, 2014. Subjectivism and the Mental. Giovanni Merlo, Dialectica. 703Giovanni Merlo. Subjectivism and the Mental. Dialectica, 70(3):311-342, 2016. The Routledge Handbook of Philosophy of Imagination. Dilip Ninan, Amy KindRoutledge20Imagination and the SelfDilip Ninan. Imagination and the Self. In Amy Kind, editor, The Routledge Handbook of Philosophy of Imagination, chapter 20. Routledge, 2016. The Rate of Time's Passage. Eric T Olson, Analysis. 691Eric T. Olson. The Rate of Time's Passage. Analysis, 69(1):3-9, January 2009. The Inverted Spectrum. Sydney Shoemaker, The Journal of Philosophy. 797Sydney Shoemaker. The Inverted Spectrum. The Journal of Philosophy, 79(7):357-381, 1982. On the meaning of the question "How fast does time pass?. Bradford Skow, Philosophical Studies. 1553Bradford Skow. On the meaning of the question "How fast does time pass?" Philosophical Studies, 155(3):325-344, September 2011. Dream, Death, and the Self. J J Valberg, Princeton University PressJ. J. Valberg. Dream, Death, and the Self. Princeton University Press, April 2007. Imagination and the Self. Bernard Williams, Problems of the Self. Bernard WilliamsCambridge University Press3Bernard Williams. Imagination and the Self. In Bernard Williams, editor, Problems of the Self, chapter 3, pages 26-45. Cambridge University Press, 1973. W Dean, Zimmerman, The A-Theory of Time, The B-Theory of Time, and 'Taking Tense Seriously'. Dialectica. 59Dean W. Zimmerman. The A-Theory of Time, The B-Theory of Time, and 'Taking Tense Seriously'. Dialectica, 59(4):401-457, 2005.

The aim of this paper is to improve the previous work on the relativistic Vlasov-Maxwell system, one of the most important equations in plasma physics. Recently in [3], C. Bardos et al. presented a proof of an Onsager type conjecture on renormalization property and the entropy conservation laws for the relativistic Vlasov-Maxwell system. Particularly, authors proved that if the distribution function u ∈ L ∞ (0, T ; W α,p (R 6 )) and the electromagnetic field E, B ∈ L ∞ (0, T ; W β,q (R 3 )), with α, β ∈ (0, 1) such that αβ + β + 3α − 1 > 0 and 1/p + 1/q ≤ 1, then the renormalization property and entropy conservation laws hold. To determine a complete proof of this work, in the present paper we improve their results under a weaker regularity assumptions for weak solution to the relativistic Vlasov-Maxwell equations. More precisely, we show that under the similar hypotheses, the renormalization property and entropy conservation laws for the weak solution to the relativistic Vlasov-Maxwell system even hold for the end point case αβ + β + 3α − 1 = 0. Our proof is based on the better estimations on regularization operators.

1905.05973

cfaae4e7c1da46d4828174da406c184a6b9ac1cf

On the renormalization property and entropy conservation laws for the relativistic Vlasov-Maxwell system 23 May 2019 May 24, 2019 Minh-Phuong Tran [email protected] Faculty of Mathematics and Statistics Applied Analysis Research Group Ton Duc Thang University Ho Chi Minh City, Viet Nam Thanh-Nhan Nguyen [email protected] Department of Mathematics University of Education Ho Chi Minh City, Viet NamHo Chi Minh City On the renormalization property and entropy conservation laws for the relativistic Vlasov-Maxwell system 23 May 2019 May 24, 2019Relativistic Vlasov-Maxwell systemOnsager type conjecturerenor- malization propertyentropy conservation laws The aim of this paper is to improve the previous work on the relativistic Vlasov-Maxwell system, one of the most important equations in plasma physics. Recently in [3], C. Bardos et al. presented a proof of an Onsager type conjecture on renormalization property and the entropy conservation laws for the relativistic Vlasov-Maxwell system. Particularly, authors proved that if the distribution function u ∈ L ∞ (0, T ; W α,p (R 6 )) and the electromagnetic field E, B ∈ L ∞ (0, T ; W β,q (R 3 )), with α, β ∈ (0, 1) such that αβ + β + 3α − 1 > 0 and 1/p + 1/q ≤ 1, then the renormalization property and entropy conservation laws hold. To determine a complete proof of this work, in the present paper we improve their results under a weaker regularity assumptions for weak solution to the relativistic Vlasov-Maxwell equations. More precisely, we show that under the similar hypotheses, the renormalization property and entropy conservation laws for the weak solution to the relativistic Vlasov-Maxwell system even hold for the end point case αβ + β + 3α − 1 = 0. Our proof is based on the better estimations on regularization operators. Introduction In recent years, mathematicians have devoted much attention to the relativistic Vlasov-Maxwell system, the most important equation describes the distribution of particles in phase space of a monocharged plasma under relativistic effects. There has been an increasing activity that studied the Vlasov-Maxwell system in kinetic plasma physics. It is well-known that the Vlasov equation describes the time evolution of particles in a plasma, how the plasma response to electromagnetic fields. This equation finds the unknown distribution function of particles u = u(t, x, ξ) satisfies: ∂ t u + v · ∇ x u + F · ∇ ξ u = 0, (1.1) where (t, x, ξ) ∈ R + × R 3 × R 3 represent time, position and momentum of particles, respectively. The relativistic velocity v of a particle with momentum ξ ∈ R 3 is given by v = ξ 1 + |ξ| 2 . (1. 2) The consideration of problem may be under electromagnetic, in which the Lorentz force F = E +v ×B corresponds to the self-consistent electric field E = E(t, x) and magnetic field B = B(t, x) generated by the charged particles in the plasma. They are coupled satisfying Maxwell's equations ∂ t E − curl B = −j, ∂ t B + curl E = 0, (1.3) div E = ρ, div B = 0, (1.4) where the quantities ρ = ρ(t, x) and j = j(t, x) are the charge density and electric current density of the plasma, respectively, defined by div E 0 = ρ 0 =ˆR 3 u 0 dξ, div B 0 = 0. (1.8) There are many interesting problems that related to the Vlasov-Maxwell system (1.1)-(1.5) that make the range of its application has been considerably extended. For instance, the existence and uniqueness of analytical solutions to this, especially for high dimensions; regularity results for the system in some spaces; the conduction of sharp estimates for solutions; some numerical methods and simulations on the solutions, etc, are at the core of many researching topics at the moment. ρ(t, x) =ˆR 3 u(t, x, ξ)dξ; j(t, x) =ˆR 3 v(ξ)u(t, x, ξ)dξ. The global existence of solution to this earlier has been studied intensively by several authors, such as R.J. Diperna and P.-L. Lions in [8], Y. Guo in [9,10] or G. Rein in [12] and several references therein. Later, different approaches to the results related to this system were recently achieved and reviewed by other authors. In our knowledge, there has been a few results on the regularity of this system. Recently in 2018, N. Besse et al. have showed in [5] that if the macroscopic kinetic energy is in L 2 , then the electric and magnetic fields belong to the Sobolev space H s loc (R + × R 3 ) with s = 6/(13 + √ 142). Moreover, in [4,11], authors have established the critical regularity of weak solutions to a general system of entropy conservation laws which are related to the famous Onsager exponent 1/3. In the nearest research paper [3], Bardos et. al. gave a proof of an Onsager type conjecture on renormalization property and entropy conservation laws for the Vlasov-Maxwell equations. More precisely, their work devoted to the results that if the distribution function u ∈ L ∞ (0, T ; W α,p (R 6 )) and the electromagnetic fields E, B ∈ L ∞ (0, T ; W β,q (R 3 )), with α, β ∈ (0, 1) satisfying αβ + β + 3α − 1 > 0 and 1/p + 1/q ≤ 1, then the renormalization property holds. As there have been too few results concerning to regularity of this system, such extensions have been promising to discussed under various assumptions and conditions of problem formulation. In the present paper, based on the regularity assumptions of weak solution to the Vlasov-Maxwell equations, a small portion of that result is improved, where the conclusion of this property holds even for αβ + β + 3α − 1 ≥ 0, the renormalization property and entropy conservation laws hold under the same hypotheses. To our knowledge, from the mathematical point of view, the end point case αβ + β + 3α − 1 = 0, the proof is more challenging than what obtained in [3]. Compare to the previous study for the case αβ + β + 3α − 1 > 0, ours have the advantage that for αβ + β + 3α − 1 ≥ 0, we work on the weaker regularity assumptions, and the effective technique is applied to extend the proof. The key idea comes from the better estimations on regularization operators that will be described later. The rest of the paper is organized as follows. Next section 2 is devoted to some notations and definitions about the renormalization property and entropy conservation laws, and our main result of this paper is also stated therein. We then introduce in Section 3 some regularization operators and properties are also presented for later use. Finally, the last section gives a brief proof of the renormalization property and entropy conservation laws for Diperna-Lions weak solution to the Vlasov-Maxwell equations. Main result At the beginning of this section, let us recall some notations and definitions concerning to the problem. Throughout the paper, we denote by D(R n ), with n ≥ 1, the space of infinitely differentiable function with compact support and by D ′ (R n ) the space of distribution. For α ∈ (0, 1), 1 ≤ p ≤ ∞, the generalized fractional order Sobolev spaces W α,p (R n ) is defined for any function f belonging to W α,p (R n ) if and only if the following Gagliardotype norm is finite: f W α,p (R n ) := ˆR n |f (x)| p dx 1/p + ˆR nˆRn |f (x) − f (y)| p |x − y| n+αp dxdy 1/p < +∞, in the case 1 ≤ p < ∞ and f W α,∞ (R n ) := max f L ∞ (R n ) , sup x =y∈R n |f (x) − f (y)| |x − y| α < +∞, for p = ∞. Here and subsequently, L 1 (R 6 ) denotes the set of non-negative almost everywhere function f such that f L 1 (R 6 ) :=ˆR 6 f (x, ξ) 1 + |ξ| 2 dx dξ < +∞. (2.1) In addition, the notation S stands for the set of non-decreasing function G ∈ C 1 (R + ; R + ) such that lim t→+∞ G(t) t = +∞. The weak solution of a coupled set of relativistic Vlasov-Maxwell equations involves the distribution function u (describes plasma components), electric and magnetic fields E, B (self-consistenly modified by particles). Here, we say that (u, E, B) is a weak solution to relativistic Vlasov-Maxwell equations (1.1) if (u, E, B) satisfies the following weak formulationˆT 0 dtˆR 3 dxˆR 3 u(∂ t ϕ + v · ∇ x ϕ + F · ∇ ξ ϕ) dξ = 0, for all ϕ ∈ D((0, T ) × R 6 ). The existence result of a global in time weak solution to the relativistic Vlasov-Maxwell equations proposed by DiPerna-Lions is stated in the following theorem, where we refer the reader to [8] for details. Theorem 2.1 Let u 0 ∈ L 1 ∩ L ∞ (R 6 ) and E 0 , B 0 ∈ L 2 (R 3 ) be initial conditions with satisfy the constraints div B 0 = 0, div E 0 =ˆR 3 u 0 dξ, in D ′ (R 3 ). Then there exists a global in time weak solution of the relativistic Vlasov-Maxwell system, i.e., there exist functions u ∈ L ∞ (R + ; L 1 ∩ L ∞ (R 6 )), E, B ∈ L ∞ (R + ; L 2 (R 3 )), and ρ, j ∈ L ∞ (R + ; L 4/3 (R 3 )), (2.2) such that (u, E, B) satisfy (1.1)-(1.4) in the sense of distributions, where ρ and j are defined in (1.5). Let (u, E, B) be a weak solution to the relativistic Vlasov-Maxwell system (1.1)-(1.5), as in Theorem 2.1. Then for any smooth function G ∈ C 1 (R + ; R + ), we say that (u, E, B) satisfies the renormalization property if ∂ t (G(u)) + ∇ x · (vG(u)) + ∇ ξ · (FG(u)) = 0, in D ′ ((0, T ) × R 6 ) (2.3) in the sense of distribution, that means, T 0 dtˆR 3 dxˆR 3 G(u) (∂ t ϕ + v · ∇ x ϕ + F · ∇ ξ ϕ) dξ = 0, for all ϕ ∈ D ′ ((0, T ) × R 6 ). Otherwise, solution (u, E, B) is said to satisfy the local in space entropy conservation law, if 4) and the local in momentum entropy conservation law, if ∂ t ˆR 3 G(u)dξ + ∇ x · ˆR 3 vG(u)dξ = 0, in D ′ ((0, T ) × R 3 ),(2.∂ t ˆR 3 G(u)dx + ∇ ξ · ˆR 3 FG(u)dx = 0, in D ′ ((0, T ) × R 3 ), (2.5) in the sense of distribution. In this way, we can state that (u, E, B) satisfies the global entropy conservation law, if we havê R 6 G(u(t, x, ξ)) dξ dx =ˆR 6 G(u(s, x, ξ)) dξ dx, for 0 < s ≤ t < T. (2.6) Let us state our main result about the renormalized property and entropy conservation laws for the global weak solution of relativistic Vlasov-Maxwell equations. Related to the present note, it emphasizes that the regularity assumptions on the weak solution in our work are weaker than in the paper of C. Bardos et. al. [3], our improved results thus are more general. In particular, we prove that the renormalization property and entropy conservation laws for the global weak in time solution to the relativistic Vlasov-Maxwell's system (1.1)-(1.5) even hold for the end point case αβ + β + 3α − 1 = 0, as described in the following theorem. u ∈ L ∞ (0, T ; W α,p (R 6 )) and E, B ∈ L ∞ (0, T ; W β,q (R 3 )), (2.7) where α, β ∈ (0, 1) such that αβ + β + 3α − 1 ≥ 0, and p, q ∈ N * such that 1 p + 1 q = 1 r ≤ 1 if 1 ≤ p, q < ∞, and 1 ≤ r < ∞ is arbitrary if p = q = ∞. (2.8) Then for any entropy function G ∈ C 1 (R + ; R + ), the global weak solution (u, E, B) satisfies the renormalization property (2.3). Moreover, if G ∈ S and the mapping t → u(t, ·, ·) is uniformly integrable in R 6 , for almost everywhere t ∈ [0, T ], then the local entropy conservation laws (2.4)-(2.5) and the global entropy conservation law (2.6) hold. Regularization operators In this section, let us mention the important consequences of this work, that leading to the proof of our main result. It is devoted to study the standard regularization operators and their properties, that gives us the idea to prove main result in this paper. We will now show their descriptions and prove some preparatory lemmas that are necessary for later use. Let ̺ ∈ D(R + ; R + ) be a smooth non negative function such that supp(̺) ⊂ [1, 2],ˆR ̺(τ )dτ = 1. (3.1) For every δ > 0 and n ∈ N * , the radially-symmetric compactly-supported Friedrichs mollifier ̺ δ : R n → R + , x → ̺ δ (x), is given by ̺ δ (x) = δ −n ̺ δ −1 |x| , x ∈ R n . (3.2) Let η, ε, δ be positive numbers and for any distribution f ∈ D ′ (R n ), g ∈ D ′ (R + × R n ) and h ∈ D ′ (R + × R n × R n ) , we define their C ∞ -regularization by f δ (x) = ̺ δ (x) * f (x), (3.3) g ε,δ (t, x) = ̺ ε (t) * t ̺ δ (x) * x g(t, x),(3.4) and h η,ε,δ (t, x, ξ) = ̺ η (t) * t ̺ ε (x) * x ̺ δ (ξ) * ξ h(t, x, ξ), (3.5) where the operator * denotes the standard convolution product. We first establish two basic estimations for the relativistic velocity v in (1.2) by the following lemmas. |v(ξ − w) − v(ξ)| ≤ 2|w|, (3.6) and v(ξ) − v δ (ξ) ≤ 4δ, (3.7) for all ξ, w ∈ R 3 . Proof. By a simple computation, we firstly get that |∇ ξ v| = I 3 1 + |ξ| 2 − ξ ⊗ ξ (1 + |ξ| 2 ) 3 ≤ 2,(3.8) where I 3 denotes the identity matrix of size 3. Combining inequality (3.8) and the fundamental theorem of calculus, we obtain the first basic estimate (3.6), |v(ξ − w) − v(ξ)| ≤ |w|ˆ1 0 |∇v(ξ − sw)|ds ≤ 2|w|. By using this estimation, we obtain the second basic estimate (3.7) as follows v(ξ) − v δ (ξ) = ˆR 3 ̺ δ (w)(v(ξ) − v(ξ − w))dw ≤ 2 ˆR 3 ̺ δ (w)|w|dw ≤ 4δ. Remark 3.2 In fact, one may obtain the better estimate for v − v δ than that of Lemma 3.1, i.e., there exists a constant C depending only on the smooth function ̺ given by (3.1), such that v(ξ) − v δ (ξ) ≤ Cδ 2 , (3.9) Proof. Using the fundamental theorem of calculus twice, for any i ∈ {1, 2, 3}, in this way we obtain componentwise v i − v δ i =ˆR 3 dw̺ δ (w)(v i (ξ) − v i (ξ − w)) = 3 j=1ˆR 3 dw̺ δ (w)w jˆ1 0 dτ ∂ j v i (ξ − τ w) = 3 j=1 ∂ j v i (ξ)ˆR 3 dw̺ δ (w)w j + 3 j,k=1ˆR 3 dw̺ δ (w)w j w kˆ1 0 dτˆ1 0 ds∂ 2 jk v i (ξ − sτ w). (3.10) Since the smooth function ̺ δ is radially symmetric, we havê R 3 dw̺ δ (w)w j = 0, ∀j ∈ {1, 2, 3}, which deduces that the first term of the right hand side of (3.10) vanishes. Therefore, (3.10) becomes v i − v δ i = 3 j,k=1ˆR 3 dw̺ δ (w)w j w kˆ1 0 dτˆ1 0 ds∂ 2 jk v i (ξ − sτ w). (3.11) Moreover, by the definition of the smooth function ̺, there exists a constant C depending only on the function ̺ such that R 3 ̺ δ (w)|w j ||w k |dw ≤ˆR 3 ̺ δ (w)|w| 2 dw ≤ Cδ 2 , ∀j, k ∈ {1, 2, 3},(3.12) On the other hand, by using δ ij as the Kronecker notation and directly computation on the relativistic velocity v, for all j, k ∈ {1, 2, 3}, one obtains ∇ 2 jk v i (ξ) = δ ij ξ k (1 + |ξ| 2 ) 3 + δ jk ξ i (1 + |ξ| 2 ) 3 + δ ik ξ j (1 + |ξ| 2 ) 3 − 3ξ i ξ j ξ k (1 + |ξ| 2 ) 3 , which implies that ∇ 2 jk v i (ξ) ≤ 6. Combining this estimation together with the inequality (3.12), and from (3.11), it completes the proof. We next present some well-known properties for C ∞ -regularization in the next lemma. For the proof of (ii) and (iii), it refers the reader to some papers found in [6,Proposition 4.2] (i) For any distribution f ∈ D ′ (R n ) and ε > 0, we have f ε , g = f, g ε , g ∈ D(R n ), where ·, · denotes the dual bracket between spaces D ′ and D. (ii) Let ε > 0, α ∈ (0, 1) and 1 ≤ p ≤ ∞. Then for any function f ∈ L 1 ∩ L ∞ ∩ W α,p (R n ), we have f ε L p (R n ) ≤ f L p (R n ) , and f ε W α,p (R n ) ≤ f W α,p (R n ) . iii) Let α ∈ (0, 1) and 1 ≤ p ≤ ∞. Then for any function f ∈ W α,p (R n ), there exists a constant C such that f (· − w) − f (·) L p (R n ) ≤ C|w| α f W α,p (R n ) ,(3. 13) for all w ∈ R n . Proof. For any distribution functions f, g ∈ D ′ (R n ) and ε > 0, we write f ε , g =ˆR n f ε (x) g(x)dx =ˆR nˆRn ̺ ε (x − y)f (y)dy g(x)dx =ˆR n f (y)ˆR n ̺ ε (x − y)g(x)dx dy =ˆR n f (y)g ε (y)dy = f, g ε , that yields the proof of (i). Lemma 3.4 Let ε > 0, α ∈ (0, 1) and 1 ≤ p ≤ ∞. Then for any function f belongs to L 1 ∩ L ∞ ∩ W α,p (R n ), there exists a constant C such that f ε − f L p (R n ) ≤ Cε α ˆR nˆRn ½ ε≤|x−y|≤2ε |f (x) − f (y)| p |x − y| n+αp dxdy 1 p ,(3. 14) and ∇f ε L p (R n ) ≤ Cε α−1 ˆR nˆRn ½ ε≤|x−y|≤2ε |f (x) − f (y)| p |x − y| n+αp dxdy 1 p . (3.15) Proof. For any y ∈ R n , by the definition of ̺ ε in (3.2) and using Hölder inequality, we obtain the following estimation |f ε (y) − f (y)| = ˆR n ̺ ε (y − x)(f (x) − f (y))dx ≤ Cε −nˆR n ½ ε≤|x−y|≤2ε |f (x) − f (y)|dx ≤ Cε − n p ˆR n ½ ε≤|x−y|≤2ε |f (x) − f (y)| p dx 1 p , where the constant C depends only on the function ̺ given in (3.1). It follows that f ε − f p L p (R n ) ≤ Cε −nˆR nˆRn ½ ε≤|x−y|≤2ε |f (x) − f (y)| p dxdy. Otherwise, by multiplying two sides of this inequality by ε −αp , it gives ε −αp f ε − f p L p (R n ) ≤ C ε n+αpˆR nˆRn ½ ε≤|x−y|≤2ε |f (x) − f (y)| p dxdy ≤ CˆR nˆRn ½ ε≤|x−y|≤2ε |f (x) − f (y)| p |x − y| n+αp dxdy, which deduces the first inequality (3.14). In order to obtain the second estimation, it will be necessary to remark that |∇f ε (y)| = ˆR n ∇̺ ε (y − x)(f (x) − f (y))dx ≤ C ε n+1ˆR n ½ ε≤|x−y|≤2ε |f (x) − f (y)|dx. and the same proof of (3.14), we obtain (3.15) the desired result. Remark 3.5 For all f ∈ W α,p (R n ), one can see that R nˆRn ½ ε≤|x−y|≤2ε |f (x) − f (y)| p |x − y| n+αp dxdy ≤ f W α,p (R n ) . (3.16) Therefore, as the consequences of Lemma 3.4, one also obtains f ε − f L p (R n ) ≤ Cε α f W α,p (R n ) , and ∇f ε L p (R n ) ≤ Cε α−1 f W α,p (R n ) . For every function f ∈ W α,p (R n × R n ), we define a function Θ f as Θ f (ε) := ˆR nˆRnˆRn ½ ε≤|x−y|≤2ε |f (x, ξ) − f (y, ξ)| p |x − y| n+αp dxdydξ 1 p , (3.17) the following Lemma is then stated and proved to give us a very important property related to this function. Corollary 3.6 Let α ∈ (0, 1), 1 ≤ p ≤ ∞ and the function f ∈ W α,p (R n × R n ). Then for any ε, δ > 0, there exists a constant C such that ∇ x f ε (x, ξ − w)−∇ x f ε (x, ξ) L p (R n ×R n ) ≤ Cε α−1 |w| α Θ f (ε), (3.18) with w ∈ R n , and (∇ x f ε ) δ − ∇ x f ε L p (R n ×R n ) ≤ Cε α−1 δ α Θ f (ε), (3.19) where the function Θ f is defined by (3.17). Proof. From Lemma 3.3, there exists a constant C such that ∇ x f ε (x, ξ − w)−∇ x f ε (x, ξ) L p (R n ×R n ) ≤ C|w| α ∇ x f ε L p (R n ;W α,p (R n )) . The inequality (3.15) in Lemma 3.4 is then applied to get ∇ x f ε (x, ξ − w)−∇ x f ε (x, ξ) L p (R n ×R n ) ≤ Cε α−1 |w| α Θ f (ε). To deal with the second estimation (3.19), by what obtained in Lemma 3.4 and Remark 3.5, there exists a constant C such that (∇ x f ε ) δ − ∇ x f ε L p (R n ×R n ) ≤ Cδ α ∇ x f ε L p (R n ;W α,p (R n )) . Repeated application of the inequality (3.15) in Lemma 3.4 enables us to write ( ∇ x f ε ) δ − ∇ x f ε L p (R n ×R n ) ≤ Cε α−1 δ α Θ f (ε), and the proof is complete. Lemma 3.7 Let α ∈ (0, 1), 1 ≤ p ≤ ∞ and the function f ∈ L 1 (0, T ; W α,p (R n × R n )). Then ω f (ε, δ) defined by ω f (ε, δ) :=ˆT 0 (Θ f (t) (ε) + Θ f (t) (δ))dt,(3. 20) vanishes as ε and δ tend to 0. Proof. By the definition of Θ f in (3.17) and Remark 3.5, we have Θ f (t) (ε) ≤ f (t, ·, ·) W α,p (R n ×R n ) < ∞, ∀t ∈ [0, T ]. Apply Lebesgue dominated convergence theorem, it is clear that Θ f (t) (ε) tends to 0 as passing ε goes to 0 for all t ∈ [0, T ]. The same conclusion is obtained for Θ f (t) (δ), and this guarantees that ω f (ε, δ) given by (3.20) vanishes as (ε, δ) goes to 0. Proof of Theorem 2.2 In this section, we consider a global in time weak solution (u, E, B) of Vlasov-Maxwell equations. The weak formulation for the Vlasov equation (1.1) readŝ T 0 dtˆR 3 dxˆR 3 u(∂ t ϕ + v · ∇ x ϕ + F · ∇ ξ ϕ) dξ = 0, for all ϕ ∈ D((0, T ) × R 6 ). Let us choose a test function ϕ as follows ϕ = (G ′ (u η,ε,δ )ψ) η,ε,δ ∈ D((0, T ) × R 6 ), where ψ ∈ D((0, T ) × R 6 ) and G ∈ C 1 (R + ; R + ). Integrating by parts this weak formulation yields that for all ψ ∈ D((0, T ) × R 6 ), there holdŝ T 0 dtˆR 3 dxˆR 3 dξG(u η,ε,δ ) ∂ t ψ + v δ · ∇ x ψ + F η,ε,δ · ∇ ξ ψ + ψG ′ (u η,ε,δ ) ∇ x · (vu) η,ϕ,δ − v δ u η,ε,δ + ∇ ξ · (Fu) η,ε,δ − F η,ε,δ u η,ε,δ = 0. (4.1) Following the renormalization property of solution (u, E, B), it is sufficient to show that the second term in the left hand side of (4.1) vanishes as (η, ε, δ) tends to 0, for all ψ ∈ D((0, T ) × R 6 ). To do so, we firstly establish some commutator estimations which are presented in the next lemma. For simplicity, the problem is considered with α ∈ (0, 1) and 1 ≤ p, r ≤ ∞, with n = 3 or n = 6 and s = 1 or s = ∞, we will use the following notations in the remain part of our paper, L s,p n := L s (0, T ; L p (R n )), L s,p,r := L s (0, T ; L p (R 3 ; L r (R 3 ))), L s W α,p n := L s (0, T ; W α,p (R n )), L s,p W α,p := L s (0, T ; L p (R 3 ; W α,p (R 3 ))). ∇ x · (vu) η,ε,δ − v δ u η,ε,δ L 1,p 6 ≤ Cε α−1 δ α+1 ω u (ε, δ). (4.2) Moreover, there exists a constant C F > 0 depending on ̺, u L 1 W α,p 6 , E L ∞ W β,q 3 and B L ∞ W β,q 3 such that ∇ ξ · (Fu) η,ε,δ − F η,ε,δ u η,ε,δ L 1,p,r ≤ C F ε α+β δ α−1 ω u (ε, δ) + δ α , (4.3) where F := E + v × B is the Lorentz force field and the function ω u is given by (3.20). Proof. We first consider the commutator estimate (4.2) for the free streaming term. It is easy to check that (vu) η,ε,δ − v δ u η,ε,δ = K δ (v, u η,ε ) − (u η,ε,δ − u η,ε )(v − v δ ), (4.4) where K δ is defined by K δ (v, g)(t, x, ξ) =ˆR 3 ̺ δ (w) (v(ξ − w) − v(ξ)) (g(t, x, ξ − w) − g(t, x, ξ)) dw. (4.5) Passing to the limit η → 0 on the right hand side of (4.4) which can be justified by the Lebesgue dominated convergence theorem and regularity assumptions (2.7), we thus get that ∇ x · (vu) η,ε,δ − v δ u η,ε,δ L 1,p 6 ≤ ∇ x · K δ (v, u ε ) L 1,p 6 + ∇ x · ((u ε,δ − u ε )(v − v δ )) L 1,p 6 . (4.6) By the definition of K δ in (4.5), one has ∇ x · K δ (v, u ε ) L 1,p 6 ≤ˆT 0 dtˆR 3 dw̺ δ (w) (v(ξ − w) − v(ξ)) · (∇ x u ε (t, x, ξ − w) − ∇ x u ε (t, x, ξ)) L p (R 6 ) . In the use of (3.6) in Lemma 3.1 and (3.18) in Corollary 3.6, we obtain that ∇ x · K δ (v, u ε ) L 1,p 6 ≤ Cε α−1ˆT 0ˆR 3 ̺ δ (w)|w| α+1 Θ u(t) (ε)dwdt ≤ Cε α−1 δ α+1ˆT 0 Θ u(t) (ε)dt ≤ Cε α−1 δ α+1 ω u (ε, δ),(4.7) where ω u given in (3.20). Additionally, from (3.7) and (3.19), there holds ∇ x · ((u ε,δ − u ε )(v − v δ )) L 1,p 6 ≤ |v − v δ |(∇ x u ε,δ − ∇ x u ε ) L 1,p 6 ≤ Cε α−1 δ α+1ˆT 0 Θ u(t) (ε)dt ≤ Cε α−1 δ α+1 ω u (ε, δ). (4.8) From what have already been proved, we obtain commutator estimate (4.2). It remains to prove the estimate in (4.3). To establish this commutator estimate for the Lorentz force term, it is possible for us to make the following decomposition as follows (Fu) η,ε,δ − F η,ε,δ u η,ε,δ = K η,ε (E, u δ ) − (E − E η,ε )(u δ − (u δ ) η,ε ) + (v × Bu) ε,δ − v δ × B ε u ε,δ ,(4.9) where K η,ε is given by K η,ε (E, g)(t, x, ξ) =ˆR dτˆR 3 dy ̺ η (τ )̺ ε (y) · (E(t − τ, x − y, ξ) − E(t, x, ξ)) (g(t − τ, x − y, ξ) − g(t, x, ξ)) . (4.10) For the sake of simplicity, in this work we will denote T E := K η,ε (E, u δ ) − (E − E η,ε )(u δ − (u δ ) η,ε ), (4.11) T B := (v × Bu) ε,δ − v δ × B ε u ε,δ ,(4.12) and make the effective use of the Lebesgue dominated convergence theorem together with regularity assumptions (2.7), passing to the limit η → 0 in T E , yields that ∇ ξ · T E L 1,p,r ≤ ∇ ξ · K ε (E, u δ ) L 1,p,r + ∇ ξ · ((E − E ε )(u δ − (u δ ) ε )) L 1,p,r . (4.13) By Hölder inequality, there holds ∇ ξ · K ε (E, u δ ) L 1,p,r ≤ˆR 3 ̺ ε (y) (E(t, x − y) − E(t, x)) · (∇ ξ u δ (t, x − y, ξ) − ∇ ξ u δ (t, x, ξ)) L 1,p,r dy ≤ˆR 3 ̺ ε (y) E(t, x − y) − E(t, x) L ∞,q 3 · ∇ ξ u δ (t, x − y, ξ) − ∇ ξ u δ (t, x, ξ) L 1,p 6 dy. Applying the estimate (3.13) in Lemma 3.3 and the regularity assumptions (2.7), we obtain that ∇ ξ · K ε (E, u δ ) L 1,p,r ≤ CˆR 3 ̺ ε (y)|y| α+β E L ∞ W β,q 3 ∇ ξ u δ (t, x, ξ) L 1,p W α,p dy ≤ Cε α+β E L ∞ W β,q 3 ∇ ξ u δ (t, x, ξ) L 1,p W α,p . Thanks to (3.15) from Lemma 3.4, we have ∇ ξ · K ε (E, u δ ) L 1,p,r ≤ Cε α+β δ α−1 E L ∞ W β,q 3ˆT 0 Θ u(t) (δ)dt ≤ Cε α+β δ α−1 E L ∞ W β,q 3 ω u (ε, δ). (4.14) And the second term on the right hand side of (4.13) is then proved thanks to Hölder inequality, ∇ ξ · ((E − E ε )(u δ − (u δ ) ε )) L 1,p,r ≤ E − E ε L ∞,q 3 ∇ ξ u δ − (∇ ξ u δ ) ε L 1,p 6 . Then, from (3.14) and (3.19), it deduces that ∇ ξ · ((E − E ε )(u δ − (u δ ) ε )) L 1,p,r ≤ Cε α+β δ α−1 E L ∞ W β,q 3ˆT 0 Θ u(t) (ε)dt ≤ Cε α+β δ α−1 E L ∞ W β,q 3 ω u (ε, δ). (4.15) From what have already been proved in (4.13), (4.14) and (4.15), we get ∇ ξ · T E L 1,p,r ≤ Cε α+β δ α−1 E L ∞ W β,q 3ˆT 0 (Θ u(t) (δ) + Θ u(t) (ε))dt ≤ Cε α+β δ α−1 E L ∞ W β,q 3 ω u (ε, δ). (4.16) We next consider the term T B given by (4.12), which can be decomposed as T B = T B1 + T B2 + T B3 ,(4.17) where T B1 :=ˆT 0 dτˆR 3 dyˆR 3 dw ̺ η (τ )̺ ε (y)̺ δ (w) [(v(ξ − w) − v(ξ)) × B(t − τ, x − y)]u(t − τ, x − y, ξ − w), T B2 := v × ((Bu δ ) η,ε − B η,ε (u δ ) η,ε ),T B3 := (v − v δ ) × B η,ε (u δ ) η,ε . Let us now denote byˆd X =ˆT 0 dτˆR 3 dyˆR 3 dw for simplicity of notations, the first term of (4.17) can be decomposed and rewritten as follows ∇ ξ · T B1 =ˆdX ̺ η (τ )̺ ε (y)∇ w ̺ δ (w) · [(v(ξ − w) − v(ξ)) × B(t − τ, x − y)]u(t − τ, x − y, ξ) +ˆdX ̺ η (τ )̺ ε (y)∇ w ̺ δ (w) · [(v(ξ − w) − v(ξ)) × B(t − τ, x − y)] · [u(t − τ, x − y, ξ − w) − u(t − τ, x − y, ξ)] =: I 1 + I 2 . (4.18) Integrating by parts and since ∇ w · [(v(ξ − w) − v(ξ)) × B(t − τ, x − y)] = 0, one observes that the first term also vanishes: (4.19) and by Hölder inequality and estimate (3.6) in Lemma 3.1, it is easy to obtain that I 1 =ˆdX ̺ η (τ )̺ ε (y)̺ δ (w) ∇ w · [(v(ξ − w) − v(ξ)) × B(t − τ, x − y)]u(t − τ, x − y, ξ) = 0,I 2 L 1,p,r ≤ 2ˆdX ̺ η (τ )̺ ε (y)|∇ w ̺ δ (w)||w| B L ∞,q 3 u(t − τ, x − y, ξ − w) − u(t − τ, x − y, ξ) L 1,p 6 . We then apply the estimate (3.13) in Lemma 3.3, the restriction property for Sobolev spaces W α,p (R n ) and regularity assumptions (2.7), it deduces from the above inequality that I 2 L 1,p,r ≤ CˆR 3 |∇ w ̺ δ (w)||w| α+1 dw B L ∞,q 3 u L 1,p W α,p ≤ Cδ α B L ∞ W β,q 3 u L 1 W α,p 6 . (4.20) It follows easily that from (4.18), (4.19) and (4.20), one has ∇ ξ · T B1 L 1,p,r ≤ Cδ α B L ∞ W β,q 3 u L 1 W α,p 6 . (4.21) To estimate ∇ ξ · T B 2 , we can now proceed analogously to what we have obtained in (4.16) for ∇ ξ · T E , giving ∇ ξ · T B 2 L 1,p,r ≤ Cε α+β δ α−1 B L ∞ W β,q 3ˆT 0 (Θ u(t) (δ) + Θ u(t) (ε))dt ≤ Cε α+β δ α−1 B L ∞ W β,q 3 ω u (ε, δ). (4.22) Hölder inequality is used repeatedly to obtain ∇ ξ · T B 3 L 1,p,r ≤ |v − v δ | B η,ε L ∞,q 3 ∇ ξ u η,δ,ε L 1,p 6 . Applying estimate (3.7) in Lemma 3.1 and Lemma 3.3 to this inequality, we have ∇ ξ · T B 3 L 1,p,r ≤ Cδ B η,ε L ∞,q 3 ∇ ξ u η,δ,ε L 1,p 6 ≤ Cδ α B L ∞ W β,q 3 u L 1 W α,p 6 . (4.23) Gathering estimates (4.21)-(4.23), we obtain from (4.17) that ∇ ξ · T B L 1,p,r ≤ Cε α+β δ α−1 B L ∞ W β,q 3ˆT 0 (Θ u(t) (δ) + Θ u(t) (ε))dt + Cδ α B L ∞ W β,q 3 u L 1 W α,p 6 ≤ C B L ∞ W β,q 3 ε α+β δ α−1 ω u (ε, δ) + δ α u L 1 W α,p 6 . (4.24) Finally, by estimates (4.16) and (4.24), we obtain (4.3) from (4.9) and therefore, the proof of Lemma is then complete. Proof of Theorem 2.2. We firstly use the notation dX =ˆT 0 dtˆR 3 dxˆR 3 dξ for simplicity, the weak formulation for the Vlasov equation (1.1) readŝ dX u(∂ t ϕ + v · ∇ x ϕ + F · ∇ ξ ϕ) = 0, ∀ϕ ∈ D((0, T ) × R 6 ),(4.25) where F = E +v ×B denotes the Lorentz force field. We remark that integrals in (4.25) are finite since for DiPena-Lions weak solutions in [8], it is known that u ∈ L ∞,2 6 and E, B ∈ L ∞,2 3 . For every positive numbers η, ε and δ, let us take the test function in (4.25) as ϕ = (G ′ (u η,ε,δ )ψ) η,ε,δ ∈ D((0, T ) × R 6 ), (4.26) with ψ ∈ D((0, T ) × R 6 ) and G ∈ C 1 (R + ; R + ). By using Lemma 3.3 and successive integrations by parts, we obtain from (4.25) and (4.26) that dX G(u η,ε,δ ) ∂ t ψ + v δ · ∇ x ψ + F η,ε,δ · ∇ ξ ψ + ψG ′ (u η,ε,δ ) ∇ x · (vu) η,ϕ,δ − v δ u η,ε,δ + ∇ ξ · (Fu) η,ε,δ − F η,ε,δ u η,ε,δ = 0,(4.27) for all ψ ∈ D((0, T ) × R 6 ). We now establish the renormalized Vlasov equation (2.3). Using regularity assumptions (2.7), Lemma 3.3, 3.6 and 4.1, we obtain that ˆd X G(u η,ε,δ ) ∂ t ψ + v δ · ∇ x ψ + F η,ε,δ · ∇ ξ ψ ≤ C ε α−1 δ α+1 + ε α+β δ α−1 ω u (ε, δ) + Cδ α , (4.28) where the function ω given in (3.20) and the constant C depends on u L 1 W α,p 6 , B L ∞ W β,q 3 , E L ∞ W β,q 3 , G and ψ. We see that ε α−1 δ α+1 + ε α+β δ α−1 = ε α−1 δ α−1 δ 2 + ε β+1 . Therefore, to balance contributions coming from the free streaming and Lorentz force terms in the right hand side of (4.28), we may choose δ 2 = ε β+1 , which guarantees that ˆd X G(u η,ε,δ ) ∂ t ψ + v δ · ∇ x ψ + F η,ε,δ · ∇ ξ ψ ≤ C ε αβ+β+3α−1 2 ω u (ε, δ) + δ α . Under our general assumption αβ + β + 3α − 1 ≥ 0, we deduce that ˆd X G(u η,ε,δ ) ∂ t ψ + v δ · ∇ x ψ + F η,ε,δ · ∇ ξ ψ ≤ C (ω u (ε, δ) + δ α ) . (4.29) Thanks to Lemma 3.7 and α ∈ (0, 1), the right hand side of (4.29) vanishes as (ε, δ) goes to 0. So we obtain the renormalization property (2.3) of the Vlasov equation. We next establish the local in space entropy conservation law (2.4). For this purpose, we restrict entropy function G ∈ S, this means G is non decreasing function in C 1 (R + ; R + ) such that lim t→∞ G(t) t = ∞. Let us first take a function Γ ∈ D(R 3 ) such that supp(Γ) ⊂ B 2 (0), Γ ≡ 1 on B 1 (0) and 0 ≤ Γ ≤ 1 on B 2 (0) \ B 1 (0), where B r (0) denotes the ball of radius r and centered at 0 in R 3 . Then we introduce a function Γ by Γ R (ξ) = Γ ξ R , with R > 0. It is obvious to check that Γ R ∈ D(R 3 ) and Γ R −→ 1 and ∇ ξ Γ R −→ 0, a.e. as R → ∞. (4.30) Now we choose a test function ψ in (4.29) such that ψ(t, x, ξ) = µ(t, x)Γ R (ξ), with µ ∈ D((0, T ) × R 3 ). By assumption that the map t → u(t, ·, ·) is uniformly integrable in R 6 , for almost everywhere t ∈ [0, T ], and the de La Vallee Poussin theorem, there exists a constant C G only depending on the entropy G such that R 3 dxˆR 3 dξ G(u η,ε,δ ) ≤ C G < ∞. (4.31) Applying the Lebesgue dominated convergence theorem under the estimate (4.31), (4.30) and regularity assumptions (2.7), we obtain that dX G(u η,ε,δ )∂ t µΓ R −→ˆdX G(u η,ε,δ )∂ t µ, as R → ∞, (4.32) dX G(u η,ε,δ )v δ · ∇ x µΓ R −→ˆdX G(u η,ε,δ )v δ · ∇ x µ, as R → ∞, (4.33) andˆd X G(u η,ε,δ )F η,ε,δ L u η,ε,δ · ∇ ξ Γ R µ −→ 0, as R → ∞. , µ and Γ such that ˆd X G(u η,ε,δ )F η,ε,δ u η,ε,δ · ∇ ξ Γ R µ ≤ c 1 R −1 . Gathering estimates (4.32)-(4.35), one obtains from (4.29) that ˆd X ∂ t µ + v δ · ∇ x µ G(u η,ε,δ ) ≤ C(ω u (ε, δ) + δ α ) + c 1 R −1 ,(4.36) under the assumption αβ + β + 3α − 1 ≥ 0. Thanks to Lemma 3.7 again, the right hand side of (4.36) vanishes as (ε, δ) → 0 and R → ∞. It deduces that the local in space conservation law (2.4) holds. The local momentum conservation law (2.5) can be obtained in a similar way. The final task is now to establish the global entropy conservation law (2.6). Let us take a test function µ in (4.36) such that µ(t, x) = σ(t)Γ R (x), with σ ∈ D((0, T )), where Γ R is defined the the previous proof for local conservation laws. By (4.31)-(4.30) and regularity assumption (2.7), we may apply the Lebesgue dominated convergence theorem to obtain that dX G(u η,ε,δ )∂ t σΓ R −→ˆdX G(u η,ε,δ )∂ t σ, as R → ∞, (4.37) dX G(u η,ε,δ )v δ · ∇ x Γ R σ −→ 0, as R → ∞. Limits (4.37)-(4.38) are uniform in (η, ε, δ) and there exists a constant c 2 > 0 only depending on C G , σ and Γ such that ˆd X G(u η,ε,δ )v δ · ∇ x Γ R σ ≤ c 2 R −1 . (4.39) Combining between (4.36) to (4.37)-(4.39), we obtain that ˆd X ∂ t σG(u η,ε,δ ) ≤ C(ω u (ε, δ) + δ α ) + (c 1 + c 2 )R −1 , (4.40) under the condition αβ+β+3α−1 ≥ 0. The global entropy conservation law (2.5) holds since the right hand side of (4.40) vanishes as (ε, δ) → 0 and R → ∞ by Lemma 3.7. The proof of Theorem 2.2 is complete. s equations must be solved together with the Vlasov equation (1.1), socalled the Vlasov-Maxwell system. Here, we are interested in the Cauchy problem for system (1.1)-(1.5), where the initial data given as u(0, x, ξ) = u 0 (x, ξ) ≥ 0, (1.6) E(0, x) = E 0 (x), B(0, x) = B 0 (x), (1.7) Theorem 2. 2 2Let (u, E, B) be a weak solution of the relativistic Vlasov-Maxwell system (1.1)-(1.5) given by Theorem 2.1. Assume moreover that this weak solution satisfies the additional regularity assumptions Lemma 3. 1 1Let δ > 0 and v be the relativistic velocity given by (1.2). Then we have the following estimations Lemma 4. 1 1Let (u, E, B) be a weak solution of the relativistic Vlasov-Maxwell system (1.1)-(1.5) given by Theorem 2.1, satisfying the regularity assumptions (2.7) of Theorem 2.2, with α, β ∈ (0, 1) and p, q, r satisfy relations (2.8). Then for any positive numbers η, ε, δ > 0, there exists a constant C > 0 depending only on the smooth function ̺ given by (3.1) such that 4.32)-(4.34) are uniform in (η, ε, δ) and there exists a constant c 1 only depending on u L or [7, Proof of Theorem 2.4], or in[2]. Sobolev spaces. R A Adams, Academic PressNew YorkR.A. Adams, Sobolev spaces, Academic Press, New York, 1975. S Alinhac, P Gerard, Opérateurs pseudo-différentiels et théoreme de Nash-Moser, Savoir Actuels, InterEditions et Editions du CNRS. S. Alinhac, P. Gerard, Opérateurs pseudo-différentiels et théoreme de Nash-Moser, Savoir Actuels, InterEditions et Editions du CNRS, 1991. C Bardos, N Besse, Toan T Nguyen, arXiv:1903.04878v2Onsager type conjecture and renormalized solutions for the relativistic Vlasov-Maxwell system. C. Bardos, N. Besse, Toan T. Nguyen, Onsager type conjecture and renormalized solutions for the relativistic Vlasov-Maxwell system, arXiv:1903.04878v2. On the extension of Onsager's conjecture for general conservation laws. C Bardos, P Gwiazda, A E S Swiercrewska-Gwiazda, E Titi, Wiedemann, 10.1007/s00332-018-9496-4J. Nonlinear Sci. C. Bardos, P. Gwiazda, A. Swiercrewska-Gwiazda. E.S. Titi, E. Wiedemann, On the extension of Onsager's conjecture for general conservation laws, J. Nonlinear Sci. (2018), https://doi.org/10.1007/s00332-018-9496-4. Regularity of weak solutions for the Vlasov-Maxwell system. N Besse, P Bechouche, J. Hyperbolic Diff. Equ. 15N. Besse, P. Bechouche, Regularity of weak solutions for the Vlasov-Maxwell sys- tem, J. Hyperbolic Diff. Equ. 15 (2018), 693-719. Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO, Milan. D S Mc Cormick, J C Robinson, J L Rodrigo, J. Math. 81D.S. Mc Cormick, J.C. Robinson, J.L. Rodrigo, Generalised Gagliardo-Nirenberg inequalities using weak Lebesgue spaces and BMO, Milan J. Math. 81 (2013), 265- 289. Generalized Gagliardo-Nirenberg inequalities using Lorentz spaces, BMO, Hölder spaces and fractional Sobolev spaces. N.-A Dao, J.-I Díaz, Q.-H Nguyen, Nonlinear Analysis. 173N.-A. Dao, J.-I. Díaz, Q.-H. Nguyen, Generalized Gagliardo-Nirenberg inequalities using Lorentz spaces, BMO, Hölder spaces and fractional Sobolev spaces, Nonlinear Analysis 173 (2018), 146-153. Global weak solutions of Vlasov-Maxwell systems. R J Diperna, P.-L Lions, Commun. Pure Appl. Math. 42R.J. DiPerna, P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Com- mun. Pure Appl. Math. 42 (1989), 729-757. Global weak solutions of the Vlasov-Maxwell system with boundary conditions. Y Guo, Commun. Math. Phys. 154Y. Guo, Global weak solutions of the Vlasov-Maxwell system with boundary condi- tions, Commun. Math. Phys. 154 (1993), 245-263. Regularity for the Vlasov equation on a half space. Y Guo, Indiana Univ. Math. J. 43Y. Guo, Regularity for the Vlasov equation on a half space, Indiana Univ. Math. J. 43 (1994), 255 -320. A note on weak-solutions of conservation laws and energy/entropy conservation. P Gwiazda, M Michalek, A Swierczewska-Gwiazda, Arch. Rational Mech. Anal. 229P. Gwiazda, M. Michalek, A. Swierczewska-Gwiazda, A note on weak-solutions of conservation laws and energy/entropy conservation, Arch. Rational Mech. Anal. 229 (2018), 1223-1238. Global weak solutions to the relativistic Vlasov-Maxwell system revisited. G Rein, Commun. Math. Sci. 2G. Rein, Global weak solutions to the relativistic Vlasov-Maxwell system revisited, Commun. Math. Sci. 2 (2004), 145-158. Theory of function spaces I, Birkhuser Basel. H Triebel, H. Triebel, Theory of function spaces I, Birkhuser Basel, 1983. The structure of functions. H Triebel, Birkhuser BaselH. Triebel, The structure of functions, Birkhuser Basel, 2001.

Spacelike and timelike generalized parton distributions (GPDs) have been investigated in chargedlepton scattering and electron-positron collisions via deeply virtual Compton scattering and twophoton processes, respectively. Furthermore, we expect that hadron-accelerator-facility measurements will be performed in future. The GPDs will play a crucial role in clarifying the origins of hadron spins and masses in terms of quarks and gluons. It is also possible to probe internal pressure within hadrons for understanding their stability. Gravitational form factors of hadrons used to be considered as a purely academic subject because gravitational interactions are too weak to be measured in microscopic systems. However, due to the development of hadron-tomography field, it became possible to extract the gravitational form factors from the actual GPD measurements without relying on direct gravitational interactions. Neutrino reactions can also be used for GPD studies in future, for example, by using the Long-Baseline Neutrino Facility at Fermilab. The neutrino GPD measurements are valuable especially for finding the flavor dependence of the GPDs in a complementary way to the charged-lepton experiments. We give an overview of the GPDs and discuss possible neutrino GPD measurements using the single-pion production processes + → ℓ − + ′ + and¯ + → ℓ + + ′ + .

10.22323/1.402.0092

2203.00848

aa63186c2f4a59b1c4178e7fa01f45719883381e

Possible studies on generalized parton distributions and gravitational form factors in neutrino reactions Possible studies on GPDs and gravitational form factors in neutrino reactions 2 Mar 2022 S Kumano R Petti S Kumano KEK Theory Center Institute of Particle and Nuclear Studies J-PARC Branch KEK Theory Center Institute of Particle and Nuclear Studies KEK, and Theory Group, Particle and Nuclear Physics Division KEK Oho 1-1305-0801TsukubaIbarakiJapan Department of Physics and Astronomy J-PARC Center 203-1, 319-1106ShirakataTokai, IbarakiJapan University of South Carolina 29208ColumbiaSouth CarolinaUSA Possible studies on generalized parton distributions and gravitational form factors in neutrino reactions Possible studies on GPDs and gravitational form factors in neutrino reactions 2 Mar 2022* Speaker Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Spacelike and timelike generalized parton distributions (GPDs) have been investigated in chargedlepton scattering and electron-positron collisions via deeply virtual Compton scattering and twophoton processes, respectively. Furthermore, we expect that hadron-accelerator-facility measurements will be performed in future. The GPDs will play a crucial role in clarifying the origins of hadron spins and masses in terms of quarks and gluons. It is also possible to probe internal pressure within hadrons for understanding their stability. Gravitational form factors of hadrons used to be considered as a purely academic subject because gravitational interactions are too weak to be measured in microscopic systems. However, due to the development of hadron-tomography field, it became possible to extract the gravitational form factors from the actual GPD measurements without relying on direct gravitational interactions. Neutrino reactions can also be used for GPD studies in future, for example, by using the Long-Baseline Neutrino Facility at Fermilab. The neutrino GPD measurements are valuable especially for finding the flavor dependence of the GPDs in a complementary way to the charged-lepton experiments. We give an overview of the GPDs and discuss possible neutrino GPD measurements using the single-pion production processes + → ℓ − + ′ + and¯ + → ℓ + + ′ + . Introduction Unpolarized structure functions 2 and 3 of the nucleon were measured by neutrino deep inelastic inelastic scattering (DIS) from heavy nuclei with appropriate nuclear corrections [1]. These structure functions are expressed by collinear parton distribution functions (PDFs), which indicate longitudinal momentum distributions of partons. In recent years, three-dimensional structure functions have been investigated extensively for clarifying the transverse structure of the nucleon in addition to the longitudinal distributions and for understanding the origin of the nucleon spin including the partonic orbital-angular-momentum (OAM) contribution. The OAM contribution should be determined by generalized parton distributions (GPDs) [2], which have been measured by deeply virtual Compton scattering (DVCS) and meson productions at charged-lepton accelerator facilities in the spacelike region. The DVCS has been investigated by the HERMES and COMPASS collaborations and also at the Thomas Jefferson National Accelerator Facility (JLab). In 2030's, it will be investigated at electron-ion colliders in US and China (EIC, EicC) [3]. There are also possibilities of measuring the GPDs at hadron facilities by using high-energy exclusive reactions such as at the Japan Proton Accelerator Research Complex (J-PARC) [4]. All of these are spacelike GPD studies, whereas it is possible to investigate the timelike GPDs [5], which are also called generalized distribution amplitudes (GDAs), by two-photon processes in + − annihilation, for example, at the KEK-B factory. There is another important purpose to investigate the GPDs for understanding hadron masses and their internal pressures in terms of quark and gluon degrees of freedom [5]. The studies on the origin of hadron masses by hadron-mass decomposition are now becoming one of major purposes for building the future EICs. The GPD measurements have been done mainly at chargedlepton accelerator facilities, and there is no GPD measurement in neutrino reactions at this stage. However, we may recollect that the neutrino DIS experiments have been important in determining the unpolarized PDFs, especially on the strange-quark distribution via the opposite-sign dimuon events and valence-quark distributions via the structure function 3 . Considering these past experiences, we expect that future neutrino experiments could provide valuable information on the GPDs in a complementary way to the charged-lepton and hadron-facility measurements [3]. The Long-Baseline Neutrino Facility (LBNF) at Fermilab can supply (anti)neutrino beams in the energy region of 2-15 GeV [6] allowing to measure the GPDs, for example, by the pion-production reaction + → + + ′ [7][8][9]. In general, (anti)neutrino Charge Current (CC) interactions are sensitive to the quark flavor, offering a valuable tool to study the flavor dependence of the GPDs together with charged-lepton data, and to investigate the origin of hadron spins and masses. In this article, we discuss such a possibility. Generalized parton distribution functions The spacelike GPDs of the nucleon are measured, for example, by the deeply virtual Compton scattering (DVCS) at charged-lepton accelerator facilities as shown in Fig. 1. The photon momenta are and ′ , and the nucleon momenta are and ′ . We define average momenta (¯ ,¯ ) and momentum transfer Δ as¯ = ( + ′ )/2,¯ = ( + ′ )/2, and Δ = ′ − = − ′ . Three variables for expressing the GPDs are the Bjorken variable , the skewness parameter , and the momentum-transfer squared are defined by = 2 /(2 · ), =¯ 2 /(2¯ ·¯ ), and = Δ 2 . By the lightcone momentum notations, and are expressed as = + / + and = −Δ + /(2 + ) with = + ′ . If the kinematical condition 2 ≫ | |, Λ 2 QCD , where Λ QCD is the QCD scale parameter, is satisfied, the DVCS process is factorized into the hard part and the soft one by the GPDs and defined in the matrix element ∫ − 4 + − ′ ¯ (− /2) + ( /2) + = ì ⊥ =0 = 1 2 +¯ ( ′ ) ( , , ) + + ( , , ) + Δ 2 ( ). (1) In pion-production and neutrino cross sections, there are other GPDs˜ and˜ associated with the matrix element of the axial-vector current as ∫ − 4 + − ′ ¯ (− /2) + 5 ( /2) + = ì ⊥ =0 = 1 2 +¯ ( ′ ) ˜ ( , , ) + 5 +˜ ( , , ) 5 Δ + 2 ( ). (2) The unique features of the GPDs are that they become the unpolarized and longitudinally-polarized PDFs in the forward limit: ( , 0, 0) = ( ),˜ ( , 0, 0) = Δ ( ), that their first moments are the corresponding form factors: ∫ 1 −1 ( , , ) = 1 ( ), ∫ 1 −1 ( , , ) = 2 ( ), ∫ 1 −1 ˜ ( , , ) = ( ), ∫ 1 −1 ˜ ( , , ) = ( ), and that the second moment is the quark contribution to the nucleon spin: = ∫ [ ( , , = 0) + ( , , = 0)]/2 = Δ + /2+ . Here, is a quark orbital-angular-momentum contribution ( ) to the nucleon spin. Since we know the quark-spin contribution Δ + from experimental measurements, it is possible to determine from the GPD measurements. The timelike GPDs are often called the GDAs, and they are measured by the -crossed process of the DVCS, so called the two-photon process, as shown in Fig. 2. The GDAs or timelike GPDs are defined by the matrix element similar to Eqs. (1) and (2) between the vacuum and the final hadron pair ℎh [5]. For example, they are defined for the 0 pair as Φ 0 0 ( , , 2 ) = ∫ − 2 (2 −1) + − /2 0 ( ) 0 ( ′ ) |¯ (− /2) + ( /2) | 0 + = ì ⊥ =0 .(3) The GDAs are expressed by three variables, the momentum fractions and in Fig. 2 and the invariant-mass squared 2 as = + / + , = + / + = (1 + cos )/2, and 2 = , where is defined by = | ì |/ 0 = 1 − 4 2 / 2 , and is the scattering angle in the center-of-mass frame of the final pions. The two-photon process is factorized if the condition 2 ≫ 2 , Λ 2 QCD is satisfied to express it in terms of the GDAs. The corresponding spacelike GPDs for the pion are given as 0 ( , , ) = ∫ − 4 ¯ + − 0 ( ′ ) ¯ (− /2) + ( /2) 0 ( ) + = ì ⊥ =0 .(4) The spacelike and timelike GPDs are related with each other by the -crossing as Φ 0 0 ( ′ , , 2 ) Electromagnetic and weak form factors of hadrons and nuclei have been measured in lepton scatterings, whereas gravitational form factors used to be considered purely theoretical quantities until recently, because gravitational interactions are too weak to be used for measuring interactions with microscopic particles. However, it became possible to measure them without direct gravitational interactions by hadron tomography techniques as illustrated in Fig. 3. In order to understand why the gravitational form factors can be obtained from electromagnetic and weak interactions, we consider moments of the nonlocal operators in Eqs. (1), (2), (3), and (4) as ↔ 0 = (1 − 2 ′ )/(1 − 2 ), = 1/(1 − 2 ), = 2 . Gravitational form factors of hadrons from spacelike and timelike GPDs + 2 ∫ −1 ∫ − 2 + − /2¯ (− /2) + ( /2) + = ì ⊥ =0 =¯ (0) + ← → + −1 (0), (5) where ← → is defined by 1 ← → 2 = [ 1 ( 2 ) − ( 1 ) 2 ]/2. For = 1, it is the ordinary vector-type electromagnetic current; however, we notice that the operator is the energy-momentum tensor of a quark for = 2 [5]. It indicates that the GPDs contain the information on the gravitational form factors. Therefore, the second moments of the spacelike and timelike GPDs are given by the matrix elements of the energy-momentum tensor , and they are expressed by the spacelike and timelike gravitational form factors Θ 1 and Θ 2 as ∫ 1 −1 0 ( , , ) = 1 ( + ) 2 0 ( ′ ) | ++ (0) | 0 ( ) = 1 2 ( + ) 2 ++ − + + Θ 1, ( ) + + + Θ 2, ( ) ,(6)∫ 1 0 (2 − 1) Φ 0 0 ( , , 2 ) = 2 ( + ) 2 0 ( ) 0 ( ′ ) | ++ (0) | 0 = 1 ( + ) 2 ++ − + + Θ 1, ( ) + + + Θ 2, ( ) .(7) Here, the energy-momentum tensor is given by ( ) = ( ) ( ( ′ ) ∫ 1 −1 ( , , ) + + ∫ 1 −1 ( , , ) + Δ 2 ( ) = 1 + ( ′ ) | ++ (0) | ( ) = 1 +¯ ( ′ ) ( ) +¯ + + ( )¯ + + Δ 2 + ( ) Δ + Δ + − ++ Δ 2 +¯ ( ) ++ ( ).(8) Therefore, these various GPD measurements enable extraction of the gravitational form factors of hadrons [5] and also clarifications of the origins of hadron spins and masses. Figure 4: GPDs in neutrino scattering. GPDs in neutrino reactions Pion-production cross section and GPDs in neutrino reactions GPD ′ N N π W ν µ µ g q There have been a number of works on the GPDs in neutrino reactions [7,8]. Instead of introducing all of these results, we explain the formalism and numerical results by Pire, Szymanowski, and Wagner [8] as a recent work in the following. Instead of the virtual-Compton-like process, it is appropriate to rely on larger meson-production cross sections in neutrino reactions. For example, the pion production process → ℓ − ′ is expressed by a typical subprocess with the GPDs in Fig. 4. In the pion production, quark and gluon GPD contributions to the amplitude are generally given as [8] = − 2 ¯ ( ′ ) H / −H / 5 + E Δ 2 −Ẽ 5 Δ · 2 ( ), = − 2 ¯ ( ′ ) H / + E Δ 2 ( ).(9) Here, charged-current reactions probe the difference between the and -quark GPDs: ( , , ) = ( , , ) − (− , , ), where = ,˜ , ,˜ . The and are coupling constants with color and flavor factors defined as = 2 /3 with = 4/3, and = /3 with = 1/2. The momentum factor is defined as 2 = − 2 by the momentum transfer , / is given by / ≡ = + with = (1, 0, 0, −1)/ √ 2, Δ is Δ = Δ , and is the nucleon mass. The functions F and F are defined by including the pion distribution amplitude as F = 2 ∫ 1 0 ( ) 1 − ∫ 1 −1 ( , , ) − + , F = 8 ∫ 1 0 ( ) (1 − ) ∫ 1 −1 ( , , ) − + ,(10) where is the pion decay constant. In terms of these GPDs, the cross section is written as 4 → − ′ 2 = 2 2 32(2 ) 4 (1 − ) ( − 2 ) 2 1 + 4 2 2 / 2 , = 1 2 [ | H¯ + H | 2 + | H¯ | 2 ] (1 − 2 ) + 4 1 − 2 [ | E¯ + E | 2 + | Ẽ¯ | 2 ] − 2 2 R [ H¯ + H ] [ E¯ + E ] * − 2 2 R [ H¯ ] [ Ẽ¯ ] * ,(11) where = · / · , 2 = ( − 2 ), and ≃ (1 − )/(1 − + 2 /2). The longitudinal crosssection is defined by the hadron tensor and the photon-polarization vector as = * . The obtained cross sections are shown in Figs. 5 and 6 for the + and 0 productions, respectively, at = 20 GeV 2 . As shown in these figures, both quark and gluon processes contribute to the + production and the gluon contribution is much larger, whereas there is no contribution to the 0 production from the gluon GPD. Therefore, the neutrino GPD measurement is valuable for clarifying the quark and gluon GPDs and the flavor dependence in the quark GPDs. → ℓ − 0 cross section at = 20 GeV 2 [8]. The solid, dashed, and dotted curves indicate the corss sections at = 0.7, 0.5, and 0.3, respectively. There is no gluon contribution for the 0 production. Opportunities for GPD Measurements at LBNF The future LBNF at the Fermi National Laboratory will deliver neutrino and antineutrino beams of unprecedented intensity with broad energy spectra. In addition to the default beam optimized for long-baseline oscillation measurements in the energy range 0.5-5 GeV, a higher energy option (mostly in the 2-15GeV range) is possible for precision measurements and searches for new physics beyond the Standard Model. The near detector complex of the Deep Underground Neutrino Experiment (DUNE) will include a high resolution on-axis detector which can address some of main limitations of (anti)neutrino experiments providing an accurate control of the targets and fluxes [10]. In particular, it will allow precision measurements of and¯ interactions on both hydrogen (H) and various nuclear targets (A) in combination with the high intensity and the energy spectra of the LBNF beams. The kinematic coverage is dominated by inelastic interactions -more than 54% of the events with the default low energy beam and most of the events with the high energy option have > 1.4 GeV -offering a good sensitivity to the GPD measurements via the pion production processes (¯ ) → ′ . The availability of a free proton target H will give access to high statistics measurements of the following channels: (a) → − + ; (b)¯ → + − ; (c)¯ → + 0 . While both quark and gluon GPDs contribute to the ± production, no gluon contribution is present for the 0 production. The last two measurements with also provide information about the GPDs in a free neutron target since it is expected (¯ → + − ) = ( → − + ) and (¯ → + 0 ) = ( → − 0 ) [8]. The study of the flavor dependence of the GPDs in free nucleons will be complemented by similar measurements performed simultaneously on a variety of nuclear targets (C, Ar, etc.) within the same detector. A comparison between measurements on H and on the nuclear targets can provide valuable information about the nuclear modifications of the GPDs. The nuclear targets will also extend the study of the flavor dependence of the GPDs by giving access to additional channels: (a) → − + ; (b) → − 0 ; (c)¯ → + − . A sizable statistics for the various single pion production channels is expected to be collected from both H and nuclear targets with the default low energy LBNF beam [10,11]. The high energy beam options will significantly enhance the sensitivity to the GPD measurements increasing the kinematic overlap with complementary EIC measurements. Summary The GPD studies will be crucial in understanding the origins of hadron spins and masses in terms of quarks and gluons. The spacelike GPDs are measured in charged-lepton scattering processes, deeply virtual Compton scattering and meson productions, and timelike GPDs are investigated by two-photon processes. Using the LBNF neutrino beam, we can access the spacelike GPDs in neutrino reactions. As the neutrino DIS measurements played an important role in establishing the flavor-dependence of the PDFs and the valence-quark distribution functions, the neutrino GPD measurements should be complementary to the charged-lepton ones. The high resolution on-axis detector in the DUNE near detector complex will be capable of detailed GPD studies on both free protons and nuclei in future, providing insights on the hadron spins and masses. GPDFigure 1 :Figure 2 : 12Spacelike Timelike GPDs (GDAs) in two-photon process. Figure 3 : 3µ q vector − axial-vector qγ µ (1 − γ 5 )q tensor qγ µ ∂ ν q g Neutrino reactions DVCS, two-photon process, · · · Gravitational form factors from electromagnetic and weak interactions. the covariant derivative = − , /2 with the QCD coupling constant and the SU(3) Gell-Mann matrix . For the spin-1/2 nucleons, the spacelike GPDs are related to the gravitational form factors , , , and¯ in the same way as Figure 5 : 5→ ℓ − + cross section at = 0.7 and = 20 GeV 2[8]. The dashed and dotted curves indicate gluon and quark contributions, respectively. The solid curve is their summation. Figure 6 : 6Figure 6: → ℓ − 0 cross section at = 20 GeV 2 [8]. The solid, dashed, and dotted curves indicate the corss sections at = 0.7, 0.5, and 0.3, respectively. There is no gluon contribution for the 0 production. Acknowledgments . See M For Example, NuTeV collaborationTzanov, NuTeV collaborationPhys. Rev. D. 7412008For example, see M. Tzanov et al. (NuTeV collaboration), Phys. Rev. D 74, 012008 (2006). For an introductory review, see M. Diehl. Phys. Rept. 38841For an introductory review, see M. Diehl, Phys. Rept. 388, 41 (2003). R , Abdul Khalek, arXiv:2103.05419see Sec. 7.5.2 Neutrino physics for GPD studies in neutrino reactions. R. Abdul Khalek et al., arXiv:2103.05419, see Sec. 7.5.2 Neutrino physics for GPD studies in neutrino reactions. . S Kumano, M Strikman, K Sudoh, Phys. Rev. D. 8074003S. Kumano, M. Strikman, and K. Sudoh, Phys. Rev. D 80, 074003 (2009); . T Sawada, Phys. Rev. D. 93114034T. Sawada et al., Phys. Rev. D 93, 114034 (2016). . S Kumano, Qin-Tao Song, O V Teryaev, Phys. Rev. D. 9714020S. Kumano, Qin-Tao Song, and O. V. Teryaev, Phys. Rev. D 97, 014020 (2018). . J Rout, Phys. Rev. D. 102116018J. Rout et al., Phys. Rev. D 102, 116018 (2020). . B Lehmann-Dronke, A Schafer, Phys. Lett. B. 52155B. Lehmann-Dronke and A. Schafer, Phys. Lett. B 521, 55 (2001); . P Amore, C Coriano, M Guzzi, J. High Energy Phys. 0238P. Amore, C. Coriano, and M. Guzzi, J. High Energy Phys. 02, 038 (2005); . C Coriano, M Guzzi, Phys. Rev. D. 7153002C. Coriano and M. Guzzi, Phys. Rev. D 71, 053002 (2005); . A Psaker, W Melnitchouk, A V Radyushkin, Phys. Rev. D. 7554001A. Psaker, W. Melnitchouk, and A. V. Radyushkin, Phys. Rev. D 75, 054001 (2007); . G R Goldstein, O G Hernandez, S Liuti, T Mcaskill, AIP Conf. Proc. 1222248G. R. Goldstein, O. G. Hernandez, S. Liuti, and T. McAskill, AIP Conf. Proc. 1222, 248 (2010); . B Z Kopeliovich, I Schmidt, M Siddikov, Phys. Rev. D. 86113018B. Z. Kopeliovich, I. Schmidt, and M. Siddikov, Phys. Rev. D 86, 113018 (2012); . B Pire, L Szymanowski, Phys. Rev. Lett. 11592001B. Pire and L. Szymanowski, Phys. Rev. Lett. 115, 092001 (2015); . M Siddikov, I Schmidt, Phys. Rev. D. 9513004M. Siddikov and I. Schmidt, Phys. Rev. D 95, 013004 (2017); . B Pire, L Szymanowski, J Wagner, Phys. Rev. D. 9594001B. Pire, L. Szymanowski, and J. Wagner, Phys. Rev. D 95, 094001 (2017). . B Pire, L Szymanowski, J Wagner, Phys. Rev. D. 95114029B. Pire, L. Szymanowski, and J. Wagner, Phys. Rev. D 95, 114029 (2017). . S Kumano, EPJ Web Conf. 2087003S. Kumano, EPJ Web Conf. 208, 07003 (2019). R Petti, PoS (DIS2019) 235, Proceedings of of the 27th International Workshop on Deep Inelastic Scattering and Related Subjects. Torino, ItalyR. Petti, PoS (DIS2019) 235, Proceedings of of the 27th International Workshop on Deep Inelastic Scattering and Related Subjects, Torino, Italy, April 8-12, 2019. . H Duyang, B Guo, S R Mishra, R Petti, arXiv:1809.08752hep-phH. Duyang, B. Guo, S. R. Mishra, and R. Petti, arXiv:1809.08752 [hep-ph].

Propagation of the surface waves on the interface between an uniaxial hyperbolic material and an isotopical topological insulator is studied. The cases of the anisotropy axes is normal to interface or one is coplanar to interface are discussed. The dispersion relations are derived and analyzed. The conditions of the existence of the surface waves are established.

1706.05951

5a5ae95501418466c6561bbbc983b2f995f4e43e

Surface Waves on the Interface Between Hyperbolic Material and Topological Insulator 19 Jun 2017 Ekaterina I Lyashko Department of General and Applied Physics Moscow Institute of Physics and Technology Moscow region141700DolgoprudnyRussia Andrei I Maimistov s:[email protected] Department of Solid State Physics and Nanostructures Engineering Physics Institute National Nuclear Research University 115409Moscow, MoscowRussia Department of General Physics, Moscow Institute of Physics and Technology Dolgoprudny, Moscow region141700 Russia Ildar R Gabitov s:[email protected] Department of Mathematics University of Arizona 85721TucsonArizonaUSA Skolkovo Institute of Science and Technology Skolkovo Innovation Center 143026MoscowRussia Surface Waves on the Interface Between Hyperbolic Material and Topological Insulator 19 Jun 2017(Dated: June 20, 2017)arXiv:1706.05951v1 [physics.optics]numbers: 4282-m4282Et4279Gn7867Pt4225Bs Propagation of the surface waves on the interface between an uniaxial hyperbolic material and an isotopical topological insulator is studied. The cases of the anisotropy axes is normal to interface or one is coplanar to interface are discussed. The dispersion relations are derived and analyzed. The conditions of the existence of the surface waves are established. I. INTRODUCTION During last several years considerable attention of scientific community was attracted to new materials known in condensed matter physics as topological insulators (TI) [1][2][3]. Electrons in TI are mobile only on the surface of the material and surface states are protected by time reversal symmetry and conservation law for number of particles. Crystals of Bi 2 Te 3 , Bi 2 Se 3 , Sb 2 Te 3 doped with Fe are representing examples of such material. Investigation of TI [4,5], as well as study of optical/electrodynamic characteristics of these materials become a challenging problem of optics. Magneto-electric effect [6,8,[10][11][12]14], which manifests itself as a giant Faraday and Kerr effect [13][14][15], is a characteristic feature of the interaction of electromagnetic field with TI. Refraction and electromagnetic wave scattering at the surface of TI are considered in [4,6,16,17]. Results of the study of Goos-Hänchen and Fedorov's shifts on the surface of TI are presented in the papers [4,18]. It is known that surface waves do not exist on the interface of topological insulator and conventional dielectric [19]. However, this limitation can be removed by replacing of conventional dielectric on other materials. For example surface wave in the form of surface plasmon can propagate along the interface of the topological insulator and metal [19]. Surface waves can also exist on the interface of topological insulator and metamaterial with negative index of refraction. These examples indicate that presence of surface waves can be achieved by use of material with negative dielectric permittivity ε < 0. Hyperbolic materials [20,21,23,26,27] represent the broad class of such metamaterials [29][30][31]. They have a strong uniaxial anisotropy -principal dielectric permittivities have opposite signs. The result of this uniaxial anisotropy is a hyperbolic shape in a k-space of the surface corresponding to a constant frequency ω( k) = const (the iso-frequency surface). Note that this shape is an ellipsoid for conventional dielectrics This paper considers propagation of the surface wave along the interface between two homogeneous media: the uniaxial hyperbolic material and topological insulator. The principal equations describing this wave are presented in Sec. II. In accordance with planar symmetry of considered problem, electromagnetic waves can be studied separately as TE or TM waves depending on their polarization. However, TE and TM waves are mixing at the interface due to nontrivial boundary conditions. In other words, the wave propagating along the interface is the result of hybridization of TE and TM waves. The dispersion relation for surface wave is derived in Sec. III. In the particular case, when the constant of magnetoelectric interaction is set to zero, the dispersion relation transforms to usual dispersion relations separately for each type of polarization. II. MODEL OF THE INTERFACE AND THE PRINCIPAL EQUATIONS In the absence of free charges and currents the macroscopic equations describing the electromagnetic field in TI, take the following form [7,8] rot H − 1 c ∂D ∂t = −α ∇θ × E + 1 c ∂θ ∂t B , rot E + 1 c ∂B ∂t = 0,(1)div D = α(∇θ · B), div B = 0, where α is equal to e 2 / c and denotes the coupling constant of electromagnetic field with the axion field θ [7]. The product αθ plays a role of the magnetoelectrical susceptibility [8]. In these equations vectors D = E + 4πP and H = B − 4πM in the absence of topological effects are equivalent to vectors of electric and magnetic inductions respectively. They specify the electromagnetic waves propagation in the conventional dielectric where electromagnetic wave dynamic is governed by the following system of equations rot E = − 1 c ∂B ∂t , div B = 0,(2)rot H = 1 c ∂D ∂t , div D = 0. It should be pointed out that if the electric and magnetic inductions are defined as follows D a = D − α θB, H a = H + α θE,(3) then the Maxwell equations (1) are taking form similar to the system of equations (2) rot E = − 1 c ∂B ∂t , div B = 0,(4)rot H a = 1 c ∂D a ∂t , div D a = 0. However, if θ is constant the Maxwell equations (1) does not contain the additional terms. The system of equations (1) (as well as (4)) is completely equivalent to the conventional Maxwell equations (2). In this case the electric and magnetic fields into TI can be derived from the Maxwell equations (2). The continuity conditions on the interface TI-dielectric contain the normal components of the inductions D a and H a because they are result of (4). The electromagnetic waves in an anisotropic medium are described by the Maxwell equations (2), where the vector of electric induction D depends only on (and noncollinear with) the E. In the case of nonmagnetic media the magnetic field and magnetic induction are equivalent. Fourier transform of the the electric induction vector in the case of homogeneous uniaxial anisotropic medium has the following form D = ε o E + (ε e − ε o ) (l · E)l.(5) Here l is the unit vector determined by the optical axis, ε 0 = ε 0 (ω) is the permittivity for ordinary wave, and ε e = ε e (ω) is the permittivity for extraordinary wave. If the sings of the principal permittivities are opposite, then the iso-frequency surface is a hyperboloid: the single sheeted hyperboloid under condition ε e > 0, ε o < 0 and two sheeted hyperboloid under condition ε e < 0, ε o > 0. Such materials are referred to the hyperbolic metamaterial [21,23,26,27]. Both equations for the TI and the hyperbolic material can be separately solved using standard methods. Obtained solutions must be matched on the interface. The matching procedure is determined by continuity condition for tangent components of the electric fields and the normal components of the inductions [32]. It is important to remember that these continuity conditions follows from the system of equations (4). Let us assume that coordinate axis X is directed along the vector n normal to the interface and axes Y and Z are directed along two orthogonal vectors t y and t z which are tangential to this interface. Let the region x < 0 is filled by the hyperbolic material and the topological insulator fills the region x > 0, see Fig. 1 . Under these conditions the continuity conditions take the form (D (1) − D (2) ) · n = −αθB (1) · n, (H (1) − H (2) ) · t z,y = αθE (1) · t z,y ,(6)(B (1) − B (2) ) · n = 0, (E (1) − E (2) ) · t z,y = 0. Here the upper index 1 marks the region x < 0, and upper index 2 marks the region x > 0. For TI θ is known to be θ = 2n + 1, where n ∈ Z [8] and θ = 0 is for dielectric. Surface wave propagating along Z does not depend on variable y due to translation symmetry (see Fig 1). In this case the system of Maxwell equations reads as ik 0 H x = − ∂E y ∂z , ik 0 H z = ∂E y ∂x , −ik 0 D y = ∂H x ∂z − ∂H z ∂x ,(7)ik 0 D x = ∂H y ∂z , ik 0 D z = − ∂H y ∂x , ik 0 H y = ∂E x ∂z − ∂E z ∂x ,(8) In the hyperbolic medium at x < 0 the relation between components of the induction vector and components of the electric field vector depends on the orientation of the optical axis l. Two particular cases for this orientation will be considered further. III. DISPERSION RELATIONS. ELECTRIC AND MAGNETIC FIELD DISTRIBUTIONS Direction of the optical axis in hyperbolic medium is determined by the method of its fabrication. In the most cases such materials are structured as alternating flat sheets of metal and dielectric [24,25,27,28] and as 3D array of metallic rods in dielectric matrix [20][21][22]. According to theses the method of the hyperbolic medium fabrication optical axis are orthogonal or tangential to the media interface. Both cases cases will be analyzed . A. Optical axis is tangential to the interface. In the domain occupied by anisotropic medium, when optical axis is tangential to the interface and orthogonal to propagation direction l = t y , components of the induction vector take the following form D x = ε o E x , D y = ε e E y , D z = ε o E z . The first group of equations from the (7), that can be named as equations of the TE-type, takes the following form ∂ 2 E y ∂z 2 + ∂ 2 E y ∂x 2 + k 2 0 ε e E y = 0,(9)H x = i k 0 ∂E y ∂z , H z = − i k 0 ∂E y ∂x , The second group of equations named as TM-type equations can be rewritten as ∂ 2 H y ∂z 2 + ∂ 2 H y ∂x 2 + k 2 0 ε o H y = 0,(10)E x = − i k 0 ε o ∂H y ∂z , E z = i k 0 ε o ∂H y ∂x , The surface wave is characterized by the boundary conditions E → 0 and H → 0 at x → ∓∞. The solutions of the TE-type equations obeying to these conditions when x → −∞ is possible if p 2 1 = β 2 − k 2 0 ε e > 0. The electric and magnetic fields read H (1) x (x, z) = − β k 0 Ae p1x+iβz , H (1) z (x, z) = − ip 1 k 0 Ae p1x+iβz ,(11)E (1) y (x, z) = Ae p1x+iβz . The solutions of the TM-type equations obeying to the boundary conditions is possible if p 2 2 = β 2 − k 2 0 ε o > 0. The electric and magnetic fields are E (1) x (x, z) = β k 0 ε o Be p2x+iβz , E (1) z (x, z) = ip 2 k 0 ε o Be p2x+iβz ,(12)H (1) y (x, z) = Be p2x+iβz . In the domain occupied by TI, which is assumed to be an isotropic medium, the components of the induction vector take the following form D x = ε 2 E x , D y = ε 2 E y , D z = ε 2 E z . The wave equations in this case have the form similar to equations (9) and (10), where ε o and ε e are replaced by ε 2 . Implementation of the boundary conditions E → 0 and H → 0 at x → +∞ for solutions of the TE-and TMtype of equations is possible if q 2 = β 2 − k 2 0 ε 2 > 0. The electric and magnetic fields have the following form H (2) x (x, z) = − β k 0 Ce −qx+iβz , H (2) z (x, z) = iq k 0 Ce −qx+iβz ,(13)E (2) y (x, z) = Ce −qx+iβz . E (2) x (x, z) = β k 0 ε 2 F e −qx+iβz , E (2) z (x, z) = − iq k 0 ε 2 F e −qx+iβz ,(14)H (2) y (x, z) = F e −qx+iβz . The amplitudes A, B, C and F can be found from the continuity conditions on the interface (6). Substitution of (11)- (14) into (6) results in following system of the algebraic equations qC + p 1 A = κ p 2 ε o B, B − F = −κA, p 2 ε o B + q ε 2 F = 0, A − C = 0, where κ = αθ. Nontrivial solutions of these equations exist if determinant of this system of equations is zero. This requirement results in the dispersion relation (q + p 1 ) 1 + p 2 ε 2 qε o + κ 2 p 2 ε o = 0, that can be rewritten as (q + p 1 ) q ε 2 + p 2 ε o + κ 2 p 2 q ε o ε 2 = 0.(15) According to definitions the conditions q > 0, p 1 > 0 and p 2 > 0 are held. If ε o > 0 and ε 2 > 0 the equation (15) has no solution, as it is sum of the positive terms. However, if ε o < 0 and ε 2 > 0 the equation (15) takes the form of (q + p 1 ) q ε 2 − p 2 |ε o | − κ 2 p 2 q |ε o |ε 2 = 0 (16) and this equation is soluble. Thus, in the case of hyperbolic material with ε o < 0 and ε e > 0, surface wave can exist, provided that the optical axis is directed normally to the direction of propagation and lies in interface. B. Optical axis is normal to interface. The components of the electric induction vector take the following form D x = ε e E x , D y = ε o E y , D z = ε o E z . Having the relation D y = ε o E y the TE-type equations take the form ∂ 2 E y ∂z 2 + ∂ 2 E y ∂x 2 + k 2 0 ε o E y = 0,(17)H x = i k 0 ∂E y ∂z , H z = − i k 0 ∂E y ∂x . Solutions satisfying to the surface wave conditions read as follows H (1) x (x, z) = − β k 0 Ae p1x+iβz , H (1) z (x, z) = − ip 1 k 0 Ae p1x+iβz ,(18)E (1) y (x, z) = Ae p1x+iβz , where p 2 1 = β 2 − k 2 0 ε o > 0 and p 1 > 0. Surface wave solutions of TM-type are 1 ε e ∂ 2 H y ∂z 2 + 1 ε o ∂ 2 H y ∂x 2 + k 2 0 H y = 0,(19)E x = − i k 0 ε e ∂H y ∂z , E z = i k 0 ε o ∂H y ∂x , have s similar form E (1) x (x, z) = β k 0 ε e Be p2x+iβz , E (1) z (x, z) = ip 2 k 0 ε o Be p2x+iβz ,(20)H (1) y (x, z) = Be p2x+iβz . where p 2 2 = (ε o /ε e )(β 2 − k 2 0 ε e ) > 0 and p 2 > 0. The solutions of the Maxwell equations in the region filled with TI (x > 0) were found in Sec.III A, see (13) and (14), and can be used here. The continuity conditions taking into account expressions (13), (14), (18) and (20) lead to the dispersion relation for the case under consideration (q + p 1 ) 1 + p 2 ε 2 qε o + κ 2 p 2 ε o = 0. Note, if ε o > 0, the equations has no solution. On the other hand in case of hyperbolic material with ε o < 0 and ε e > 0, this dispersion relation admits solution if p 2 1 = β 2 + k 2 0 |ε o | > 0, |ε o | ε e (k 2 0 ε e − β 2 ) > 0. It follows that surface wave exists under condition β 2 /k 2 0 < ε e . IV. ANALYSIS OF THE DISPERSION RELATIONS In the following analysis the dimensionless parameter n ef f = β/k 0 will be used. The inhomogeneous material under consideration can be considered as the uniform one, which is characterized by the effective index n ef f . It is standard practice to treat the waveguide or fiber systems [33]. In the case of a transparent medium n ef f defines the phase velocity of the guided wave according to expression v ph = c/n ef f . A. Optical axis is tangential to the interface. l = ty The dispersion relation (16) obtained earlier will take a form: 1 + n 2 ef f − ε e n 2 ef f − ε 2 1 − ε 2 |ε o | n 2 ef f + |ε o | n 2 ef f − ε 2 = = κ 2 |ε o | n 2 ef f + |ε o | n 2 ef f − ε 2 .(21) This relation connects effective index n ef f and radiation frequency ω implicitly. We suppose that radiation frequency is far away from all resonance frequencies specific to the insulator, thus ε 2 is a constant. But we take into account dispersion of the hyperbolic medium parameters, ε e (ω) and ε o (ω). The simplest for realization type of hyperbolic medium is material formed from alternating conductive and dielectric layers of sub-wavelength width. In effective medium approach dielectric constants of such medium can be presented as ε o = f m ε m + (1 − f m )ε d ,(22)ε e = ε d ε m f m ε d + (1 − f m )ε m , where f m is a volume fraction of metal, the filling factor. Usually resonance frequencies of dielectric are lying in the ultraviolet region. So considering infrared and optical frequencies of radiation in hyperbolic material one could take into account the frequency dispersion only of metal layers: ε d = const, ε m = ε ∞ − ω 2 p ω 2 + iγω , where the Drude-Lorentz model was used. ω p is plasma frequency, γ is collision frequency, ε ∞ is permittivity at height frequencies. By constitution of these relations to the (22) one can receive the frequency dependencies ε e (ω) and ε o (ω). They are presented in Fig 2. Parameters were chosen as follows: ω 2 p = 1.38 · 10 16 rad/c, ε ∞ = 5 (Ag), γ = 0 rad/c, ε d = 4.6 (SiO 2 ), f m = 0.4. Metamaterial described by permittivities (22) is hyperbolic medium with ε o < 0 and ε e > 0 for chosen frequency range up to approximately 620 THz. If κ = 0 it is possible to consider two types of surface wave independently. Wave of TE type is described by first multiplier in the left side of equation (21): 1 + n 2 ef f − ε e (ω) n 2 ef f − ε 2 = 0. This equation has no real solutions. Thus, surface wave of TE type is not propagate in the case of linear electrodynamics. Second multiplier in the left side of equation (21) describe TM wave: at κ = 0) is presented at Fig. 3. The value of topological insulator permittivity is ε 2 = 2.25. So condition |ε o | > ε 2 is hold everywhere in the considering frequency interval. At low frequencies n 2 ef f tends to ε 2 . Also critical value of frequency, ω c , at which n 2 ef f → ∞ exists. Critical value ω c can be defined from equation ε o (ω c ) = −ε 2 . 1 − ε 2 |ε o (ω)| Let us consider a topologically nontrivial insulator, i.e. κ = (2n + 1)(e 2 / c) = (2n + 1)α, n = 0, 1, ..,. As follows from the equation (21) a surface wave of hybrid polarization propagates on the media interfaces. A "topological" term in the right side of this equation connects wave polarizations of TE and TM type to one hybrid polarization. This implies additional condition on possible n 2 ef f values: n 2 ef f > ε e besides inequality n 2 ef f > ε 2 . Critical frequency value is a function ω(κ) in considering situation. The definition of ω(κ) could be obtained from relation (21) in case n 2 ef f → ∞: ω 2 c = f m ω 2 p f m ε ∞ + (1 − f m )ε d + ε 2 + κ 2 /2 .(23) From equation (23) follows that value of ω c shifts to the lower frequencies with κ. It is interesting to note, that by substitution f m = 1 and κ = 0 the definition (23) reduces to standard plasmon-polariton critical frequency value: ω 2 c = ω 2 p /(ε ∞ + ε 2 ). Solutions of dispersion equation (21) at different values of κ are presented in Fig. 4. In situation presented in Fig. 4 an inequality ε e (ν) > ε 2 is hold for all frequencies (ε 2 = 2.25). Thus, additional condition, n 2 ef f > ε e (ν), leads to the decrease of frequency interval of surface wave existence. At κ > 0 only narrow enough frequency domain where n ef f is defined exists. From pictures presented in Fig. 4 one can notice that ω c (or ν c for THz instead of radians per second) shifts to lower frequencies. So connection of TM and TE waves at topological insulator/hyperbolic material interface lead to the rigid re- striction on the frequency domain of surface wave existence at ε e > ε 2 . If ε e < ε 2 for all considering frequencies, the surface wave will exist in interval (0, ω c ) as shown in Fig. 3. This situation is similar with the case of topological insulator/metal interface considered in [19]. In these cases an existence of nonzero parameter κ will lead only to ω c (κ) shift to lower frequencies. B. Optical axis is tangential to the interface, l = tz When optical axis is aligned with Z axis the wave propagation parameters used earlier have the following definitions: p 2 1 = k 2 0 (n 2 ef f − ε o ), p 2 2 = k 2 0 ε e ε o (n 2 ef f − ε o ),q 2 = k 2 0 (n 2 ef f − ε 2 ). All presented parameters must be real and positive. Using the same technique as presented in previous section a dispersion relation for situation l = t z can be achieved. It has a form: 1 + n 2 ef f − ε o n 2 ef f − ε 2 1 + ε 2 ε e ε e ε o n 2 ef f − ε o n 2 ef f − ε 2 = = − κ 2 ε e ε e ε o n 2 ef f − ε o n 2 ef f − ε 2 .(24) When κ = 0 relation (24) splits into two equations describing TE surface wave: 1 + n 2 ef f − ε o n 2 ef f − ε 2 = 0, that has no real solutions, and TM wave: 1 + ε 2 ε e ε e ε o n 2 ef f − ε o n 2 ef f − ε 2 = 0. The second equation have solutions in case of hyperbolic material with permittivities ε e < 0, ε o > 0. In case of ε e > 0 both summands are positive and the sum can't be equal to zero. Taking into account definitions of p 2 and q effective refractive index must satisfy conditions: ε 2 < n 2 ef f < ε o . But when a topologically nontrivial insulator is under consideration, one must take into account additional condition, comes from the TE wave description, n 2 ef f > ε o . This inequality is in the contradiction with previous one. Thus, the relation (24) at κ > 0 has no real solutions. The surface wave is impossible on boundary between topological insulator and hyperbolic media with optical axis aligned with propagation direction. C. Optical axis is normal to the interface When optical axis of hyperbolic medium is aligned with normal to the interface the dispersion relation has solutions at ε o < 0, ε e > 0 and has a form: 1 + n 2 ef f + |ε o | n 2 ef f − ε 2 1 − ε 2 |ε o | |ε o | ε e ε e − n 2 ef f n 2 ef f − ε 2 = = κ 2 |ε o | |ε o | ε e ε e − n 2 ef f n 2 ef f − ε 2 ,(25) where definitions of parameters p 1 , p 2 , q were used: p 2 1 = k 2 0 (n 2 ef f + |ε o |), p 2 2 = k 2 0 |ε o | ε e (ε e − n 2 ef f ), q 2 = k 2 0 (n 2 ef f − ε 2 ). To satisfy conditions p 2 1 > 0, p 2 2 > 0 and q 2 > 0 a value of effective refractive index must lie in the interval ε 2 < n 2 ef f < ε e . In contrast to previous cases an additional condition from the TE wave description, n 2 ef f + |ε o | > 0, is always held. Thus a case of hyperbolic material/topological insulator boundary has no additional restrictions in comparison with case of hyperbolic material/dielectric boundary. The frequency dispersion of hyperbolic medium permittivities ε e (ω), ε o (ω) here is taken into account with the same parameters as in subsection A. Solutions of equation (25) for different values of κ are presented in Fig. 5, and 6. As follows from figures n 2 ef f exists for whole frequency range, where composite material described by (22) is hyperbolic one with ε o (ω) < 0 and ε e (ω) > 0 (compare with Fig. 2). One can notice from Fig. (5) (a), (b) and Fig. 6 (a) that point, where n 2 ef f = ε e (ω), is the same as that, were ε o (ω) change its sign. The considered situation is differ from the cases of surface waves at topological insulator/ metal and dielectric/ metal interfaces. The main feature is that frequency interval of surface wave existence is limited before the critical frequency ω c could be achieved. The magnetoelectric constant κ is taking equal to (2 × 100 + 1) α. The big value of the axion field θ is The hyperbolic materials are composite materials that contain conductive components such as metallic layers or nanoroads. The metallic inclusions lead to the energy dissipation in these materials. For real hyperbolic materials dielectric constants ε o and ε e are complex values. For example, ε o = −2.78 + i0.13 and ε = 6.31 + i0.09 at 465 nm [23]. The imaginary parts of ε o and ε e can be small enough, if the radiation frequency ω is far away from all typical for material resonances. The derivation of the dispersion relations in the case of complex values of permittivities can be carried out as in the previous sections. Metamaterial was described by permittivities (22). The contribution of the metallic inclusions was taking into account according to Drude-Lorentz model with γ = 5.07 · 10 13 rad/c. The equations defining the dispersion relations have the forms that are similar to (21), (24) and (25). However, now the constants ε o and ε e are complex values. Solutions of the obtained equations n 2 ef f (ω) are complex ones. Re n ef f describes the velocity of the surface wave, and Im n ef f describes the dumping this wave. The reciprocal of dumping length is equal to 2(ω/c)Im n ef f . The Fig.7 and Fig.8 demonstrate the dependencies of the real and imaginary parts of the effective index in the different cases of the optical axis orientation. The real parts of n 2 ef f in case of nonzero γ have the same shape as presented in figures 4, 5 and 6. The dissipative loss in the hyperbolic medium results in the finite dumping length that is usual for plasmonpolariton surface waves. VI. CONCLUSION Here the surface wave propagation along the interface between a hyperbolic material and a topological insulator was considered. The description of the electrodynamic of topological insulators is based on the generalized Maxwell equations [7,8]. The continuity conditions for tangent components of electric field vectors and the normal components of induction vectors were used. The dispersion relations for surface waves are derived. The cases of the anisotropy axes is normal to interface or one is coplanar to interface are discussed. If the optical axis of the hyperbolic material is tangential to the interface and directed across the propagation direction, and the magnetoelectric constant κ is not zero, the surface wave exists in the narrow frequency domain, if ε e > ε 2 . In the case of trivial insulator (κ = 0) the surface wave corresponds to the conventional surface plasmon. It should be note that the surface wave is impossible if the optical axis aligned with propagation direction. This means that some critical direction of the optical axis exists. If the optical axis is normal to the interface the surface wave exists for whole frequency range, where composite material is hyperbolic one with ε o (ω) < 0 and ε e (ω) > 0. Due to mixing of TM and TE waves at topological insulator/hyperbolic material interface the surface wave propagation conditions becomes more strict comparing to topological insulator/metal or dielectric/hyperbolic material cases. At last ones the surface wave exists in the frequency interval (0, ω c ), at ω c propagation constant tends to infinity. It is interesting to note, that in case of topological insulator/hyperbolic material with optical axis is tangential to the interface and directed across the propagation direction this interval narrows to (ω 1 , ω c ) if ε e > ε 2 . At ω 1 the propagation constant is β(ω 1 ) = k 0 √ ε e . A value of ω c is a function of magnetoelectric constant. In case of the normal direction to inter-face of the optical axis interval (0, ω c ) narrows to (0, ω 2 ). At ω 2 value of propagation constant is finite β = k 0 √ ε e . Here the topological insulator is considered. This material is characterized by the axion field, which is static field. However, axion field is static in a time-reversal invariant topological insulator. In [34] the antiferromagnetic long-range order in a topological insulator was discussed. As the result of time-reversal symmetry breaking θ becomes a dynamical axion field taking continuous values from 0 to 2π. As the dynamic axion field couples nonlinearly to the electromagnetic field, this term in the generalized Maxwell equations is the origin of the nonlinear responses of the topological magnetic insulators. If there is an externally applied static and uniform magnetic field, parallel to the electric of the electromagnetic wave, the linear approximation for the generalized Maxwell equations is constructible. New type of the bulk polariton referred to as the axionic polariton was proposed [34]. The surface axionic polariton would be expected to be propagating along the interface between a topological magnetic insulator and a non-topological material. FIG. 1 : 1A schematic illustration of the interface between TI slab on hyperbolic substrate. FIG. 2 : 2Dependencies of dielectric permittivities of hyperbolic medium on frequency. FIG. 3 : 3+ |ε o (ω)| n 2 ef f − ε 2 = 0.This equation can be solved if |ε o | > ε 2 . That is because an expression under the square root symbol is always more than one. Solution of this equation (or of eq.(21) Dependence of n 2 ef f on frequency at κ = 0, l = ty. FIG. 4 : 4Dependence of n 2 ef f on frequency for interface with topologically nontrivial insulator, εe(ν) > ε2, l = ty. FIG. 5 : 5Dependence of n 2 ef f on frequency for interface with topologically nontrivial insulator, l = n. (c) κ = 20 FIG. 6 : 206Dependence of n 2 ef f on frequency for interface with topologically nontrivial insulator, l = n.used to obtain better illustration of the dispersion curves.Figures 4, 5 and 6 indicate that the main features of the dispersion curves depend only slightly on the κ. The magnetoelectric effect generate the TE-TM wave coupling independently of the value κ = 0. V. THE ROLE OF DISSIPATION OF THE HYPERBOLIC MATERIAL FIG. 7 : 7Dependence of Re n ef f and Im n ef f on frequency for the surface wave with regard to the dissipation. Optical axis is tangential to the interface. FIG. 8 : 8Dependence of Re n ef f and Im n ef f on frequency for the surface wave with regard to the dissipation. Optical axis is normal to the interface. AcknowledgementThis investigation is funded by the Russian Foundation for Basic Research (Grant No. 15-02-02764). . Liang Fu, C L Kane, Phys.Rev. B. 761745302Liang Fu and C. L. Kane, Phys.Rev. B. 76, 045302 (17 pp) (2007). . 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As an emerging medical diagnostic technology, photoacoustic imaging has been implemented for both preclinical and clinical applications. For clinical convenience, a handheld free-scan photoacoustic tomography (PAT) system providing 3D imaging capability is essentially needed, which has potential for surgical navigation and disease diagnosis. In this paper, we proposed a free-scan 3D PAT (fsPAT) system based on a hand-held linear-array ultrasound probe. A global positioning system (GPS) is applied for ultrasound probe's coordinate acquisition. The proposed fsPAT can simultaneously realize real-time 2D imaging, and large field-of-view 3D volumetric imaging, which is reconstructed from the multiple 2D images with coordinate information acquired by the GPS. To form a high-quality 3D image, a dedicated space transformation method and reconstruction algorithm are used and validated by the proposed system. Both simulation and experimental studies have been performed to prove the feasibility of the proposed fsPAT. To explore its clinical potential, in vivo 3D imaging of human wrist vessels is also conducted, showing clear subcutaneous vessel network with high image contrast.

2203.09048

f6772fdd73985d8dee9ace3c5b6ccc79c111f667

Hand-held 3D Photoacoustic Imager with GPS Daohuai Jiang Shanghai Engineering Research Center of Intelligent Vision and Imaging School of Information Science and Technology ShanghaiTech University 201210ShanghaiChina Chinese Academy of Sciences Shanghai Institute of Microsystem and Information Technology 200050ShanghaiChina University of Chinese Academy of Sciences 100049BeijingChina Hongbo Chen Shanghai Engineering Research Center of Intelligent Vision and Imaging School of Information Science and Technology ShanghaiTech University 201210ShanghaiChina Chinese Academy of Sciences Shanghai Institute of Microsystem and Information Technology 200050ShanghaiChina University of Chinese Academy of Sciences 100049BeijingChina Yuting Shen Shanghai Engineering Research Center of Intelligent Vision and Imaging School of Information Science and Technology ShanghaiTech University 201210ShanghaiChina Yifan Zhang Shanghai Engineering Research Center of Intelligent Vision and Imaging School of Information Science and Technology ShanghaiTech University 201210ShanghaiChina Feng Gao Shanghai Engineering Research Center of Intelligent Vision and Imaging School of Information Science and Technology ShanghaiTech University 201210ShanghaiChina Rui Zheng Shanghai Engineering Research Center of Intelligent Vision and Imaging School of Information Science and Technology ShanghaiTech University 201210ShanghaiChina Fei Gao [email protected] Shanghai Engineering Research Center of Intelligent Vision and Imaging School of Information Science and Technology ShanghaiTech University 201210ShanghaiChina Hand-held 3D Photoacoustic Imager with GPS As an emerging medical diagnostic technology, photoacoustic imaging has been implemented for both preclinical and clinical applications. For clinical convenience, a handheld free-scan photoacoustic tomography (PAT) system providing 3D imaging capability is essentially needed, which has potential for surgical navigation and disease diagnosis. In this paper, we proposed a free-scan 3D PAT (fsPAT) system based on a hand-held linear-array ultrasound probe. A global positioning system (GPS) is applied for ultrasound probe's coordinate acquisition. The proposed fsPAT can simultaneously realize real-time 2D imaging, and large field-of-view 3D volumetric imaging, which is reconstructed from the multiple 2D images with coordinate information acquired by the GPS. To form a high-quality 3D image, a dedicated space transformation method and reconstruction algorithm are used and validated by the proposed system. Both simulation and experimental studies have been performed to prove the feasibility of the proposed fsPAT. To explore its clinical potential, in vivo 3D imaging of human wrist vessels is also conducted, showing clear subcutaneous vessel network with high image contrast. Introduction In the past two decades, photoacoustic (PA) imaging (also called optoacoustic imaging) has been widely applied for biomedical applications. PA imaging is a kind of hybrid imaging modalities that combines spectroscopic optical absorption contrast and deep acoustic penetration [1,2]. The basic principle of photoacoustic imaging includes laser excitation, optical absorption and temperature elevation, thermoelastic expansion, PA wave emission, signals detection and image reconstruction [3,4]. Generally, the implementation of PA imaging can be categorized as three modalities: photoacoustic microscopy (PAM), photoacoustic tomography (PAT) and photoacoustic endoscopy [5]. The PAM system has high spatial resolution suitable for skin and capillary imaging. To further classify PAM, it can be categorized as optical-resolution PAM (OR-PAM) and acoustic-resolution PAM (AR-PAM) [6,7]. OR-PAM can achieve micrometer-level resolution with optical focusing, but its imaging deep is limited within 1 mm, which is difficult to realize 3D imaging with deep penetration. On the other hand, AR-PAM has better penetration than OR-PAM due to deeper acoustic focusing. However, its imaging speed is limited by the mechanical scanning shown in Fig. 1(a). Recently, microelectromechanical system (MEMS) based mirror has been applied for improving the imaging speed [8,9], which is still suffering quilted limited field-of-view. Therefore, the PAM system is normally used for small region of interest (ROI) imaging with slow imaging speed, which cannot satisfy the real-time and large ROI requirements in clinical applications [10]. On the other hand, PAT system can realize real-time imaging with much larger ROI than PAM system [11][12][13]. For hand-held PAT system, a linear-array ultrasound transducer (UT) is applied for PA wave detection as shown in Fig. 1(b). This PAT system can only reconstruct 2D image of the UT's receiving plane. To realize 3D imaging, two conventional approaches are: 1). Using custom-designed 3D UT, such as spherical array shown in Fig. 1(c) [14][15][16][17]; 2). Mechanical scanning of the linear-array UT [18,19]. Regarding the first approach, although custom-designed spherical-array UT can easily achieve 3D PA imaging, it is limited to fixed ROI, and expensive nonstandard UT fabrication cost. Regarding the second approach, it suffers: 1). The mechanical scanning part includes a stepper motor and a fixed movement rail, which is clumsy and inflexible for clinical application; 2). The imaging area is dependent on the mechanical stepper motor with limited movement distance; 3). The UT detection track is also fixed and cannot dynamically adjust during the scanning. Therefore, a free-scan 3D PAT (fsPAT) system based on conventional linear-array UT is highly needed to address these issues, which can realize flexible handheld scanning with large ROI in 3D. To achieve above-mentioned performance, a clinically-available hand-held linear-array UT is applied for PA wave detection and image formation in 2D. Meanwhile, a global positioning system records every coordinate of the linear-array UT for each 2D frame, which will be used for 3D image reconstruction from all the acquired 2D frames. Moreover, a dedicated space transformation and reconstruction method for 3D image realization is used in this paper, followed by phantom study and experimental validation in vivo. Method fsPAT System Architecture The architecture of the proposed fsPAT system is shown in Fig. 2. The laser source of the PAT system is an optical parametric oscillator (OPO) laser (PHOCOUS MOBILE, OPOTEK, USA) with a 10 Hz repetition rate and 690 to 950 nm wavelength tunable rage. The 128-element linear-array UT (Doppler Inc., China) is used for PA signal's detection, whose central frequency is 7.5 MHz with 73% bandwidth. A data acquisition (DAQ) card with 128 channels (PhotoSound, USA) is used for PA data acquisition with sampling rate of 40 MSPS. The 3D global spatial positioning system (G4, Polhemus, USA) is comprised of three parts: system electronics unit (SEU), standard sensor (SS), and electric-magnetic field source (EFS). The EFS generates the magnetic field so that the sensor's trajectory can be precisely tracked as it is moving within the magnetic field. The SEU embedded hardware and software compute the position and orientation of the sensor, and transmit the coordinate data to personal computer (PC). For the proposed fsPAT, the SS is fixed to the ultrasonic probe that tracks its movement during the handheld scanning process. To be more specific, the SS of positioning system is attached in the same plane behind UT probe to avoid the presence of magnetic distortion. Then, the positioning system captures the 6 degrees of freedom (6-DOF) coordinate with 120 Hz sampling rate for each 2D PA image. The acquired 6-DOF coordinates include 3-DOF translation with unit of mm ( , , ) and 3-DOF rotation with unit of degree ( , , ). The whole system is synchronized by proper triggering, controlled by the PC. fsPAT Working Flow The working flow of the proposed fsPAT is presented in Fig. 3. The procedure can be summarized as: (1) Synchronizing signal triggers the laser output to illuminate the imaging target. Following that, PA wave is generated on account of the PA effect; (2) The ultrasound probe captures the PA signals in real time, followed by data acquisition of DAQ system. At the meantime, the standard sensor of the positioning system captures the 6-DOF coordinates of the ultrasound probe; (3) After data acquisition, 2D PA image can be reconstructed by applying universal back-projection algorithms in the computer [20]. (4) When sufficient 2D PA images are collected revealing continuous multiple cross-sections of the imaging target, 3D PA image reconstruction can be performed based on the collected 6-DOF coordinates. Fig. 3 The free-scan handheld 3D PAT imaging system working flow. DOF: degrees of freedom. 3D Space Building and Transformation In order to build the geometric relationship for 3D PA image reconstruction, four coordinate spaces in this system are separated: (1) the 2D image space I to express each pixel based on imaging plane; (2) the positioning SS space P to represent pixels according to the installation of sensor; (3) the 3D EFS space S to record 6-DOF position; (4) the volume space V to describe the region of 3D PA imaging. Once the 2D scan is finished, a constant translation matrix is measured to calibrate the center of probe receiving plane in image space to the sensor space. A matrix is calculated using acquired translation and rotation for the transformation from sensor space to EFS field space. The minimum of transformed coordinate values in x, y, z axes are regarded as the origin of volume space. An affine transformation is implemented to move all the acquired location in the EFS space into the view of volume space. The above processes can be expressed by the following formula: = * * * () = [ 1] () = [ 0 1] () where and are the homogeneous coordinates of pixel and voxel in the 2D image and volume space, respectively. T denotes the operation of transposition. Finally, by applying Eq. (1), the exact coordinate in volume space of each 2D image can be calculated for the following 3D geometric reconstruction. Fig. 4 shows the visualization result before and after space transformation of a typical PA image scan. Fig. 4(a) is the probe trajectory before transformation. The black rectangle stands for each frame plane of 2D image combined with spatial information in EFS space. Fig. 4(b) is the probe trajectory in volume space after transformation. After all the coordinates transform into volume space, the coordinates can be further used for 3D PA image reconstruction. 3D PA Image Reconstruction The realization of 3D PA image reconstruction is based on each 2D image acquisition with corresponding spatial position. Fig. 5 shows the pipeline of 3D PA image reconstruction procedure. During the reconstruction process, the fsPAT system will record 128 channels' raw PA data and a 6-DOF spatial coordinate for each 2D PA image, which are then combined for 3D image reconstruction. To be more specific, a regular grid volume is constructed through the coordinates of all the pixels after transformation by Eq. (1). The voxel size of this volume is set as 0.2 mm × 0.2 mm × 0.6 mm (x direction, y direction and z direction). Then, 2D PA images labeled with position information are fused into the grid volume for recovering the mapping connection between pixels and voxels. Here, the reconstruction algorithm applied in this paper is a Forward mapping algorithm with acceleration, the Pixel Nearest Neighbor based Fast-Dot Projection (FDP-PNN) [21,22]. The mapping result of a pixel point 0 in the 2D image can be acquired by the nearest voxel in 3D volume around it: 0 = min ( * * * 0 ) () where 0 is a 1×3 coordinate vector of the voxel, which is mapped by 0 . The corresponding voxels of other pixels in the same 2D image can be computed as following: = * ∆ + ℎ * ∆ℎ + () where, for a 2D image, and ℎ are the normalized direction vector in both axes of width and height. ∆ and ∆ℎ are the difference between the index of each pixel and 0 . Lastly, the intensity of mapped voxel is assigned by the value of its matching pixel. Experimental Results Phantom Study The phantom of human hair embedded in agar gel is used for experimental feasibility study. Fig. 6(a) shows the photograph of the phantom, where the hair is about 7~10 cm long. The imaging frame rate is limited by the laser repetition rate at 10 Hz. The whole scanning time consumption is about half-minute, capturing 300 frames of the 2D PA images in total. Each 2D PA image is reconstructed with 256×256 pixel size. After the space building and transformation, and by applying the 3D image reconstruction algorithms mentioned above, the 3D PA imaging of the hair phantom is shown in Fig. 6 Quantitative Analysis A pencil leads made phantom is used for quantitative analysis for the performance evaluation of the proposed system. Three pencil leads are placed parallel with different depths in the agar gel, and the other three are standing upright in agar gel. A total of nine physical distances d1~d9 are measured using a caliper indicated in Fig. 7(a)-(b), including three lengths of leads (d1, d2 and d3) and six interval distances (d4 ~ d9). Fig. 7(c) shows the reconstructed 3D structure of the parallel placed three pencil leads. The projection images on the top view of the 3-D reconstruction result for the leads phantom are shown in Fig. 7 (d)-(e). The same nine distances are measured by selecting the pixels on the projection image, calculating the Euclidean distances between these pixels with multiplication by voxel size. Table 1 presents the measurement results of these distances. The absolute error and relative error are applied to analyze the accuracy between the physical distance of the phantom and the PA imaging result. The mean and standard deviation of these two errors for all the distances are 0.22mm±0.21mm and 2.11%±2.18%, respectively. In-vivo Human Study The 3D PA imaging of human left-wrist vascular is also performed by the proposed fsPAT. To get better penetration of light, the wavelength is changed to 1064 nm with another high-power laser (LPS-1064-L, CNI, China) with 120 mJ pulse energy at the fiber output with 10 Hz repetition rate. The light is transmitted by an optical fiber and illuminated on the surface of skin with illumination area of about 8 cm 2 . The fluence is less than 20 mJ/cm 2 at the skin surface within the safety limit [23]. In this experiment, the hand wrist and forearm are immersed in water with a constant temperature of 28 ℃. The photograph of the experimental operation is shown in Fig. 8(a). There are totally 240 frames captured in the ROI from the middle of the forearm to wrist with scanning range of about 50 mm. The reconstructed 3D PA image of vessels is shown in Fig. 8(b)-(d). Fig. 8(b) shows the top view of the reconstructed 3D structure of the blood vessel, where the outline of the blood vessel is obvious with striking contrast to the background. Fig. 8(c) shows the lateral view of the blood vessel, where the depth of the blood vessel is revealed in this view. Fig. 8(d) is the left top view of the reconstructed 3D PA image. The arrows labelled ① to ④ in Fig. 8 [24]. Generally, the human wrist imaging results show clear outline of the blood vessel with complete structure, which well proved its feasibility for clinical applications in the near future. Discussion In this paper, we proposed fsPAT system, which contains 128-channel UT probe for 2D PA image acquisition, and a global positioning system for spatial coordinates collection of the 2D image. The fsPAT system realized 3D PAT imaging with free-scanning handheld probe. The imaging ROI of the system is significantly improved with handheld scanning. The global positioning system captures the spatial information with 120 Hz, which records the coordinates of 2D image fast enough. The 2D imaging is with 10 Hz rate and 38.4 mm imaging width. Therefore, the fsPAT system's ROI has cross-sectional width of 38.4 mm, and the length of the ROI is based on the handheld probe scanning region. With 10 Hz laser repetition rate and 0.1~0.4 mm scanning interval, the fsPAT can realize 60~240 mm per minute scanning speed for 3D imaging. Moreover, the scanning speed can be further improved by applying the laser source with higher repetition rate. In the phantom study of human hair embedded in agar gel, the feasibility of the proposed fsPAT system for 3D photoacoustic imaging is verified. The fsPAT realized 7~10 cm length of large ROI imaging by handheld free-scanning, which is flexible for clinical applications. The imaging results in Fig. 6(b)-(d) reveal the 3D structure and size of the phantom with strong contrast to the background. Furthermore, the quantitative analysis of the fsPAT for 3D imaging is verified by a pencil leads made phantom. There are nine Euclidean distances measured in the phantom as shown Fig. 7(a)-(b), and we calculated the corresponding distances in the reconstructed imaging results. Table 1 shows the measurement results, where the utmost relative error is 5.88%. The mean absolute error of the nine measurements is 0.22 mm with 2.11% relative error, which shows satisfactory accuracy of the reconstructed 3D structure. The in-vivo imaging results of human wrist are shown in Fig. 8, showing that the fsPAT system with handheld free-scanning realized a large ROI 3D imaging, where the reconstructed 3D image shows the size and shape of the blood vessel. The in-vivo imaging results of the human wrist blood vessel show high potential of the fsPAT system for clinical 3D photoacoustic imaging applications. However, the proposed fsPAT also has some limitations and need to be further optimized: 1) the 3D imaging speed for large ROI is limited by the 2D image realization, e.g. the repetition rate of the laser source; 2) the cross-sectional width of the 3D image is limited by the UT probe receiving width; 3) the 2D image distortion induce the challenge for exactly 3D reconstruction. To further advance the fsPAT for 3D imaging, the reconstruction algorithms from raw PA signals directly to 3D image will be developed to achieve isotropic-resolution 3D PA imaging. Conclusion The integration of the global positioning system and PAT system enables the proposed fsPAT to realize large ROI of 3D imaging by freely handheld scanning, which greatly improves the flexibility of the 3D PA imaging compared with the conventional 3D PA imaging methods. In this paper, both phantom study and in-vivo imaging are conducted that verified the feasibility and performance of the proposed system. Furthermore, in-vivo human wrist blood vessel imaging shows potential for clinical applications such as peripheral vessel visualization, and breast cancer detection. In the future work, fsPAT can be further optimized by employing laser with high repetition rate for fast imaging scanning, and multi-wavelength for functional 3D imaging e.g. blood oxygen saturation quantification and imaging. In addition, the fsPAT can be also combined with ultrasound imaging to visual more details of vessels. Fig. 1 1Conventional 3D photoacoustic imaging implementations. (a) the PAM system with 2D mechanical scanning; (b) 3D PAT realization with UT array mechanical scanning; (c) 3D PAT realization with hemispherical UT array. Fig. 2 2The setup of the 3D PAT imaging system. DAQ: data acquisition; EFS: electric-magnetic field source; MF: magnetic field; SS: standard sensor; SEU: system electronics unit; CD: coordinate data; UT: ultrasound transducer; LLL: linear laser light; PC: personal computer; Fig. 4 4The visualization of transformation. Each frame plane is described as a rectangle (black). (a) The probe trajectory in EFS space before transformation. (b) The probe trajectory in volume space after transformation. Fig. 5 5The pipeline of fsPAT 3D image reconstruction procedure. (b)-(d) corresponding to the top view, lateral view and front view, respectively. The 3D imaging result reveals the structure of the phantom clearly with high contrast to the background. The black arrows in Fig. 6(a) mark several positions of the phantom, and their corresponding positions in the PA images are pointed by the white arrows in Fig. 6(b)-(d). Fig. 6 6The phantom experimental result. (a) The photograph of the phantom (human hair embedding in agar gel); (b)-(d) the reconstructed 3D PA imaging of the hair phantom corresponding to the top view, lateral view and front view, respectively. Fig. 7 7The photograph of pencil leads phantom and experimental results. (a) The photograph of the pencil leads phantom (illustration of three distances of leads' length (d1, d2 and d3) and two distances of horizontal interval (d4 and d5)); (b) The pencil leads phantom with illustration of two distances of oblique interval (d6 and d7) and two distances of horizontal interval (d8 and d9); (c) the reconstructed 3D structure of the parallel placed three pencil leads; (d) Projection image of three standing leads on top view of 3-D imaging volume. (e) Projection image of three paralleled leads on top view of 3-D imaging volume. Fig. 8 8In-vivo experimental results of human wrist. (a) the photograph of the experimental operation with the proposed fsPAT. (b)-(d) the 3D PA imaging of human wrist blood vessel in top view, lateral view and left top view. ①: Ulnar vein;②: Superficial Palmar venous arch; ③: Median antebrachial vein;④: Radial vein. Table 1 . 1The Analysis of Measurement ResultDistance (mm) Physical distance Reconstructed distance Absolute error Relative error d1 18.70 18.91 0.21 1.10% d2 18.17 18.04 0.13 0.72% d3 18.22 18.07 0.15 0.82% d4 5.70 5.80 0.10 1.75% d5 6.80 6.40 0.40 5.88% d6 20.45 20.53 0.08 0.37% d7 20.22 20.24 0.02 0.11% d8 7.40 7.60 0.20 2.74% d9 12.90 13.61 0.71 5.53% Mean Error 0.22 2.11% Standard deviation 0.21 2.18% (b)-(d) correspond to the structures of Ulnar vein, a part of Superficial Palmar venous arch, Median antebrachial vein and Radial vein in anatomy Photoacoustic imaging in biomedicine. M Xu, L V Wang, 10.1063/1.2195024Review of Scientific Instruments. 774M. Xu and L. V. Wang, "Photoacoustic imaging in biomedicine," Review of Scientific Instruments, vol. 77, no. 4, 2006, doi: 10.1063/1.2195024. 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Advancements in ultra-low-power tiny machine learning (TinyML) systems promise to unlock an entirely new class of smart applications. However, continued progress is limited by the lack of a widely accepted and easily reproducible benchmark for these systems. To meet this need, we present MLPerf Tiny, the first industry-standard benchmark suite for ultra-low-power tiny machine learning systems. The benchmark suite is the collaborative effort of more than 50 organizations from industry and academia and reflects the needs of the community. MLPerf Tiny measures the accuracy, latency, and energy of machine learning inference to properly evaluate the tradeoffs between systems. Additionally, MLPerf Tiny implements a modular design that enables benchmark submitters to show the benefits of their product, regardless of where it falls on the ML deployment stack, in a fair and reproducible manner. The suite features four benchmarks: keyword spotting, visual wake words, image classification, and anomaly detection.

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MLPerf Tiny Benchmark Colby Banbury Vijay Janapa Reddi Peter Torelli Jeremy Holleman Nat Jeffries Csaba Kiraly Pietro Montino David Kanter Sebastian Ahmed Danilo Pau Urmish Thakker I Antonio Torrini Ii Peter Warden Jay Cordaro Giuseppe Di Guglielmo Iii IVJavier Duarte Stephen Gibellini Videet Parekh V Honson Tran V Nhan Tran Vi Niu Wenxu Vii Xu Xuesong Vii MLPerf Tiny Benchmark Advancements in ultra-low-power tiny machine learning (TinyML) systems promise to unlock an entirely new class of smart applications. However, continued progress is limited by the lack of a widely accepted and easily reproducible benchmark for these systems. To meet this need, we present MLPerf Tiny, the first industry-standard benchmark suite for ultra-low-power tiny machine learning systems. The benchmark suite is the collaborative effort of more than 50 organizations from industry and academia and reflects the needs of the community. MLPerf Tiny measures the accuracy, latency, and energy of machine learning inference to properly evaluate the tradeoffs between systems. Additionally, MLPerf Tiny implements a modular design that enables benchmark submitters to show the benefits of their product, regardless of where it falls on the ML deployment stack, in a fair and reproducible manner. The suite features four benchmarks: keyword spotting, visual wake words, image classification, and anomaly detection. Introduction Machine learning (ML) inference on the edge is an increasingly attractive prospect due to its potential for increasing energy efficiency [4], privacy, responsiveness, and autonomy of edge devices. Thus far, the field edge ML has predominantly focused on mobile inference, but in recent years, there have been major strides towards expanding the scope of edge ML to ultra-low-power devices. The field, known as "TinyML" [1], achieves ML inference under a milliWatt, and thereby breaks the traditional power barrier preventing widely distributed machine intelligence. By performing inference on-device, and near-sensor, TinyML enables greater responsiveness and privacy while avoiding the energy cost associated with wireless communication, which at this scale is far higher than that of compute [5]. Furthermore, the efficiency of TinyML enables a class of smart, batterypowered, always-on applications that can revolutionize the real-time collection and processing of data. Deploying advanced ML applications at this scale requires the co-optimization of each layer of the ML deployment stack to achieve the maximum efficiency. Due to this complex optimization, the direct comparison of solutions is challenging and the impact of individual optimizations is difficult to measure. In order to enable the continued innovation, a fair and reliable method of comparison is needed. In this paper, we present MLPerf Tiny, an open-source benchmark suite for TinyML systems. The MLPerf Tiny inference benchmark suite provides a set of four standard benchmarks, selected by more than 50 organizations in academia and industry. In order to capture the tradeoffs inherent to TinyML, the benchmark suite measures latency, energy, and accuracy. MLPerf Tiny is designed with flexibility and modularity in mind to support hardware and software users alike and provides complete reference implementations to act as open-source community baselines. Challenges TinyML systems present a number of unique challenges to the design of a performance benchmark that can be used to measure and quantify performance differences between various systems systematically. We discuss the four primary obstacles and postulate how they might be overcome. Low Power Power consumption is one of the defining features of TinyML systems. Therefore, a useful benchmark should profile the energy efficiency of each device. However, there are many challenges in fairly measuring energy consumption. TinyML devices can consume drastically different amounts of power,which makes maintaining accuracy across the range of devices difficult. Also, determining what falls under the scope of the power measurement is difficult to determine when data paths and pre-processing steps can vary significantly between devices. Other factors like chip peripherals and underlying firmware can impact the measurements. Limited Memory While traditional ML systems like smartphones cope with resource constraints in the order of a few GBs, tinyML systems are typically coping with resources that are two orders of magnitude smaller. Traditional ML benchmarks use inference models that have drastically higher peak memory requirements (in the order of gigabytes) than TinyML devices can provide. This also complicates the deployment of a benchmarking suite as any overhead can make the benchmark too big to fit. Individual benchmarks must also cover a wide range of devices; therefore, multiple levels of quantization and precision should be represented in the benchmarking suite. Finally, a variety of benchmarks should be chosen such that the diversity of the field is supported. Hardware Heterogeneity Despite its nascency, TinyML systems are already diverse in their performance, power, and capabilities. Devices range from general-purpose MCUs to novel architectures, like in event-based neural processors [2] or memory compute citekim20191. This heterogeneity poses a number of challenges as the system under test (SUT) will not necessarily include otherwise standard features, like a system clock or debug interface. Furthermore, creating a standard interface while minimizing porting effort is a key challenge. Today's state-of-the-art benchmarks are not designed to handle these challenges readily. They need reengineering to be flexible enough to handle the hardware heterogeneity that is commonplace in the TinyML ecosystem. Software Heterogeneity TinyML systems are often tightly coupled with their inference stack and deployment tools. To achieve the highest efficiency, users often develop their own tool chains to optimally deploy and execute a model on their hardware systems. This becomes increasingly critical on systems with multiple compute units, like accelerators and DSPs. This poses a challenge when designing a benchmark for these systems because any restriction, posed by the benchmark, on the inference stack, would negatively impact performance on these systems and result in unrepresentative results. Therefore we must balance optimality with portability, and comparability with representativeness. A TinyML benchmark should support many options for model deployment and not impose any restrictions that may unfairly impact users with a different deployment stack. Cross-product Figure 1 illustrates the diversity of options at every level in the TinyML stack. Each option and each layer has its own impact on performance and TinyML software users can provide an improvement to the overall system at any layer. A TinyML benchmark should enable these users to demonstrate the performance benefits of their solution in a controlled setting. Related Work There are a few ML related hardware benchmarks, however, none that accurately represent the performance of TinyML workloads on tiny hardware. EEMBC(r)'s CoreMark(r) benchmark [9] has become the standard benchmark for MCU-class devices due to its ease of implementation and use of real algorithms. Yet, CoreMark does not profile full programs, nor does it accurately represent machine learning inference workloads. EEMBC's MLMark(r) benchmark [19] addresses these issues by using actual ML inference workloads. However, the supported models are far too large for MCU-class devices and are not representative of TinyML workloads. They require far too much memory (GBs) and have significant runtimes. Additionally, while CoreMark supports power measurements with the EEMBC ULPMark(tm)-CM benchmark, MLMark does not, which is critical for a TinyML benchmark. MLPerf, a community-driven benchmarking effort, has recently introduced a benchmarking suite for ML inference [18] and has plans to add power measurements. However, much like MLMark, the current MLPerf inference benchmark precludes MCUs and other resource-constrained platforms due to a lack of small benchmarks and compatible implementations.As Table 1 summarizes, there is a clear and distinct need for a TinyML benchmark that caters to the unique needs of ML workloads, makes power a first-class citizen and prescribes a methodology that suits TinyML. Benchmarks All machine learning benchmarks fall somewhere on the continuum between low level and application level evaluation. Low level benchmarks target kernels that are core to many ML workloads, like matrix multiply, but they gloss critical elements like memory bandwidth or model level optimizations. On the other hand, application level benchmarks can obscure the target of the benchmark behind other stages of the application pipeline. MLPerf Tiny specifically targets model inference and does not include pre-or post-processing in the measurement window. Table 1 shows the v0.5 benchmarks. In this section, we describe the benchmarks that form the MLPerf Tiny benchmark suite. Each benchmark targets a specific usecase and specifies a dataset, model, and quality target. Additionally, each benchmark has a reference implementation that includes training scripts, pre-trained models, and C code implementations. The reference implementations run the reference models in the TFLite format using TFLite for Microcontrollers (TFLM) [7] on the NUCLEO-L4R5ZI board. A knowngood snapshot of the TFLM runtime is used to ensure stability, and the reference implementation is built using a bare-metal MBED project with the GCC-ARM toolchain. Visual Wake Words Rational Tiny image processing models are becoming increasingly widespread, primarily for simple image classification tasks. The Visual Wakewords challenge [6] tasked submitters with detecting whether at least one person is in an image. This task is directly relevant to smart doorbell and occupancy applications, and the reference network provided as part of the challenge fits on most 32-bit embedded microcontrollers. Dataset The Visual Wakewords Challenge uses the MSCOCO 2014 dataset [15] as the training, validation and test datasets for all person detection models. The dataset is preprocessed to train on image which contain at least one person occupying more than 2.5% of the source image. Images are also resized to 96x96 for model training. Model The Visual Wakewords model is a MobilenetV1 [11] which takes 96x96 input images with an alpha of 0.25 and two output classes (person and no person). The TFLM model is 325KB in size. Quality Target Based on validation and training accuracy, the model reaches about 86% accuracy across the preprocessed MSCOCO 2014 test dataset. In order to accommodate changes in accuracy due to quantization and rounding differences between platforms, submissions to the closed category should reach at least 80% accuracy across the same dataset. Image Classification Rational Novel machine vision techniques offer a cross-industry potential for breakthroughs and advances in autonomous and low power embedded solutions. Compact vision systems performing image classification at low cost, high efficiency, low latency and high performances are pervading manufacturing, IoT devices, and autonomous agents and vehicles. New hardware platforms, algorithms and development tools form a wide variety of deep embedded vision systems requiring a point of reference for scientific and industrial evaluation. Dataset CIFAR-10 [14] is a labeled subset of the 80 Million Tiny Images dataset [20]. The low resolution of the images make CIFAR-10 the most suitable source of data for training tiny image classification models. It consists of 60000 32x32x3 RGB images, with 6000 images per class. The 10 different classes represent airplanes, cars, birds, cats, deers, dogs, frogs, horses, ships and trucks. The dataset is divided into five training batches and one testing batch, each with 10000 images. A significant amount of prior work in TinyML has used CIFAR-10 as a target dataset [8], by continuing this trend, we create a point of reference in the benchmark suite that can be used to relate future results to historical data points. Model The Image Classification model is a customized ResNetv1 [10] which takes 32x32x3 input images and outputs a probability vector of size 10. The custom model is made of fewer residual stacks than the official ResNet: three compared to four. Moreover, the first convolutional layer is not followed by the pooling layer due to the low resolution of the input data. The number of convolution filters and the convolution strides dimension are lower as well compared to the official ResNet. The TFLite model for IC is 96KB in size and fits on most 32-bit embedded microcontrollers. Quality Target A set of 200 images from the CIFAR-10 test set are selected to evaluate the performances of the IC reference implementation. The performance evaluation test is performed with the benchmark framework software, i.e. the runner (see Appendix). The model reaches 86.5% accuracy across the 200 testing raw images. To accommodate minor differences in quantization and various other optimizations, we set the quality target to 85% top-1 accuracy. Keyword Spotting Rational Recognition of specific words and brief phrases, known as keyword spotting, is one of the primary use cases of ultra-low-power machine learning. Voice is an important mode of human-machine interaction. Wakeword detection, a specific case of keyword spotting wherein a detector continuously monitors for a specific word or phrase (e.g. "Hey Siri"™, "Hey Google"™, "Alexa"™) in order to enable a larger processor, requires continuous operation and therefore low power consumption. For example, a 100 mA current drain would deplete a typical phone battery in one day without any other activity. Command phrases such as "volume up" or "turn left" are the provide a simple and natural interface to embedded devices. Both cases require low latency and low power consumption and must typically run on small, low-cost devices. Dataset We used the Speech Commands v2 dataset [21], a collection of 105,829 utterances collected from 2,618 speakers with a variety of accents. It is freely available for download under the Creative Commons BY license. The dataset contains 30 words and a collection of background noises and is divided into training, validation, and test subsets such that any individual speaker only appears in one subset. Following the typical usage of this dataset, we used 10 words and combined the background noise with the remaining 20 words to approximate an open set labeled "unknown," which, along with "silence", results in 12 output classes. This approach exercises a combination command-phrase systems' need for multiple words with wakeword systems' need for an open set of background noise. Model For the closed division model, we used the small depthwise-separable CNN described in [22]. We chose this model because with 38.6K parameters, it fits within the available memory of most microcontrollers and similarly-scaled devices while achieving accuracy of 92.2% in our experiments. The model utilizes standard layers that can be expected of most neural network hardware. For this version of the benchmark, we opted to exclude feature extraction from the measurement and provide three choices of pre-computed features for the open division. Feature extraction is typically a small fraction of the overall compute cost, so the impact on the measurements is minor. While the pre-selected feature choices somewhat limit innovation in the overall KWS system, the features chosen represent the features most commonly used in KWS systems. Quality Target To evaluate accuracy on the device under test, we randomly selected 1000 utterances from the speech commands. The quantized reference model demonstrated 91.6% accuracy on the full test set and 91.7% on the 1000-utterance subset. To allow for slight variations in quantization strategies, we set an accuracy requirement of 90%. Anomaly Detection Rational Anomaly detection is broadly the task of separating normal samples from anomalous samples and has applications in a large number of fields. It's unsupervised variant, which is used in our benchmark, is of particular importance to industrial use cases such as early detection of machine anomalies, where failure types are many and failure events can be rare, and thus data is often only available on normal operation for training. Anomalies can be observed in various data sources (or combinations thereof), including power, temperature, vibration, with the most intuitive being the detection of anomalies based on sound captured by microphones. The most important distinctive features of this benchmark are the use of unsupervised learning and an autoencoder (AE). Dataset For this benchmark we have selected to use the dataset from the DCASE2020 [13] competition, which is itself a combination of two publicly available sources: the ToyADMOS [12], and the MIMII [17] datasets. The DCASE dataset contains data for six machine types (slide rail, fan, pump, valve, toy-car, toy-conveyor), and competition rules allowed for the training of separate models for each. Benchmarking would not benefit from this type of complexity, thus we choose to use the Toy-car machine type only. For training, normal sound samples of seven different toy-cars are provided, each having 1000 samples mixing the machine sound with environmental noise. Model While several models are openly available from the competition, the competition itself did not focus on TinyML, and only a few of these fit our target model size. Among these, we have selected the AE based model also used in the reference implementation of DCASE2020 for multiple reasons. First, it is the reference for a large part of the literature around the audio anomaly detection problem Quality Target The output of the autoencoder is an anomaly score, which assumes a further threshold to be set before the binary normal/anomalous decision can be made. For this reason the parameterless AUC-ROC (Area Under The Curve, Receiver Operating Characteristics) fits better for the quality evaluation for this problem than metrics requiring the selection of a threshold. To ensure models have generalization characteristics, we have selected an evaluation dataset composed of normal and anomalous sounds from four different machines totalling 248 samples. On this set, the fp32 version of the reference model achieves an AUC of 0.88 while the AUC after quantization is 0.86. Based on these two numbers, we've set the threshold for the benchmark to AUC 0.85. Run Rules In this section we describe the modular design of the benchmark harness, as well as the closed and open benchmark divisions, and the overall run rules of the benchmark suite to ensure reproducibility. Modular Design TinyML applications often require cross stack optimization to meet the tight constraints. Hardware and software users can provide value by targeting specific components of the pipeline, like quantization, or offer end-to-end solutions. Therefore, to enable users to demonstrate the competitive advantage of their specific contribution, we employ a modular approach to benchmark design. Each benchmark in the suite has a reference implementation that contains everything from training scripts to a reference hardware platform. This reference implementation not only provides a baseline result, it can be modified by a submitter wishing to show the performance of a single hardware or software component. Figure 2 illustrates the modular components of a MLPerf Tiny reference implementation. Closed and Open Divisions MLPerf Tiny has two divisions for submitting results: a stricter closed division and a more flexible open division. A submitting organization can submit to either or both divisions. This two division design allows the benchmark to balance comparability and flexibility. The closed division enables a more direct comparison of systems. Submitters must use the same models, datasets, and quality targets as the reference implementation. The closed division permits post training quantization using the provided calibration datasets, but prohibits any retraining or weight replacement. Figure 2 demonstrates how submissions to the closed division allow the comparison of an inference stack to another in a controlled setting. The open division is designed to broaden the scope of the benchmark and allow submitters to demonstrate improvements to performance, energy, or accuracy at any stage in the machine learning pipeline. The open division allow submitters to change the model, training scripts, and dataset. A submission to the open division will still use the same test dataset to benchmark the accuracy but is not required to meet the accuracy threshold, which allow submitters to demonstrate tradeoffs between accuracy, latency, and energy consumption. Each submission to the open division must document how it deviates from the reference implementation. Figure 2 demonstrates how submissions to the open division allow users to demonstrate the specific value add of their product. Measurement Procedure The platforms targeted by TinyML typically do not contain the resources required to run a complete benchmark locally, e.g., file IO, standard input and output, interactivity, etc. As a result, a benchmark framework ( Figure 3) is required to facilitate controlling and measuring the device under test (DUT). The benchmark framework used to implement the MLPerf Tiny benchmark is based on EEMBC's software development platform. The resulting program coordinates execution of the benchmark's behavioral model among the various components in the system as described in the Appendix. The framework's measurement procedure is as follows: 1. Latency -Perform this five times: download an input stimulus, load the input tensor (converting the data as needed), run inference for a minimum of 10 seconds and 10 iterations, measure the inferences per second (IPS); report the median IPS of the five runs as the score. 2. Accuracy -Perform a single inference on the entire set of validation inputs, and collect the output tensor probabilities. The number of inputs vary, depending on the model. From the individual results, compute the Top-1 percent and the AUC. Each model has a minimum accuracy that must be met for the score to be valid. 3. Energy -Identical to latency, but in addition to measuring the number of inferences per second, measure the total energy used in the timing window and compute micro-Joules per inference. The same method of taking the median of five measurements is used. Results are stored in a folder and can be reloaded again. An energy viewer allows the user to examine the energy trace. Figure 5 illustrates the reference implementation results. The four benchmarks cover a wide scope in terms of latency and energy and each reference meets the minimum accuracy. Benchmark Assessment In this section we assess the success of the benchmark at meeting the needs of the community. The suite is designed to provide standardization and compatibility while enabling a diverse set of organizations to submit results. Submissions The MLPerf Tiny benchmark suite accepts results from submitting organizations twice a year. The submitting organizations must implement the benchmarks on their hardware/software stack and submit their results and implementations at the deadline. All results are then transparently peerreviewed by the group of submitters and a review committee to ensure they conform to the rules. Table 2. The results from the first round submissions are currently available at https://mlcommons. org/en/inference-tiny-05/. Each submission is publicly available on GitHub with instructions on how to reproduce each result at https://github.com/mlcommons/tiny_results_v0.5. Insights The v0.5 submissions were diverse and demonstrated the desired flexibility of the benchmark suite. There were submissions to the open and closed divisions, as well as from hardware and software vendors. Each submitter had a specific element of their submission they wished to demonstrate and the benchmark suite was able to accommodate this diverse set of goals due to its modular design. The submissions indicate general trends in TinyML. For example, the most common numerical format is 8-bit integer for inference as it offers a performance boost with little impact to the model accuracy. ML frameworks range from open source interpreters (TFLite Micro) to hardware specific inference compilers, indicating that there is still often a trade-off between optimization and portability. There were results collected on a wide variety of hardware platforms, including MCUs, accelerators, and FPGAs. The re-configurable hardware (FPGA) is able to utilize variable precision models for increased performance. The power consumption of these platforms ranged from µWatts to Watts. More specifically, one organization used the benchmark to demonstrate that their SDK developer tools are hardware agnostic, easy to use and uniquely designed to optimize neural network inferences for compute, energy and memory, while preserving algorithmic accuracy. Another organization, a hardware vendor, used the benchmark to demonstrate outstanding performance on their NN accelerator, enabled by their novel microarchitecture design and optimized datapath, which avoids wait-states and maintains high computational efficiency. An academic research institute used the benchmark to demonstrate AI compute capability and potential applications for RISC-V-based AI micro-controllers. RISC-V is a free, open standard instruction set architecture (ISA) [3]. Finally, a multi-disciplinary team of scientists and engineers showcased their "hls4ml" (high-level synthesis for ML) open-source workflow. It was designed to enable researchers and engineers to codesign optimized neural networks for efficient dataflow architectures on a multitude of accelerator hardware platforms. Originally developed for the Large Hadron Collider to make ultrafast sub-microsecond ML inference, the workflow aims to serve the wider ML community to accelerate both the design process and neural network implementation across a broad range from low-power to high-performance devices. These results indicate a snapshot of this emerging field but future submission rounds can be used to indicate the evolution of TinyML over time. For instance, despite a general trend in AI towards data-centric design, none of the submissions in the first round modified the training dataset. While we anticipate this will shift in future versions, TinyML design is still largely focused on models, frameworks, and hardware. To this end, as demonstrated, the benchmark meets a variety of needs. Impact The benchmarks have already acted as a standard set of tasks for TinyML research [4] and have been made into public projects on a TinyML development platform [16]. The benchmark will standardize the nascent field of TinyML and enable future progress through competition and comparability. TinyML as a field has the potential to democratize AI by removing the barrier of expensive hardware and can preserve privacy by keeping user's data on the device that captures it. At the same time, the technology could be misused to more efficiently track and monitor unwilling individuals. Furthermore, because the devices themselves are inexpensive, TinyML can lead to increased electronic waste. By establishing a collaborative community, MLPerf Tiny can aid in the creation of standards for the responsible deployment of TinyML and mitigate the potential negative impacts of the technology. Limitations New benchmarks & Long-term stability MLPerf Tiny will continue to evolve to reflect the needs of the community. This will include new benchmarks that target new applications domains, such as wearables, medical devices, and environmental monitoring. While new benchmarks are being added and existing ones are evolved, it is also important to keep in mind that such a benchmark serves both as the comparison of state-of-the-art and for tracking historical progress. This latter goal requires benchmarks to be stable over time, eventually providing a multi-year perspective into the evolution of the field. Keeping the benchmark open source helps long-term reproducibility, while we also envision a subset of benchmarks to be kept long-term stable for this purpose. Also, thanks to MLCommons (mlcommons.org), a non-profit open engineering organization that hosts and develops the MLPerf benchmarks, the MLPerf Tiny benchmark will continue to be supported through future generations. Streaming inputs & Pre-processing Time-domain tasks, such as the keyword spotting and anomaly detection benchmarks, typically involve continuously streaming inputs. Information from previous time steps can be exploited to improve the performance-efficiency tradeoff. However, limited bandwidth between the test runner (running on e.g. a PC) and the DUT makes it difficult to recreate a streaming scenario without adding delays incurred by data transfer. The choice of whether to include pre-processing in the measured benchmark can also have distorting effects on the results. Excluding feature extraction from measurement while allowing submitter-selected variations in feature extraction creates the possibility of a degenerate case where an entire model up to the penultimate layer is defined as "feature extraction". Rigidly defined feature extraction precludes joint optimization over feature and model architecture that can be critical in the constrained systems targeted by this benchmark suite. Inclusion of feature extraction in the benchmark brings its own subtle complexities. Operating on a single discrete audio input in a non-streaming benchmark, one second of audio would involve one inference cycle and 40 feature extraction cycles, over-emphasizing the cost of feature extraction. In future versions, we aim to widen the scope of the benchmarks to include pre-processing as well. Extended coverage of layer types and model architectures The closed division of the current benchmark suite includes models based mainly on FC and CNN layers in a variety of well known architectures, allowing only open division submissions to deviate from these. While our current suite provides a reasonable baseline that can be implemented on most HW platforms and openness to more experimental submissions, future benchmarks may include reference implementations using additional architectures e.g. RNNs. Conclusion The field of TinyML is poised to drive enormous growth within the IoT hardware and software industry. However, measuring the performance of these rapidly proliferating systems and comparing them in a meaningful way presents a considerable challenge; the complexity and dynamicity of the field obscure the measurement of progress and make embedded ML application and system design and deployment intractable. To enable more systematic development while fostering innovation, we need a fair, replicable, and robust method of evaluating TinyML systems. Developed as a collaboration between academia and industry, MLPerf Tiny is a suite of benchmarks for assessing the energy, latency, and accuracy of TinyML hardware, models, and runtimes. The benchmark suite and reference implementations are open-source and available at https://github.com/mlcommons/tiny. Acknowledgments MLPerf Tiny is the work of many individuals from multiple organizations. In this section, we acknowledge all those who helped produce the first set of results or supported the overall benchmark development. This work was also sponsored in part by the ADA (Applications Driving Architectures) Center, a JUMP Center co-sponsored by SRC and DARPA. Amazon Amin Fazel Arizona State University Jae-sun Seo University of York Poonam Yadav A Benchmark Framework The benchmark framework coordinates execution of the benchmark's behavioral model among the various components in the system as described in the following sections. A.1 Framework Hardware In its most basic form, the framework consists of a host PC, a binary runner application GUI, the DUT, and a thin shim of firmware on the DUT that adheres to the framework communication protocol. The framework for this particular benchmark provides two hardware configurations: one for measuring latency and accuracy, and the other for measuring energy (see Figure 3). The former mode is a simple connection between the host PC and the DUT through a serial port. The latter configuration expands the framework to include an electrical-isolation proxy (IO Manager), and an energy monitor (EMON), which supplies and measures energy consumption. Both configurations share the same process for initializing the DUT, loading the input test data, triggering the inference, and collecting results. The only difference is the energy configuration must initialize and manage the IO Manager and EMON hardware as well. Two framework configurations are provided because energy configuration is a more complex setup, and energy scores may not be desired. In the performance configuration, the Host PC talks directly to the DUT. In energy configuration, the IO Manager passes commands from the host software to the DUT through a serial port proxy. This proxy maintains a resilient serial connection state regardless of power cycling, and level shifters provide voltage domain conversion because the IO Manager GPIOs run at 5 Volts, whereas the DUT GPIO may vary. This configuration electrically isolates the DUT. For this framework version, the IO Manager is deployed as an Arduino UNO with its own custom firmware. The energy monitor supplies and measures energy. The runner contains three EMON drivers for the following hardware: the STMicroelectronics LPM01A, the Jetperch Joulescope JS110, and the Keysight N6705. Regardless of the EMON used, it must supply one channel for the DUT and the other for the level shifters, the power used by level-shifters is not included in the total energy score because it is a cost associated with the framework and not the DUT. Only the power delivered to the core is measured. Furthermore, only one power supply is allowed to power the core; no additional energy supplies, such as batteries, supercapacitors, or energy harvesters may be used to defeat the measurement. Some energy monitors provide settings to increase the sample frequency or change the voltage. The runner GUI exposes these options to the user, so that the user can measure energy at a lower voltage. Note that the tradeoff between performance and energy can be dictated by the voltage, as higher voltage is often required to run at higher frequencies, and on-board SMPS conversion efficiency may vary considerably. This is one of the key insights of the benchmark: performance vs. energy tradeoffs. The configurable sampling frequency is the rate at which data is returned from the EMON, which is much lower than the actual ADC sampling rate for all three devices. Since these internal sampling rates are typically well-within the time constant of the decoupling DUT's power delivery decoupling capacitance, the rate has no impact on the score. A.2 Framework Firmware To provide consistency between the benchmark framework and the DUT implementation, a simple API is defined in C code that runs on the DUT. The DUT must provide two basic functions to interface to the host runner: a UART interface for receiving commands and returning status, and a timestamp function. In performance mode, the timestamp function is a local timer with a resolution of at least one millisecond (1 kHz); in energy mode, the timestamp function generates a falling edge GPIO signal which is used to trigger an external timer and synchronize data collection with the energy monitor. The API C code also provides functions to load data into a local buffer, translate and copy that data to the input tensor, trigger an inference, and read back the predictive results. All of this functionality is partitioned into boilerplate code that does not change, or internally implemented, and code that has to be ported to the particular platform, called submitter implemented. Results are stored in a folder and can be reloaded again. An energy viewer allows the user to examine the energy trace ( Figure 6) Figure 1 : 1Summary of the Tiny Machine Learning Stack. There is diversity at every level which makes standardization for benchmarking challenging. Figure 2 : 2The modular design of MLPerf Tiny enables both the direct comparison of solutions and the demonstration of an improvement over the reference. The reference implementations are fully implemented solutions that allow individual components to be swapped out. The components in green can be modified in either division and the orange components can only be modified in the open division. The reference implementations also act as the baseline the results. and thus a well known baseline. Second, it enhances the benchmark suite with a new model type based entirely on FC layers. The model has input and output sizes of 640. Both the encoder and decoder are made of four 128 unit FC layers with BatchNorm applied during the training and with ReLU activation, while the bottleneck layer is of size 8. The model itself is not applied directly on the 10 second audio. The audio is pre-processed into a log-mel-spectrogram with 128 bands and a frame size of 32 ms. Then, the model is used repeatedly over a sliding window of five frames (hence the 640 input size), and the MSE of the resulting reconstruction error is averaged over the central 6.4 second part of the spectrogram providing an overall anomaly score. Figure 3 : 3The two configuration modes of the benchmark framework for (a.) latency and accuracy measurement, or (b.) energy measurement. Figure 4 : 4The graphical user interface (GUI) for the benchmark runner. Figure 2 2illustrates which components of the reference benchmark can be modified in which division. If only the components shaded in green are modified then the submitter can submit to the open division. If any of the components shaded in orange are modified then the submitter must submit to the open division. Figure 5 : 5The energy and latency results of the reference implementations. Each reference implementation was run on the NUCLEO-L4R5ZI board which is shown in the top right. 2 : 2Summary of the MLPerf Tiny v0.5 round of submissions. The submissions are diverse and illustrate the ability of the benchmark to demonstrate an advantage in a variety of ways. The X indicates no modification was made from the reference. The check mark indicates a modification. PTQ refers to post training quantization and QAT refers to quantization aware training. The first round of submissions to MLPerf Tiny (which took place in June 2021) are summarized in Fermilab Benjamin Hawks and Jules Muhizi (Harvard) Google Ian Nappier, Dylan Zika, and David Patterson (UCB) Harvard Max Lam and William Fu KU Leuven Marian Verhelst (IMEC) ON Semiconductor Jeffrey Dods Peng Cheng Lab Yang Shazhou and Tang Hongwei Reality AI Jeff Sieracki Renesas Osama Neiroukh Silicon Labs Dan Riedler STMicroelectronics Mahdi Chtourou Synopsys Dmitry Utyansky Syntiant Mohammadreza Heydary and Rouzbeh Shirvani UCSD Rushil Roy Figure 6 : 6Energy Viewer The reference implementations are open source and available on GitHub at https://github.com/mlcommons/tinyUse Case Dataset Model Quality Target (Input Size) (TFLite Model Size) (Metric) Keyword Spotting Speech Commands (49x10) DS-CNN (52.5 KB) 90% (Top-1) Visual Wake Words VWW Dataset (96x96) MobileNetV1 (325 KB) 80% (Top-1) Image Classification CIFAR10 (32x32) ResNet (96 KB) 85% (Top-1) Anomaly Detection ToyADMOS (5*128) FC-AutoEncoder (270 KB) .85 (AUC) Table 1: MLPerf Tiny v0.5 Inference Benchmarks. Table . Energy -Identical to latency, but in addition to measuring the number of inferences per second, measure the total energy used in the timing window and compute micro-Joules per inference. The same method of taking the median of five measurements is used. The submitter implemented code is the API that connects to the particular SDK, which may consist of optimized MCU libraries, or interface to hardware accelerators.The API implemented by the submitter requires: 1. A timestamp function which prints a timestamp with a minimum resolution of one millisecond, or generates an open-drain, GPIO toggle, depending on whether the DUT is configured latency or energy measurement.2. UART Tx and Rx functionality, for communicating with the framework software.3. A function for loading the input tensor with data sent down by the framework runner UI.4.A function for performing a single inference.5.A function for printing the prediction results.The first two API functions provide external triggering and generic communication support; the latter three implement the minimum requirements to perform inference. Once these functions have been implemented, the framework software can successfully detect and communicate with the DUT.The internally implemented firmware connects the API function calls in such a way that the framework software can send down an input dataset and instruct the firmware to perform a given number of iterations. The results are then extracted from the DUT (or the energy monitor) by the framework software.A.3 Framework SoftwareThe benchmark framework includes a Host PC GUI application called the runner. The runner allows the user to perform some basic setup and configuration, and then executes a pre-defined command script which activates the different hardware components required to execute the benchmark.The runner is needed for three main reasons:1. To provide a consistent benchmark interface to the user.2. To standardize the execution of the benchmark, including measurements 3. To facilitate downloading the large number of input files required for accuracy measurements, since the typical target platform has less than a megabyte of flash memory.The runner also provides visualization of the energy data, and feedback that can be useful for debugging framework connectivity.The runner GUI, for the energy configuration, is shown inFigure 4.The runner detects and initiates a handshake with detected hardware. After initialization, the test is started by clicking a button on the GUI, and a series of asynchronous commands are issued. These commands use the DUT API firmware to load input data and perform measurements. After the test completes, the runner collects the scores from the DUT and presents them to the user. The input stimuli files are located in a directory on the host PC, and are fed into the DUT depending on the measurement.The measurement procedure is as follows:1. Latency -Perform the following five times: download an input stimulus, load the input tensor (converting the data as needed), run inference for a minimum of 10 seconds and 10 iterations, measure the inferences per second (IPS); report the median IPS of the five runs as the score.2. Accuracy -Perform a single inference on the entire set of validation inputs, and collect the output tensor probabilities. The number of inputs vary, depending on the model. From the individual results, compute the Top-1 percent and the AUC. Each model has a minimum accuracy that must be met for the score to be valid. 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A dedicated setup for the in-beam measurement of absolute cross sections of astrophysically relevant charged-particle induced reactions is presented. These, usually very low, cross sections at energies of astrophysical interest are important to improve the modeling of the nucleosynthesis processes of heavy nuclei. Particular emphasis is put on the production of the p nuclei during the astrophysical γ process. The recently developed setup utilizes the highefficiency γ-ray spectrometer HORUS, which is located at the 10 MV FN tandem ion accelerator of the Institute for Nuclear Physics in Cologne.The design of this setup will be presented and results of the recently measured 89 Y(p,γ) 90 Zr reaction will be discussed. The excellent agreement with existing data shows, that the HORUS spectrometer is a powerful tool to determine total and partial cross sections using the in-beam method with high-purity germanium detectors.

10.1016/j.nima.2014.04.025

1405.2703

ca0a36fc45fcc8efe6e0d697a453154d6cf67289

The γ-ray spectrometer HORUS and its applications for Nuclear Astrophysics 12 May 2014 L Netterdon Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany V Derya Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany J Endres Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany C Fransen Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany A Hennig Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany J Mayer Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany C Müller-Gatermann Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany A Sauerwein Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany P Scholz Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany M Spieker Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany A Zilges Institute for Nuclear Physics University of Cologne Zülpicher Straße 77D-50937CologneGermany The γ-ray spectrometer HORUS and its applications for Nuclear Astrophysics 12 May 2014γ-ray spectroscopynuclear astrophysicsmeasured cross-sectionsin-beam method with high-purity germanium detectors A dedicated setup for the in-beam measurement of absolute cross sections of astrophysically relevant charged-particle induced reactions is presented. These, usually very low, cross sections at energies of astrophysical interest are important to improve the modeling of the nucleosynthesis processes of heavy nuclei. Particular emphasis is put on the production of the p nuclei during the astrophysical γ process. The recently developed setup utilizes the highefficiency γ-ray spectrometer HORUS, which is located at the 10 MV FN tandem ion accelerator of the Institute for Nuclear Physics in Cologne.The design of this setup will be presented and results of the recently measured 89 Y(p,γ) 90 Zr reaction will be discussed. The excellent agreement with existing data shows, that the HORUS spectrometer is a powerful tool to determine total and partial cross sections using the in-beam method with high-purity germanium detectors. Introduction Reliable cross-section measurements of radiative capture-reactions are of utmost importance to understand the nucleosynthesis of nuclei heavier than iron, in particular the γ-process nucleosynthesis [1]. This process is believed to be mainly responsible for the production of the 30 -35 neutron-deficient p nuclei. Since the involved cross sections are typically in the µb range or even lower, a reliable measurement is very challenging. Various techniques are used for the measurement of radiative capture cross-sections. One of the most widely applied is the activation technique, which has provided a large amount of experimental data [2,3,4,5,6,7]. However, this technique is limited to reactions resulting in an unstable reaction product with appropriate half-lives. In order to overcome this limitation, mainly two different other methods are applied, namely the inbeam 4π-summing technique [8,9] and the in-beam technique with high-purity germanium (HPGe) detectors [10,11,12]. The γ-ray spectrometer HORUS (High efficiency Observatory for γ-Ray Unique Spectroscopy) [13] in combination with a recently developed target chamber can be used for the latter technique. With this method, the prompt γ-decays of the excited compound nucleus are observed. By measuring the angular distributions using a granular high-efficiency HPGe γ-ray detector array like HORUS at the University of Cologne, absolute reaction cross-sections can be determined. Moreover, since the individual γ-decay patterns are observed, further insight into the structure of the residual nucleus is provided. This might also lead to new nuclear structure results, e.g. spin and parity assignments, see Ref. [11]. In this paper, a new setup utilizing the γ-ray spectrometer HORUS is presented. In Section 2 the design of the HORUS spectrometer is introduced followed by the target chamber dedicated for nuclear astrophysics experiments in Section 3. In Section 4 the results of a first experiment on the 89 Y(p,γ) 90 Zr reaction are discussed. The γ-ray spectrometer HORUS The γ-ray spectrometer HORUS is located at the 10 MV FN tandem ion accelerator at the Institute for Nuclear Physics in Cologne. It consists of up to 14 HPGe detectors, where six of them can be equipped with active BGO shields in order to actively suppress the Compton background. The HORUS spectrometer has a cubic geometry and the 14 detectors are mounted on its eight corners and six faces, respectively. Thus, the detectors are placed at five different angles with respect to the beam axis, namely at 35 • , 45 • , 90 • , 135 • , and 145 • , see Fig. 1. This allows the measurement of angular distributions at five different angles, which is important to determine absolute cross sections. The distances of the HPGe detectors to the target can be freely adjusted. Due to the geometry of the target chamber and mounted BGO shields, the distances typically vary between 9 cm and 16 cm. Because of its flexible architecture, different types of HPGe can be mounted, e.g., clover-type HPGe detectors or cluster-like detectors. Data acquisition The signal processing at HORUS is performed digitally using DGF-4C Rev. F modules, manufactured by the company XIA [14,15]. Each module provides four input channels for the preamplifier signals of the semiconductor detectors as well as four channel-specific VETO inputs, which are used for the active Comptonbackground suppression with BGO shields. The preamplifier signals are digitized by ADCs with a depth of 14 bit and a frequency of 80 MHz. The modules allow the extraction of energy and time information as well as traces of the individual digitized preamplifier signals, if required by the experimentalist. An earlier revision of these modules has been successfully applied to the data acquisition of the Miniball spectrometer [16]. Using these modules, it is possible to acquire data over a wide dynamic range of up to tens of MeV, which is important for astrophysical applications, since very highenergy γ rays must be observed for this purpose. By storing the data event-by-event in a listmode format, it is possible to obtain γγ coincidence data, see Section 2.4. Proton-energy determination In order to obtain a reliable determination of the proton energy, the E p = 3674.4 keV resonance of the 27 Al(p,γ) 28 Si reaction was used [17]. By scanning this resonance, one obtains a calibration of the analyzing magnet. In a first step, the resonance was scanned by varying the projectile energy in small energy steps of down to 0.5 keV. By normalizing the E γ = 1779 keV peak volume stemming from the 27 Al(p,γ) reaction to the accumulated charge, a resonance yield curve was obtained, see Fig. 2. This yield curve shows a sharp lowenergy edge and a plateau. The width of the sharp rising edge of the resonance yield curve is determined by the energy spread of the incoming protons. The location of the plateau is important for the efficiency calibration at high γ-ray energies, see Section 2.3. Since the natural width of this resonance is smaller than 2 keV [17], the width of the plateau is determined by the proton-energy loss inside the 27 Al target. By fitting the resonance yield curve, a spread in the proton energy of ± 4 keV was found for the present setup. It is obvious from Fig. 2, that the center of the rising edge of the yield curve is not located at the literature value of E p = 3674.4 keV, but shifted by ≈ 17 keV to higher energies. This has mainly two reasons. First, the earlier calibration of the analyzing magnet, relating the NMR measurement of the analyzing magnet to the particle energy, might not be exactly valid, since this is very sensitive to the exact geometry of the beamlines and slits defining the entrance to the analyzing magnet. For this purpose, a measurement of the exact proton energy must be done during every conducted experiment, where the particle energy must be precisely known. Secondly, a finite opening angle of the slits in front of the analyzing magnet allows a geometrical uncertainty for the beam entrance into the analyzing magnet, i.e. a skew pathway of the beam. However, since these parameters do not change during the experiment, the beam uncertainty and energy offset can be considered as constant. This constant offset must be taken into account in the data analysis, especially when it comes to determine the energy straggling inside the target material, or when comparing the measured cross-section results to theoretical calculations. Si reaction. This resonance was scanned by varying the proton energy in small steps. This yield curve is used to exactly determine the energy of the incident protons as well as their energy spread. Moreover, this resonance is used for a relative full-energy peak efficiency calibration for γ-ray energies up to E γ ≈ 10.5 MeV, when measuring on top of the resonance. See text for details. Measurement of the full-energy peak efficiency The absolute full-energy peak efficiency is mandatory to derive absolute reaction cross-sections. Up to a γ-ray energy of about 3.6 MeV the full-energy peak efficiency can be determined using standard calibrated radioactive sources. However, for astrophysical applications, the absolute full-energy peak efficiencies for γ-ray energies of up to more than 10 MeV must be precisely known. For this purpose, more sophisticated techniques such as measuring resonances of capture reactions or inelastic scattering reactions must be applied. For this setup, the E p = 3674.4 keV resonance of the 27 Al(p,γ) 28 Si reaction was used [17], which allows the determination of the full-energy peak efficiency up to an energy of E γ = 10509 keV. The resonance with an excitation energy of E x = 15127 keV is populated, when measuring on top of the resonance plateau, see Fig. 2. The absolute branching ratios of the decay of this resonant state are known [17]. They can be used to determine the relative full-energy peak efficiencies, which are subsequently scaled to an absolute efficiency calibration obtained with calibration sources. For the present case, the γ rays with energies of E γ =10509 keV, 8239 keV, 6182 keV, and 4458 keV were used for the efficiency calibration. Fig. 3 shows the absolute full-energy peak efficiency for a typical HPGe detector used in the HO-RUS spectrometer as a function of γ-ray energy. The full-energy peak efficiencies for the lower energy range up to a γ-ray energy of E γ ≈ 2.5 MeV were obtained using standard calibrated 152 Eu and 226 Ra sources. Finally, the full-energy peak efficiencies were obtained by fitting a function of the form f (E γ ) = a · exp b · E γ + c · exp d · E γ(1) to the experimental efficiencies. In order to obtain a reliable efficiency determination over the whole energy region and allow for coincidence summing effects, Monte Carlo simulations were performed using the Geant4 toolkit [18]. These simulations agree very well with the experimentally obtained full-energy peak efficiencies over an energy range from E γ ≈ 350 keV up to the highest γ-ray energy of E γ ≈ 10500 keV, see Fig. 3. Due to the large distance of the HPGe detector to the target and low count rates, summing effects are negligible. Below a γ-ray energy of E γ ≈ 350 keV, the Geant4 simulation tends to overestimate the experimental data. This is due to the fact, that the very details of the detector geometry become significant at such low energies. This includes, e.g., the exact thickness of the detector end caps or dead layers of the HPGe detectors, which might not be known to a sufficient precision. However, this does not play a significant role for the efficiency determination, since the full-energy peak efficiencies for the lower energies can be well determined using standard calibration sources. γγ coincidences The combination of the high granularity of the HO-RUS spectrometer and the event-by-event data format facilitates the construction of γγ coincidence matrices. The γγ coincidence technique is a powerful tool to suppress the beam-induced background. Although it cannot be applied directly to the determination of absolute cross sections, it is most helpful to unambigiously identify the γ-ray transitions visible in the spectra. With this, it can be proved, for instance, that the observed γ-ray transition corresponding to the reaction of interest is free from contaminants from other decaying nuclei resulting from target contaminants. Figs. 4 a) to c) demonstrate the γγ coincidence method. Fig. 4 a) shows a part of a γ-ray spectrum of the 89 Y(p,γ) 90 Zr reaction using 4.7 MeV protons, where no gate was applied. The three marked transitions from higher lying states feeding the 2186 keV-state in 90 Zr can hardly be recognized due to the large beam-induced background. After a gate on the γ-ray transition from the first excited J π = 2 + 1 state to the ground state was applied, these transitions become clearly visible, see The decay of this resonance provides γ rays with energies of up to E γ = 10509 keV, which are used for a relative efficiency determination. The remaining ones were obtained using calibrated radioactive sources. The experimental full-energy peak efficiencies are compared to the efficiencies obtained with a Geant4 simulation (dashed line), which shows an excellent agreement over an energy range from E γ ≈ 350 keV up to the highest γ-ray energy of E γ ≈ 10500 keV. compound state to the first excited state, denoted as γ 1 , can be clearly identified, together with its single escape peak. Moreover, γ-ray transitions from the observed compound nucleus with a low intensity might vanish in the beam-induced background. These transitions would be missing later on, when the total cross section is determined. Using the γγ coincidence technique, the beaminduced background can be reduced so efficiently, that even the weakest γ-ray transitions become visible in the coincidence spectra. The absolute influence of such a transition on the cross section might not be determined using γγ coincidences. But it strongly supports determining an upper limit of the impact of a transition on the total cross section, which is hidden in the beam-induced background. Thus, using this method, systematic uncertainties concerning missing γ-strength during the data analysis are drastically reduced. Target chamber design The target chamber mounted inside the HORUS spectrometer was optimized for nuclear astrophysics experiments. The chamber itself measures 7 cm in length and 5.5 cm in width, see The upper panel a) shows a singles spectrum, i.e., no gate was applied. The three marked transitions are hardly visible due to the beam-induced background. Afterwards, a gate was set on the γ-ray transition from the first excited J π = 2 + 1 state to the ground state. The low-energy part b) shows the transitions from higher lying discrete states in 90 Zr feeding the J π = 2 + 1 state. In the high-energy part c) one can clearly recognize the de-excitation from the compound state to the first excited state, denoted as γ 1 , together with its single escape peak. chamber were kept as small as possible, in order to facilitate a preferably short distance from the HPGe detectors to the target, which allows a high full-energy peak efficiency. The body of the target chamber is made of aluminum with a thickness of 2 mm. It provides a tantalum coating inside with a thickness of 0.1 mm, which is used to suppress competitive reactions on the aluminum housing. The thickness of the tantalum coating was optimized in a way, that on the one hand it is thick enough to stop 5 MeV protons and 15 MeV α-particles, but thin enough not to significantly absorb γ rays down to an energy of E γ = 250 keV. Moreover, this coating can be removed for activation experiments, when γ rays with lower energies have to be observed with the HORUS 7 cm Figure 5: Illustration of the target chamber used for nuclear astrophysics experiments [19]. A cooling finger around the target ladder prevents residual gas deposits on the target. The inset b) shows the inside of the target chamber without the cooling finger. A silicon detector for RBS measurements is also available during the experiments, see Section 3.2. The chamber is coated with tantalum to avoid competetive reactions on the aluminum housing. The cup closely behind the target can be tilted to stop the beam downstream the HORUS spectrometer. spectrometer after the activation. The target is surrounded by a copper tube cooled by LN 2 . This tube serves as a cooling finger and is used to minimize residual gas deposits on the target material. Fig. 5 a) shows an overview of the target chamber including the cooling finger around the target ladder. The inset b) shows a close-up view of the chamber with the cooling finger removed, offering a view on the target ladder. Current read-out Since the total number of particles impinging on the target must be known for the cross-section determination, the beam current is read out at three different positions. Firstly, the current is measured at the target itself and at a Faraday cup, which is located at a distance of 15 cm behind the target. This Faraday cup can be tilted, i.e., one can choose to stop the beam in the Faraday cup closely behind the target or the beam dump downstream the HORUS spectrometer. Additionally, the current is separately read out at the target chamber itself, in order to measure released secondary electrons as well as of the target chamber is used to prevent secondary electrons from hitting the beamline upstream. RBS setup The target chamber additionally houses a silicon detector, which is used for Rutherford Backscattering Spectrometry (RBS) measurements, see Fig 5 a). It is placed at an angle of 135 • relative to the beam axis at a distance of 11 cm to the target to monitor target stability during the experiment and to measure the target thickness. Since the RBS setup became available after the 89 Y(p,γ) measurement, the proof of principle for this setup is shown for a measurement on the 85 Rb(p,γ) reaction in the following. A typical RBS spectrum of an 85 Rb target irradiated with protons with an energy of E p = 4 MeV is shown in Fig. 6. This target was prepared by evaporating RbCl, enriched to (99.78±0.02) % in 85 Rb, onto a 150 mg cm 2 thick gold backing. The two peaks belonging to Rb (right) and Cl (left) can be clearly identified. The measured RBS spectrum was simulated using the SIMNRA code [20], which yields a very good agreement, see Fig. 6. The thickness of this target was measured prior to the experiment at the RBS facility at the RUBION dynamitron-tandem accelerator at the Ruhr-Universität Bochum and amounts to an areal particle density of Rb atoms of (2.01 ± 0.09) × 10 18 1 cm 2 . The RBS measurement in Cologne yields (1.97±0.16)×10 18 1 cm 2 , which is 5 in excellent agreement with the Bochum results. Hence, one can conclude, that no target material was lost during the irradiation. Cross-section measurement of the 89 Y(p,γ) 90 Zr reaction The astrophysically relevant 89 Y(p,γ) 90 Zr reaction was chosen as a test case for the recently developed experimental setup in Cologne. This nucleus is located in a mass region, where the p-nuclei abundances are not well reproduced by reaction-network calculations [21]. It is well-suited as a test case, since it was recently measured in two experiments using the in-beam method with HPGe detectors [12] as well as the 4π-summing technique [8]. The reaction cross-section was measured at five different proton energies ranging from E p = 3.7 MeV to E p = 4.7 MeV. For this experiment, a natural Y target containing 99.9 % of 89 Y was used. It was prepared by vacuum evaporation on a 130 mg cm 2 thick tantalum backing, where the beam was stopped. The target had a thickness of (583 ± 24) µg cm 2 . An RBS measurement at the Ruhr-Universität Bochum before and after the measurement ensured, that no target material was lost during the irradiation. The target was bombarded for several hours with beam currents ranging from 1 nA to 60 nA. The large span of the beam current was due to technical limitations of the accelerator. A typical γ-ray spectrum for the 89 Y(p,γ) reaction using a proton energy of E p = 4.5 MeV is shown in Figs. 7 a) -c). It was obtained by summing up the γ-ray spectra of the six HPGe detectors positioned at an angle of 90 • relative to the beam axis. Despite the rather strong beam-induced background, all relevant γ-ray transitions populating the ground state in the reaction product 90 Zr can be clearly identified, marked by asterisks in Figs. 7 a) and b). Moreover, de-excitations of the compound state up to the 15 th excited state, denoted as γ i , can be observed, see Fig. 7 c). The excitation energies were adopted from Ref. [22]. Cross-section determination In order to determine the total cross section σ of the (p,γ) reaction, the number of produced compound nuclei N comp must be known, which is given by N comp = σ · N proj · m target ,(2) where N proj is the number of projectiles, and m target is the areal particle density of target nuclei. N comp is derived by measuring the angular distributions of all γ rays populating the ground state. The measured intensities of these γ rays are then corrected for the respective number of impinging projectiles for each beam energy, the full-energy peak efficiency as well as the dead time of the data-acquisition system. Subsequently, for every γray transition a sum of Legendre polynomials was fitted to the experimental angular distributions: W i (θ) = A i 0         1 + k=2,4 α k P k (cos θ)         ,(3) with the energy-dependent coefficients A 0 and α k (k = 2, 4). An example of an angular distribution for the γ-ray transition from the E x = 2186 keV [22] level to the ground state in 90 Zr for an incident proton energy of E p = 4.2 MeV is shown in Fig. 8. The cross section can be calculated from the absolute coefficients of the respective angular distributions A i 0 : σ = N i=1 A i 0 m target ,(4) where N is the total number of coefficients, i.e., the number of considered γ-ray transitions. Further details about the data-analysis procedure can be found, e.g., in Ref. [12]. In the compound nucleus 90 Zr, there is an isomeric J π = 5 − state at an excitation energy of E x = 2319 keV with a half-life of 809.2 ms [22]. Thus, one must take care, that no yield stemming from this transition is lost, if only the prompt γ-rays are detected. This was accomplished by acquiring data until ≈ 5 s after the irradiation was stopped, in order to guarantee that all γ-rays stemming from this ground-state transition are detected. By using the method of in-beam γ-ray spectroscopy with HPGe detectors, it is also possible to observe deexcitations of the compound nucleus to various excited states. In the cases of activation or 4π-summing techniques, this is only feasible for isomeric states with a sufficiently long half-life. By investigating the angular distributions for each of these γ-ray transitions, it is possible to derive partial cross sections. The possibility to measure partial cross sections is a tremendous advantage compared to the other aforementioned experimental techniques. Partial cross sections can be used to experimentally constrain the γ-strength function, which is an important input parameter for theoretically predicted astrophysical reaction rates, not only for γ-process calculations, but also for, e.g., neutron-capture reactions during the r process [23]. Experimental results The experimental total cross sections of the 89 Y(p,γ) 90 Zr reaction obtained in this work are given in Table 1 and shown in Fig. 9. The effective energies given in the first column of Table 1 were obtained from E p = E 0 − ∆E 2 ,(5) where E 0 is the incident proton energy including the aforementioned 17 keV offset, see Section 2.2, and ∆E is the average energy loss inside the target material, that was obtained using the Srim code [24] and amounts to 24-28 keV depending on the incident energy. The energy straggling inside the target material was ≈ 8 keV for all energies. The uncertainties in the measured cross sections are composed of the uncertainties in target thickness (≈ 4 %), accumulated charge (≈ 5 %), detector efficiency (≈ 8 %), and the statistical error of the fit of the angular distribution (≈ 4 − 7 %). A comparison of the experimental cross sections to previously measured data of Refs. [8,12] is given in Fig. 9. The present results are in excellent agreement with the previous data. Hence, one can conclude, that the recently developed setup at the Institute for Nuclear Physics in Cologne has become fully operational for cross-section measurements. Note, that the measured partial cross sections are not discussed here. The discussion and their possible impact on the γ-strength function will be subject of a forthcoming publication. Conclusions In this article, the recently developed setup for crosssection measurements relevant for nuclear astrophysics utilizing the high-efficiency γ-ray detector array HO-RUS was presented. HORUS is a highly flexible spectrometer, where also other detector types such as cluster-like detectors or clover-type HPGe detectors can be mounted. It is possible to obtain γγ coincidence data, which is a powerful tool to suppress the beam-induced . The results are compared to previous measurements using the 4π-summing technique (triangles) [8] and the in-beam method with HPGe detectors (cirlces) [12]. The present data are found to be in excellent agreement with the previous measurements. background and to clearly identify the γ-ray transitions of interest. This method drastically reduces systematic uncertainties concerning missing weak γ-ray transitions in the spectra. The recently developed setup for nuclear astrophysics experiments was used to determine total cross sections of the 89 Y(p,γ) 90 Zr reaction at five different energies between E p = 3.7 − 4.7 MeV. The data at hand is compared to previously measured data of Refs. [8,12] and an excellent agreement is found. Moreover, for the first time partial cross sections of this reaction were measured. The results and their discussion concerning the impact of these measurements on the γ-strength function will be presented in a forthcoming publication. The method of in-beam γ-ray spectroscopy allows studying reactions with a stable reaction product. Thus, the experimental data needed for, e.g., γ-process network calculations can be widely extended. The γ-ray spectrometer HORUS embodies an excellent tool to study charged-particle induced reactions at energies of astrophysical interest. Figure 1 : 1Drawing of the HORUS γ-ray spectrometer. The 14 HPGe detectors are mounted on the eight cornes and six faces of a cube geometry. This allows the measurement at five different angles relative to the beam axis, namely at 35 • (detectors 12 and 13), 45 • (detectors 7 and 8), 90 • (detectors 0 to 5), 135 • (detectors 6 and 9), and 145 • (detectors 10 and 11). online) Resonance yield curve of the E p = 3674.4 keV resonance of the 27 Al(p,γ) 28 Figure 3 : 3Fig. 4 b).Fig. 4 c)shows the high-energy part of the coincidence spectrum. The transition from the (Color online) Full-energy peak efficiency for one of the HPGe detectors with a distance of 14 cm to the target. The efficiencies for γ-ray energies higher than E γ = 4 MeV were obtained by using the E p = 3674.4 keV resonance of the 27 Al(p,γ) 28 Si reaction. Fig. 5 .Figure 4 : 54The Excerpt from a γγ coincidence spectrum of the 89 Y(p,γ) 90 Zr reaction using 4.7 MeV protons. Figure 6 : 6scattered beam particles. The accumulated charge is individually determined by current integrators with an overall uncertainty of about 5 %. A negatively charged aperture with a voltage of U S = −400 V at the entrance (Color online) Relevant part of the RBS spectrum of a measured RbCl target, using the built-in RBS detector with 4 MeV protons. The spectrum is very well reproduced by a simulation using the SIMNRA code (dashed line). The target thickness was additionally measured at the RUBION facility in Bochum, Germany, prior to the experiment. Both measurement yield an excellent agreement within the experimental uncertainties. The peaks belonging to Rb (right) and Cl (left) can be clearly identified on top of the Au backing. See text for details. Figure 7 : 7Typical γ-ray spectrum recorded during the irradiation of 89 Y with 4.5 MeV protons. This spectrum was obtained by summing over all HPGe detectors at an angle of 90 • relative to the beam axis. All transitions to the ground state of the reaction product 90 Zr are marked with an asterisk. The high-energy part c) shows de-excitations from the compound state to the ground state or one of the excited states. The transition to the ground state is indicated by γ 0 , to the first excited state as γ 1 , and so on. De-excitations up to the 15 th excited state were observed. For the strongest transitions, the single and / or double escape peak is marked, too. Figure 8 : 8Angular distribution for the 2 + 1 → 0 + g.s. γ-ray transition in 90 Zr. The incident proton energy was E p = 4.2 MeV. The dashed line depicts the sum of Legendre polynomials fitted to the experimental data. Figure 9 : 9et al. Harissopulos et al. This work (Color online) Total cross section for the 89 Y(p,γ) 90 Zr reaction as a function of the effective center-of-mass energy E c.m. 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When people converse about social or political topics, similar arguments are often paraphrased by different speakers, across many different conversations. Debate websites produce curated summaries of arguments on such topics; these summaries typically consist of lists of sentences that represent frequently paraphrased propositions, or labels capturing the essence of one particular aspect of an argument, e.g. Morality or Second Amendment. We call these frequently paraphrased propositions ARGUMENT FACETS. Like these curated sites, our goal is to induce and identify argument facets across multiple conversations, and produce summaries. However, we aim to do this automatically. We frame the problem as consisting of two steps: we first extract sentences that express an argument from raw social media dialogs, and then rank the extracted arguments in terms of their similarity to one another. Sets of similar arguments are used to represent argument facets. We show here that we can predict ARGUMENT FACET SIMI-LARITY with a correlation averaging 0.63 compared to a human topline averaging 0.68 over three debate topics, easily beating several reasonable baselines.

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Measuring the Similarity of Sentential Arguments in Dialog Association for Computational LinguisticsCopyright Association for Computational LinguisticsSeptember 2016. 2016 Amita Misra University of California Santa Cruz Natural Language and Dialog Systems Lab 1156 N. High. SOE-3 Santa Cruz95064CaliforniaUSA Brian Ecker University of California Santa Cruz Natural Language and Dialog Systems Lab 1156 N. High. SOE-3 Santa Cruz95064CaliforniaUSA Marilyn A Walker University of California Santa Cruz Natural Language and Dialog Systems Lab 1156 N. High. SOE-3 Santa Cruz95064CaliforniaUSA Measuring the Similarity of Sentential Arguments in Dialog Proceedings of the SIGDIAL 2016 Conference the SIGDIAL 2016 ConferenceLos Angeles, USAAssociation for Computational LinguisticsSeptember 2016. 2016 When people converse about social or political topics, similar arguments are often paraphrased by different speakers, across many different conversations. Debate websites produce curated summaries of arguments on such topics; these summaries typically consist of lists of sentences that represent frequently paraphrased propositions, or labels capturing the essence of one particular aspect of an argument, e.g. Morality or Second Amendment. We call these frequently paraphrased propositions ARGUMENT FACETS. Like these curated sites, our goal is to induce and identify argument facets across multiple conversations, and produce summaries. However, we aim to do this automatically. We frame the problem as consisting of two steps: we first extract sentences that express an argument from raw social media dialogs, and then rank the extracted arguments in terms of their similarity to one another. Sets of similar arguments are used to represent argument facets. We show here that we can predict ARGUMENT FACET SIMI-LARITY with a correlation averaging 0.63 compared to a human topline averaging 0.68 over three debate topics, easily beating several reasonable baselines. Introduction When people converse about social or political topics, similar arguments are often paraphrased by different speakers, across many different conversations. For example, consider the dialog excerpts in Fig. 1 from the 89K sentences about gun control in the IAC 2.0 corpus of online dialogs (Abbott et al., 2016). Each of the sentences S1 to S6 provide different linguistic realizations of the same proposition namely that Criminals will have guns even if gun ownership is illegal. S1: To inact a law that makes a crime of illegal gun ownership has no effect on criminal ownership of guns.. S2: Gun free zones are zones where criminals will have guns because criminals will not obey the laws about gun free zones. S3: Gun control laws do not stop criminals from getting guns. S4: Gun control laws will not work because criminals do not obey gun control laws! S5: Gun control laws only control the guns in the hands of people who follow laws. S6: Gun laws and bans are put in place that only affect good law abiding free citizens. Debate websites, such as Idebate and ProCon produce curated summaries of arguments on the gun control topic, as well as many other topics. 12 These summaries typically consist of lists, e.g. Fig. 2 lists eight different aspects of the gun control argument from Idebate. Such manually curated summaries identify different linguistic realizations of the same argument to induce a set of common, repeated, aspects of arguments, what we call ARGUMENT FACETS. For example, a curator might identify sentences S1 to S6 in Fig. 1 with a label to represent the facet that Criminals will have guns even if gun ownership is illegal. Like these curated sites, we also aim to induce and identify facets of an argument across multiple conversations, and produce summaries of all the different facets. However our aim is to do this automatically, and over time. In order to simplify the problem, we focus on SENTENTIAL ARGU-MENTS, single sentences that clearly express Pro Arguments A1: The only function of a gun is to kill. A2: The legal ownership of guns by ordinary citizens inevitably leads to many accidental deaths. A3: Sports shooting desensitizes people to the lethal nature of firearms. A4: Gun ownership increases the risk of suicide. Con Arguments A5: Gun ownership is an integral facet of the right to self defense. A6: Gun ownership increases national security within democratic states. A7: Sports shooting is a safe activity. A8: Effective gun control is not achievable in democratic states with a tradition of civilian gun owership. a particular argument facet in dialog. We aim to use SENTENTIAL ARGUMENTS to produce extractive summaries of online dialogs about current social and political topics. This paper extends our previous work which frames our goal as consisting of two tasks (Misra et al., 2015;Swanson et al., 2015). • Task1: Argument Extraction: How can we extract sentences from dialog that clearly express a particular argument facet? • Task2: Argument Facet Similarity: How can we recognize that two sentential arguments are semantically similar, i.e. that they are different linguistic realizations of the same facet of the argument? Task1 is needed because social media dialogs consist of many sentences that either do not express an argument, or cannot be understood out of context. Thus sentences that are useful for inducing argument facets must first be automatically identified. Our previous work on Argument Extraction achieved good results, (Swanson et al., 2015), and is extended here (Sec. 2). Task2 takes pairs of sentences from Task1 as input and then learns a regressor that can predict Argument Facet Similarity (henceforth AFS). Related work on argument mining (discussed in more detail in Sec. 4) defines a finite set of facets for each topic, similar to those from Idebate in Fig. 2. 3 Previous work then labels posts or sentences using these facets, and trains a classifier to return a facet label (Conrad et al., 2012;Hasan and Ng, 2014;Boltuzic andŠnajder, 2014;Naderi and Hirst, 2015), inter alia. However, this simplification may not work in the long term, both because the sentential realizations of argument facets are propositional, and hence graded, and because facets evolve over time, and hence cannot be represented by a finite list. In our previous work on AFS, we developed an AFS regressor for predicting the similarity of human-generated labels for summaries of dialogic arguments (Misra et al., 2015). We collected 5 human summaries of each dialog, and then used the Pyramid tool and scheme to annotate sentences from these summaries as contributors to (paraphrases of) a particular facet (Nenkova and Passonneau, 2004). The Pyramid tool requires the annotator to provide a human readable label for a collection of contributors that realize the same propositional content. The AFS regressor operated on pairs of human-generated labels from Pyramid summaries of different dialogs about the same topic. In this case, facet identification is done by the human summarizers, and collections of similar labels represent an argument facet. We believe this is a much easier task than the one we attempt here of training an AFS regressor on automatically extracted raw sentences from social media dialogs. The contributions of this paper are: • We develop a new corpus of sentential arguments with gold-standard labels for AFS. • We analyze and improve our argument extractor, by testing it on a much larger dataset. We develop a larger gold standard corpus for ARGUMENT QUALITY (AQ). • We develop a regressor that can predict AFS on extracted sentential arguments with a correlation averaging 0.63 compared to a human topline of 0.68 for three debate topics. 4 Corpora and Problem Definition Many existing websites summarize the frequent, and repeated, facets of arguments about current topics, that are linguistically realized in different ways, across many different social media and debate forums. For example, Fig. 2 illustrates the eight facets for gun control on IDebate. Fig. 3 illustrates a different type of summary, for the death penalty topic, from ProCon, where the argument facets are called out as the "Top Ten Pros and Cons" and given labels such as Morality, Constitutionality and Race. See the top of Fig. 3. The bottom of Fig. 3 shows how each facet is then elaborated by a paragraph for both its Pro and Con side: due to space we only show the summary for the Morality facet here. These summaries are curated, thus one would Figure 3: Facets of the death penalty debate as curated on ProCon.org not expect that different sites would call out the exact same facets, or even that the same type of labels would be used for a specific facet. As we can see, ProCon ( Fig. 3) uses one word or phrasal labels, while IDebate ( Fig. 2) describes each facet with a sentence. Moreover, these curated summaries are not produced for a particular topic once-and-for-all: the curators often reorganize their summaries, drawing out different facets, or combining previously distinct facets under a single new heading. We hypothesize that this happens because new facets arise over time. For example, it is plausible that for the gay marriage topic, the facet that Gay marriage is a civil rights issue came to the fore only in the last ten years. Our long-term aim is to produce summaries similar to these curated summaries, but automatically, and over time, so that as new argument facets arise for a particular topic, we can identify them. We begin with three debate topics, gun control (38102 posts), gay marriage (22425 posts) and death penalty (5283 posts), from the Internet Argument Corpus 2.0 (Abbott et al., 2016). We first need to create a large sample of high quality sentential arguments (Task1 above) and then create a large sample of paired sentential arguments in order to train the model for AFS (Task2 above). Argument Quality Data We extracted all the sentences for all of the posts in each topic to first create a large corpus of topicsorted sentences. See Table 1. We started with the Argument Quality (AQ) re- gressor from Swanson et al. (2015), which gives a score to each sentence. The AQ score is intended to reflect how easily the speaker's argument can be understood from the sentence without any context. Easily understandable sentences are assumed to be prime candidates for producing extractive summaries. In Swanson et al. (2015), the annotators rated AQ using a continuous slider ranging from hard (0.0) to easy to interpret (1.0). We refined the Mechanical Turk task to elicit new training data for AQ as summarized in Table 1. Fig. 8 in the appendix shows the HIT we used to collect new AQ labels for sentences, as described below. We expected to to apply Swanson's AQ regressor to our sample completely "out of the box". However, we first discovered that many sentences given high AQ scores were very similar, while we need a sample that realizes many diverse facets. We then discovered that some extracted sentential arguments were not actually high quality. We hypothesized that the diversity issue arose primarily because Swanson's dataset was filtered using high PMI n-grams. We also hypothesized that the quality issue had not surfaced because Swanson's sample was primarily selected from sentences marked with the discourse connectives but, first, if, and so. Our sample (Original column of Table 1) is much larger and was not similarly filtered. 4 plots the distribution of word counts for sentences from our sample that were given an AQ score > 0.91 by Swanson's trained AQ regressor. The first bin shows that many sentences with less than 10 words are predicted to be high quality, but many of these sentences in our data consisted of only a few elongated words (e.g. HA-HAHAHA...). The upper part of the distribution shows a large number of sentences with more than 70 words with a predicted AQ > 0.91. We discovered that most of these long sentences are multiple sentences without punctuation. We thus refined the AQ model by removing duplicate sentences, and rescoring sentences without a verb and with less than 4 dictionary words to AQ = 0. We then restricted our sampling to sentences between 10 and 40 tokens, to eliminate run-on sentences and sentences without much propositional content. We did not retrain the regressor, rather we resampled and rescored the corpus. See the Rescored column of Table 1. After removing the two tails in Fig. 4, the distribution of word counts is almost uniform across bins of sentences from length 10 to 40. As noted above, the sample in Swanson et al. (2015) was filtered using PMI, and PMI contributes to AQ. Thus, to end up with a diverse set of sentences representing many facets of each topic, we decided to sample sentences with lower AQ scores than Swanson had used. We binned the sentences based on predicted AQ score and extracted random samples across bins ranging from .55-1.0, in increments of .10. Then we extracted a smaller sample and collected new AQ annotations for gay marriage and death penalty on Mechanical Turk, using the definitions in Fig. 8 (in the appendix). See the Sampled column of Table 1. We pre-selected three annotators using a qualifier that included detailed instructions and sample annotations. A score of 3 was mapped to a yes and scores of 1 or 2 mapped to a no. We simplified the task slightly in the HIT for gun control, where five annotators were instructed to select a yes label if the sentence clearly expressed an argument (score 3), or a no label otherwise (score 1 or 2). We then calculated the probability that the sentences in each bin were high quality arguments using the resulting AQ gold standard labels, and found that a threshhold of predicted AQ > 0.55 maintained both diversity and quality. See Fig. 9 in the appendix. Table 1 summarizes the results of each stage of the process of producing the new AQ corpus of 6188 sentences (Sampled and then annotated). The last column of Table 1 shows that gold standard labels agree with the rescored AQ regressor between 77% and 88% of the time. Argument Facet Similarity Data The goal of Task2 is to define a similarity metric and train a regression model that takes as input two sentential arguments and returns a scalar value that predicts their similarity(AFS). The model must reflect the fact that similarity is graded, e.g. the same argument facet may be repeated with different levels of explicit detail. For example, sentence A1 in Fig. 2 is similar to the more complete argument, Given the fact that guns are weapons-things designed to kill-they should not be in the hands of the public, which expresses both the premise and conclusion. Sentence A1 leaves it up to the reader to infer the (obvious) conclusion. S7: Since there are gun deaths in countries that have banned guns, the gun bans did not work. S8: It is legal to own weapons in this country, they are just tightly controlled, and as a result we have far less gun crime (particularly where it's not related to organised crime). S9: My point was that the theory that more gun control leaves people defenseless does not explain the lower murder rates in other developed nations. Our approach to Task2 draws strongly on recent work on semantic textual similarity (STS) (Agirre et al., 2013;Dolan and Brockett, 2005;Mihalcea et al., 2006). STS measures the degree of semantic similarity between a pair of sentences with values that range from 0 to 5. Inspired by the scale used for STS, we first define what a facet is, and then define the values of the AFS scale as shown in Fig. 10 in the appendix (repeated from Misra et al. (2015) for convenience). We distinguish AFS from STS because: (1) our data are so different: STS data consists of descriptive sentences whereas our sentences are argumentative excerpts from dialogs; and (2) our definition of facet allows for sentences that express opposite stance to be realizations of the same facet (AFS = 3) in Fig. 10. Related work has primarily used entailment or semantic equivalence to define argument similarity (Habernal and Gurevych, 2015;Boltuzic anď Snajder, 2015;Boltuzic andŠnajder, 2015;Habernal et al., 2014). We believe the definition of AFS given in Fig. 10 will be more useful in the long run than semantic equivalence or entailment, because two arguments can only be contradictory if they are about the same facet. For example, consider that sentential argument S7 in Fig. 5 is anti gun-control, while sentences S8 and S9 are pro gun-control. Our annotation guidelines label them with the same facet, in a similar way to how the curated summaries on ProCon provides both a Pro and Con side for each facet. See Fig. 3. Figure 6: The distribution of AFS scores as a function of UMBC STS scores for gun control sentences. In order to efficiently collect annotations for AFS, we want to produce training data pairs that are more likely than chance to be the same facet (scores 3 and above as defined in Fig. 10). Similar arguments are rare with an all-pairs matching protocol, e.g. in ComArg approximately 67% of the annotations are "not a match" (Boltuzic anď Snajder, 2014). Also, we found that Turkers are confused when asked to annotate similarity and then given a set of sentence pairs that are almost all highly dissimilar. Annotations also cost money. We therefore used UMBC STS (Han et al., 2013) to score all potential pairs. 5 To foreshadow, the plot in Fig. 6 shows that this pre-scoring works: (1) the lower quadrant of the plot shows that STS < .20 corresponds to the lower range of scores for AFS; and (2) the lower half of the left hand side shows that we still get many arguments that are low AFS (values below 3) in our training data. We selected 2000 pairs in each topic, based on their UMBC similarity scores, which resulted in lowest UMBC scores of 0.58 for GM, 0.56 for GC and 0.58 for DP. To ensure a pool of diverse arguments, a particular sentence can appear in at most ten pairs. MT workers took a qualification test with definitions and instructions as shown in Fig. 10. Sentential arguments with sample AFS annotations were part of the qualifier. The 6000 pairs were made available to our three most reliable pre-qualified workers. The last row of Table 3 reports the human topline for the task, i.e. the average pairwise r across all three workers. Interestingly, the Gay marriage topic (r = 0.60) is more difficult for human annotators than either Death Penalty (r = 0.74) or Gun Control (r = 0.69). Argument Facet Similarity Given the data collected above, we defined a supervised machine learning experiment with AFS as our dependent variable. We developed a number of baselines using off the shelf tools. Features are grouped into sets and discussed in detail below. Feature Sets NGRAM cosine. Our primary baseline is an ngram overlap feature. For each argument, we extract the unigrams, bigrams and trigrams, and then calculate the cosine similarity between two texts represented as vectors of their ngram counts. Rouge. Rouge is a family of metrics for comparing the similarity of two summaries (Lin, 2004), which measures overlapping units such as continuous and skip ngrams, common subsequences, and word pairs. We use all the rouge f-scores from the pyrouge package. Our analysis shows that rouge s* f score correlates most highly with AFS. 6 UMBC STS. We consider STS, a measure of the semantic similarity of two texts (Agirre et al., 2012), as another baseline, using the UMBC STS tool. Fig. 6 illustrates that in general, STS is rough approximation of AFS. It is possible that our selection of data for pairs for annotation using UMBC STS either improves or reduces its performance. Google Word2Vec. Word embeddings from word2vec (Mikolov et al., 2013) are popular for expressing semantic relationships between words, but using word embeddings to express entire sentences often requires some compromises. In particular, averaging word2vec embeddings for each word may lose too much information in long sentences. Previous work on argument mining has developed methods using word2vec that are effective for clustering similar arguments (Habernal and Gurevych, 2015;Boltuzic andŠnajder, 2015) Other research creates embeddings at the sentence level using more advanced techniques such as Paragraph Vectors (Le and Mikolov, 2014). We take a more direct approach in which we use the word embeddings directly as features. For each sentential argument in the pair, we create a 300-dimensional vector by filtering for stopwords and punctuation and then averaging the word embeddings from Google's word2vec model for the remaining words. 7 Each dimension of the 600 dimensional concatenated averaged vector is used directly as a feature. In our experiments, this concatenation method greatly outperforms cosine similarity (Table 2, Table 3). Sec. 3.3 discusses properties of word embeddings that may yield these performance differences. Custom Word2Vec. We also create our own 300-dimensional embeddings for our dialogic domain using the Gensim library (Řehůřek and Sojka, 2010), with default settings, and a very large corpus of user-generated dialogic content. This includes the corpus described in Sec. 2 (929, 206 forum posts), an internal corpus of 1, 688, 639 tweets on various topics, and a corpus of 53, 851, 542 posts from Reddit. 8 LIWC category and Dependency Overlap. Both dependency structures and the Linguistics Inquiry Word Count (LIWC) tool have been useful in previous work (Pennebaker et al., 2001;Somasundaran and Wiebe, 2009;Hasan and Ng, 2013). We develop a novel feature set that combines LIWC category and dependency overlap, aiming to capture a generalized notion of concept overlap between two arguments, i.e. to capture the hypothesis that classes of content words such as affective processes or emotion types are indicative of a shared facet across pairs of arguments. We create partially generalized LIWC dependency features and count overlap normalized by sentence length across pairs, building on previous work (Joshi and Penstein-Rosé, 2009). Stanford dependency features are generalized by leaving one dependency element lexicalized, replacing the other word in the dependency relation with its LIWC category and by removing the actual dependency type (nsubj, dobj, etc.) from the triple. This creates a tuple of ("governor token", LIWC category of dependent token). We call these simplified LIWC dependencies. Fig. 7 illustrates the generalization process for three LIWC simplified dependencies, ("deter", "fear), ("deter", "punishment"), and ("deter", "love"). Because LIWC is a hierarchical lexicon, two dependencies may share many generalizations or only a few. Here, the tuples with dependent tokens fear and punishment are more closely related because their shared generalization include both Negative Emotion and Affective Processes, but the tuples with dependent tokens fear and love have a less similar relationship, because they only share the Affective Processes generalization. Machine Learning Regression Results We randomly selected 90% of our annotated pairs to use for nested 10-fold cross-validation, setting aside 10% for qualitative analysis of predicted vs. gold-standard scores. We use Ridge Regression (RR) with l2-norm regularization and Support Vector Regression (SVR) with an RBF kernel from scikit-learn (Pedregosa et al., 2011). Performance evaluation uses two standard measures, Correlation Coefficient (r) and Root Mean Squared Error (RMSE). A separate inner crossvalidation within each fold of the outer crossvalidation is used to perform a grid search to determine the hyperparameters for that outer fold. The outer cross-validation reports the scoring metrics. Simple Ablation Models. We first evaluate simple models based on a single feature using both RR and SVR. Table 2, Rows 1, 2, and 3 show the baseline results: UMBC Semantic Textual Similarity (STS), Ngram Cosine, and Rouge. Surprisingly, the UMBC STS measure does not perform as well as Ngram Cosine for Death Penalty and Gay Marriage. LIWC dependencies (Row 4) perform similarly to Rouge (Row 3) across topics. Cosine similarity for the custom word2vec model (Row 5) performs about as well or better than ngrams across topics, but cosine similarity using the Google model (Row 6) performs worse than ngrams for all topics except Death Penalty. Interestingly our custom Word2Vec models perform significantly better than the Google word2vec models for Gun Control and Gay Marriage, with both much higher r and lower RMSE, while performing identically for Death Penalty. Feature Combination Models. Table 3 shows the results of testing feature combinations to learn which ones are complementary. Since SVR consistently performs better than RR, we use SVR only. Significance is calculated using paired t-tests between the RMSE values across folds. We paired Ngrams separately with LIWC and ROUGE to evaluate if the combination is significant. Ngram+Rouge (Row 1) is significantly better than Ngram for Gun Control and Death Penalty (p < .01), and Gay Marriage (p = .03). Our best result using our hand-engineered features is a combination of LIWC, Rouge, and Ngrams (Row 3). Interestingly, adding UMBC STS (Row 4) gives a small but significant improvement (p < 0.01 for gun control; p = 0.07 for gay marriage). Thus we take Ngrams, LIWC, Rouge, and UMBC STS (Row 4) as our best handengineered model across all topics with a correlation of 0.65 for gun control, 0.50 for death penalty and 0.40 for gay marriage. This combination is significantly better than the baselines for Ngram baseline (p < .01), UMBC STS (p <= .02) and Rouge (p < .01) for all three topics. We then further combine the hand-engineered features (Row 4) with the Google Word2Vec features (Row 6), creating the model in Row 8. A paired t-test between RMSE values from each cross-validation fold for each model (Row 4 vs. Row 8 and Row 6 vs. Row 8) shows that the our hand-engineered features are complementary to Word2Vec, and their combination yields a model significantly better than either model alone (p < .01). We note that although the custom word2vec model performs much better for gun control and gay marriage when using cosine, it actually performs slightly but significantly (p = .05) worse when using concatenation with hand-engineered features. This may simply be due to the size of the training data, i.e. the Google model used nearly twice as much training data, while our domain-specific word2vec model achieves comparable performance to the Google model with much less training data. Analysis and Discussion Although it is common to translate word embeddings into single features or reduced feature sets for similarity through the use of clustering (Habernal and Gurevych, 2015) or cosine similarity (Boltuzic andŠnajder, 2015), we show that it is possible to improve results by directly combining word embeddings with hand-engineered features. In our task, sentences were limited to a maximum of 40 tokens in order to encourage singlefacet sentences, but this may have provided an additional benefit by allowing us to average word embeddings while still preserving useful signal. Our results also demonstrate that using concate-ID Argument 1 Argument 2 STS Ngram Rouge LIWC dep W2Vec AFS MT AFS GC1 You say that gun control must not be effective because the study's conclusions about gun control were inconclusive. You're right that gun control isn't about guns, however, but 'control' is a secondary matter, a means to an end. GC3 They do not have the expressed, enumerated power to pass any law regarding guns in the constitution. Which passed the law requireing "smart guns", if they ever become available (right now they do not exist). GM4 Technically though marriage is not discrimination, because gays are still allowed to marry the opposite sex. Everyone has the right to marry someone of the opposite sex, and with gay marriage, everyone will have the right to marry someone of the same AND opposite sex. GM5 If the state wants to offer legal protections and benefits to straight married couples, it cannot constitutionally refuse equal protections to gay ones. Same-sex couples are denied over 1,000 benefits, rights, and protections that federal law affords to married, heterosexual couples, as well as hundreds of such protections at the state level. Table 4: Illustrative Argument pairs, along with the predicted scores from individual feature sets, predicted(AFS) and the Mechanical Turk human topline (MT AFS). The best performing feature set is shown in bold. GC=Gun Control, DP=Death Penalty, GM=Gay Marriage. nation for learning similarity with vector representations works much better than the common practice of reducing a pair of vectors to a single score using cosine similarity. Previous work (Li et al., 2015;Pennington et al., 2014) also shows that all dimensions are not equally useful predictors for a specific task. For sentiment classification, Li et al. (2015) find that "too large a dimensionality leads many dimensions to be non-functional ... causing two sentences of opposite sentiment to differ only in a few dimensions." This may also be the situation for the 300-dimensional embeddings used for AFS. Hence, when using concatenation, single dimensions can be weighted to adjust for non-functional dimensions, but using cosine makes this per-dimension weighting impossible. This might explain why our custom word2vec model outperforms the Google model when using cosine as compared to concatenation, i.e. more dimensions are informative in the custom model, but overall, the Google model provides more complementary information when non-functional dimensions are accounted for. More analysis is needed to fully support this claim. To qualitatively illustrate some of the differences between our final AFS regressor model (Row 8 of Table 3) and several baselines, we apply the model to a set-aside 200 pairs per topic. Table 4 shows examples selected to highlight the strengths of AFS prediction for different models as compared to the AFS gold standard scores. MT AFS values near 1 indicate same topic but no similarity. Rows GC1 and DP2 talk about totally different facets and only share the same topic (AFS = 1). Rouge and Ngram features based on word overlap predict scores that are too high. In contrast, LIWC dependencies and word2vec based on concept and semantic overlap are more accurate. MT values near 3 indicate same facet but somewhat different arguments. Arguments in row GM4 talk about marriage rights to all, and there is some overlap in these arguments beyond simply being the same topic, however the speakers are on opposite stance sides. Both of the arguments in row GM5 (MT AFS of 3.3) reference the same facet of the financial and legal benefits available to married couples, but Arg2 is more specific. Both Word2vec and our trained AFS model can recognize the similarity in the concepts in the two arguments and make good predictions. MT values above 4 indicate two arguments that are the same facet and very similar. Row DP6 gets a high Rouge overlap score and Word2vec relates 'lower crime rate' as semantically similar to 'deter murder rates' thus yielding an accurately high AFS score. DP7 is an example where LIWC dependencies perform better as compared to other features, because it focuses in on the dependency between the death penalty and cost, but none of the models do well at predicting the MT AFS score. One issue here may be that, despite our attempts to sample pairs with more representatives of high AFS, there is just less training data available for this part of the distribution. Hence all the regressors will be conservative at predicting the highest values. We hope in future work to improve our AFS regressor by finding additional methods for populating the training data with more highly similar pairs. Related Work There are many theories of argumentation that might be applicable for our task (Jackson and Jacobs, 1980;Reed and Rowe, 2004;Walton et al., 2008;Gilbert, 1997;Toulmin, 1958;Dung, 1995), but one definition of argument structure may not work for every NLP task. Social media arguments are often informal, and do not necessarily follow logical rules or schemas of argumentation (Stab and Gurevych, 2014;Peldszus and Stede, 2013;Ghosh et al., 2014;Habernal et al., 2014;Goudas et al., 2014;Cabrio and Villata, 2012). Moreover, in social media, segments of text that are argumentative must first be identified, as in our Task1. Habernal and Gurevych (2016) train a classifier to recognize text segments that are argumentative, but much previous work does Task1 manually. Goudas et al. (2014) annotate 16,000 sentences from social media documents and consider 760 of them to be argumentative. Hasan and Ng (2014) also manually identify argumentative sentences, while Boltuzic andŠnajder (2014) treat the whole post as argumentative, after manually removing "spam" posts. Biran and Rambow (2011) automatically identify justifications as a structural component of an argument. Other work groups semantically-similar classes of reasons or frames that underlie a particular speaker's stance, what we call ARGUMENT FACETS. One approach categorizes sentences or posts using topic-specific argument labels, which are functionally similar to our facets as discussed above (Conrad et al., 2012;Hasan and Ng, 2014;Boltuzic andŠnajder, 2014;Naderi and Hirst, 2015). For example, Fig. 2 lists facets A1 to A8 for Gun Control from the IDebate website; Boltuzic andŠnajder (2015) use this list to label posts. They apply unsupervised clustering using a semantic textual similarity tool, but evaluate clusters using their hand-labelled argument tags. Our method instead explicitly models graded similarity of sentential arguments. Conclusion and Future Work We present a method for scoring argument facet similarity in online debates using a combination of hand-engineered and unsupervised features with a correlation averaging 0.63 compared to a human top line averaging 0.68. Our approach differs from similar work that finds and groups the "reasons" underlying a speakers stance, because our models are based on the belief that it is not possible to define a finite set of discrete facets for a topic. A qualitative analysis of our results, illustrated by Table 4, suggests that treating facet discovery as a similarity problem is productive, i.e. examination of particular pairs suggests facets about legal and financial benefits for same-sex couples, the claim that the death penalty does not actually affect murder rates, and an assertion that "they", implying "congress", do not have the express, enumerated power to pass legislation restricting guns. Previous work shows that metrics used for evaluating machine translation quality perform well on paraphrase recognition tasks (Madnani et al., 2012). In our experiments, ROUGE performed very well, suggesting that other machine translation metrics such as Terp and Meteor may be useful (Snover et al., 2009;Lavie and Denkowski, 2009). We will explore this in future work. In future, we will use our AFS regressor to cluster and group similar arguments and produce argument facet summaries as a final output of our pipeline. Habernal and Gurevych (2015) apply clustering in argument mining by averaging word embeddings from posts and sentences from debate portals, clustering the resulting averaged vectors, and then computing distance measures from clusters to unseen sentences ("classification units") as features. Cosine similarity between weighted and summed vector representations is also a common approach, and Boltuzic andŠnajder (2015) show word2vec cosine similarity beats bag-ofwords and STS baselines when used with clustering for argument identification. Finally, our AQ extractor treats all posts on a topic equally, operating on a set of concatenated posts. We will explore other sampling methods to ensure that the AQ extractor does not eliminate arguments made by less articulate citizens, by e.g. enforcing that "Every speaker in a debate contributes at least one argument". We will also sample by stance-side, so that summaries can be organized using "Pro" and "Con", as in curated summaries. Our final goal is to combine quality-based argument extraction, our AFS model, stance, post and author level information, so that our summaries represent the diversity of views on a topic, a quality not always guaranteed by summarization techniques, human or machine. Figure 9 shows the relation between predicted AQ score and gold-standard argument quality annotations. Figure 9: Probability of sentential argument for AQ score across bin for Death Penalty. Figure 10 provides our definition of FACET and instructions for AFS annotation. This is repeated here from (Misra et al., 2015) for the reader's convenience. Facet: A facet is a low level issue that often reoccurs in many arguments in support of the author's stance or in attacking the other author's position. There are many ways to argue for your stance on a topic. For example, in a discussion about the death penalty you may argue in favor of it by claiming that it deters crime. Alternatively, you may argue in favor of the death penalty because it gives victims of the crimes closure. On the other hand you may argue against the death penalty because some innocent people will be wrongfully executed or because it is a cruel and unusual punishment. Each of these specific points is a facet. For two utterances to be about the same facet, it is not necessary that the authors have the same belief toward the facet. For example, one author may believe that the death penalty is a cruel and unusual punishment while the other one attacks that position. However, in order to attack that position they must be discussing the same facet. We would like you to classify each of the following sets of pairs based on your perception of how SIMILAR the arguments are, on the following scale, examples follow. (5) Completely equivalent, mean pretty much exactly the same thing, using different words. (4) Mostly equivalent, but some unimportant details differ. One argument may be more specific than another or include a relatively unimportant extra fact. (3) Roughly equivalent, but some important information differs or is missing. This includes cases where the argument is about the same FACET but the authors have different stances on that facet. (2) Not equivalent, but share some details. For example, talking about the same entities but making different arguments (different facets) (1) Not equivalent, but are on same topic (0) On a different topic Figure 1 : 1Paraphrases of the Criminals will have guns facet from multiple conversations. Figure 2 : 2The eight facets for Gun Control on IDebate, a curated debate site. Figure 4 : 4Word count distribution for argument quality prediction scores > 0.91 for Swanson's original model. Fig. Fig. 4 plots the distribution of word counts for sentences from our sample that were given an AQ score > 0.91 by Swanson's trained AQ regressor. The first bin shows that many sentences with Figure 5 : 5Paraphrases of the Gun ownership does not lead to higher crime facet of the Gun Control topic across different conversations. Figure 7 : 7LIWC Generalized Dep. tuples Figure 10 : 10Definitions used for Facet and AFS in MT HIT. Table 1 : 1Sentence count in each domain. Sam- pled bin range > 0.55 and number of sentential arguments (high AQ) after annotation. GC=Gun Control, DP=Death Penalty, GM=Gay Marriage. Table 2 : 2Results for predicting AFS with individual features using Ridge Regression (RR) and Support Vector Regression (SVR) with 10-fold Cross-Validation on the 1800 training items for each topic.ID Feature Combinations with SVR Gun Control Gay Marriage Death Penalty r RMSE r RMSE r RMSE 1 Ngram-Rouge 0.59 0.85 0.29 0.89 0.40 1.11 2 Ngram-LIWC dependencies 0.61 0.83 0.34 0.88 0.43 1.10 3 Ngram-LIWC dependencies-Rouge 0.64 0.80 0.38 0.86 0.49 1.05 4 Ngram-LIWC dependencies-Rouge-UMBC 0.65 0.79 0.40 0.86 0.50 1.05 5 CustomW2Vec Concatenated vectors 0.71 0.72 0.48 0.80 0.56 0.99 6 GoogleW2Vec Concatenated vectors 0.71 0.72 0.50 0.79 0.57 0.98 7 Ngram-LIWC dependencies-Rouge-UMBC- CustomW2Vec Concatenated vectors 0.73 0.70 0.54 0.77 0.62 0.93 8 Ngram-LIWC dependencies-Rouge-UMBC- GoogleW2Vec Concatenated vectors 0.73 0.70 0.54 0.77 0.63 0.92 9 HUMAN TOPLINE 0.69 0.60 0.74 Table 3 : 3Results for feature combinations for predicting AFS, using Support Vector Regression (SVR) with 10-fold Cross-Validation on the 1800 training items for each topic.Ngram+LIWC (Row 2) is significantly better than Ngram for Gun Control, and Death Penalty (p < .01). Thus both Rouge and LIWC provide com- plementary information to Ngrams. See http://debatepedia.idebate.org/en/ index.php/Debate: Gun control, 2 See http://gun-control.procon.org/ See also the facets inFig. 3below from ProCon.org. Both the AQ and the AFS pair corpora are available at nlds.soe.ucsc.edu. This is an off-the-shelf STS tool from University of Maryland Baltimore County available at swoogle.umbc.edu/SimService/. https://pypi.python.org/pypi/pyrouge/ 7 https://code.google.com/archive/p/ word2vec/ One month sample https://www.reddit.com/ r/datasets/comments/3bxlg7/i_have_every_ publicly_available_reddit_comment AcknowledgmentsThis work was supported by NSF CISE RI 1302668. Thanks to Keshav Mathur and the three anonymous reviewers for helpful comments.A Appendix Score Scoring Criteria 3The phrase is clearly interpretable AND either expresses an argument, or a premise or a conclusion that can be used in an argument about a facet or a sub-issue for the topic of gay marriage. 2The phrase is clearly interpretable BUT does not seem to be a part of an argument about a facet or a sub-issue for the topic of gay marriage. 1The phrase cannot be interpreted as an argument. Internet argument corpus 2.0: An sql schema for dialogic social media and the corpora to go with it. Robert Abbott, Brian Ecker, Pranav Anand, Marilyn Walker, Language Resources and Evaluation Conference. 2016Robert Abbott, Brian Ecker, Pranav Anand, and Mari- lyn Walker. 2016. Internet argument corpus 2.0: An sql schema for dialogic social media and the corpora to go with it. In Language Resources and Evalua- tion Conference, LREC2016. Semeval-2012 task 6: A pilot on semantic textual similarity. 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Starting with generic Wess-Zumino type coupling to constant four-form and the dual seven-form field strengths in the ABJM theory, we obtain mass-deformed theories with N = 2, 4 supersymmetries. These theories contain massless scalar fields and allow the implementation of the Mukhi-Papageorgakis Higgsing procedure. Using this procedure, we connect the Higgsed theories to three-dimensional mass-deformed SYM theories. These are also connected by the four-dimensional N = 1 * , 2 * mass-deformed SYM theories through dimensional reduction. We classify the three-dimensional mass-deformed SYM theories of N = 1, 2, 4 supersymmetry, of which a few cases of N = 1, 2 are connected neither by MP Higgsing procedure nor dimensional reduction.

10.1007/jhep11(2012)169

1209.5817

ba56c2bc5ccda7a92f19a58a1153a03745cc6348

Partially Supersymmetric ABJM Theory with Flux 26 Sep 2012 Yoonbai Kim [email protected] Department of Physics Physics Research Division Institute of Basic Science BK21 O-Kab Kwon Department of Physics Physics Research Division Institute of Basic Science BK21 D D Tolla [email protected] Department of Physics Physics Research Division Institute of Basic Science BK21 University College Sungkyunkwan University 440-746SuwonKorea Partially Supersymmetric ABJM Theory with Flux 26 Sep 2012 Starting with generic Wess-Zumino type coupling to constant four-form and the dual seven-form field strengths in the ABJM theory, we obtain mass-deformed theories with N = 2, 4 supersymmetries. These theories contain massless scalar fields and allow the implementation of the Mukhi-Papageorgakis Higgsing procedure. Using this procedure, we connect the Higgsed theories to three-dimensional mass-deformed SYM theories. These are also connected by the four-dimensional N = 1 * , 2 * mass-deformed SYM theories through dimensional reduction. We classify the three-dimensional mass-deformed SYM theories of N = 1, 2, 4 supersymmetry, of which a few cases of N = 1, 2 are connected neither by MP Higgsing procedure nor dimensional reduction. Introduction Various three-dimensional supersymmetric gauge theories have attracted much interest as the theories describing the low energy dynamics of multiple M2/D2-branes with and without background fluxes. Much of recent interests are focused on the superconformal Chern-Simons matter theory of Aharony-Bergman-Jafferis-Maldacena (ABJM) [1], which describes the dynamics of M2-branes on C 4 /Z k orbifold singularity. It was known that the circle compactification of this theory via the Mukhi-Papageorgakis (MP) Higgsing procedure [2] leads to the three-dimensional N = 8 super Yang-Mills (SYM) theory [3,4], which is the low energy effective theory of multiple D2-branes. (See also Refs. [5,6].) Though the circle compactification of the N = 6 supersymmetry-preserving mass-deformed ABJM (mABJM) theory [7,8] can also be taken into account, the MP Higgsing procedure cannot be implemented as a method of the circle compactification. This is because in the N = 6 mABJM thoery all the scalar fields are massive and the bosonic potential does not involve any flat direction allowing an infinitely large vacuum expectation value of the scalar fields. The latter is a crucial requirement for the application of MP Higgsing procedure. The origin of the mass-deformation in the N = 6 mABJM theory is identified by the presence of Wess-Zumino (WZ) type coupling to special type of constant four-form and the dual seven-form field strengths [9,10] in the infinite M2-brane tension limit. As we discussed in the previous paragraph, one cannot apply the MP Higgsing procedure to the N = 6 mABJM theory. In this regard, we construct some supersymmetric mABJM theories with flat directions, which let the MP Higgsing procedure possible. Subsequently, we relate the resulting theories after the MP Higgsing to three-dimensional mass-deformed super Yang-Mills (mSYM) theories. To be specific, we start from the gauge-invariant WZ-type coupling [10,11] in the ABJM theory and then apply the formalism to a generic constant field strength in the infinite M2-brane tension limit. By appropriate choices of the fluxes we construct mABJM theories preserving N = 2, 4 supersymmetries. An intriguing aspect of the partially supersymmetric mABJM theories is the fact that they always contain certain number of massless scalar fields which result in some flat directions of the bosonic potential. We show that the MP Higgsing of the N = 2 mABJM theory leads to a mSYM theory, with the same number of supersymmetry. This mass-deformed theory is equivalent to one of the three distinct three-dimensional mSYM theories, which contain one massless vector multiplet and three massive matter multiplets [4]. The three distinct theories are obtained by making different choices of the mass parameters of the six massive fermionic fields of the matter multiplets. Similarly, we show that the N = 4 mABJM theory is equivalent to a unique N = 4 mSYM in three dimensions. We also notice that one of the three distinct N = 2 mSYM theories, but not the one obtained from the N = 2 mABJM theory, is equivalent to the one from the dimensional reduction of the four-dimensional N = 1 * mSYM theory studied by Polchinski-Strassler [12]. The N = 4 mSYM is also equivalent to the one from the dimensional reduction of the N = 2 * mSYM theory in four-dimensions. In the framework of gauge/gravity correspondence, three-dimensional mSYM theories have been studied in Refs. [13][14][15][16][17]. The remaining part of the paper is organized as follows. In section 2 we study the deformation of the ABJM theory with generic WZ-type couplings to constant background fluxes. For later convenience we single out only the WZ-type coupling which survives in the limit of infinite tension of M2-brane. In section 3 we appropriately choose the fluxes in order to preserve certain amount of supersymmetry. In section 4 we apply the MP Higgsing procedure to the partially supersymmetric mABJM theories and obtain the corresponding mSYM theories. We then study the classification of these theories in relation with the dimensional reductions of four-dimensional mSYM theories. The detailed procedure of the dimensional reduction of the N = 1 * , 2 * theories is included in appendix A. Section 5 is devoted to discussions and future research directions. ABJM Theory with Constant Flux The ABJM action [1] is given by a Chern-Simons matter theory with N = 6 supersymmetry and U(N)×U(N) gauge symmetry, S = d 3 xL ABJM = d 3 x (L 0 + L CS + L ferm + L bos ) ,(2.1) where L 0 = tr −D µ Y † A D µ Y A + iΨ †A γ µ D µ Ψ A , (2.2) L CS = k 4π ǫ µνρ tr A µ ∂ ν A ρ + 2i 3 A µ A ν A ρ − µ ∂ νÂρ − 2i 3 µÂνÂρ , (2.3) L ferm = − 2πi k tr Y † A Y A Ψ †B Ψ B − Y A Y † A Ψ B Ψ †B + 2Y A Y † B Ψ A Ψ †B − 2Y † A Y B Ψ †A Ψ B + ǫ ABCD Y † A Ψ B Y † C Ψ D − ǫ ABCD Y A Ψ †B Y C Ψ †D , (2.4) L bos = 4π 2 3k 2 tr Y † A Y A Y † B Y B Y † C Y C + Y A Y † A Y B Y † B Y C Y † C + 4Y † A Y B Y † C Y A Y † B Y C (2.5) − 6Y A Y † B Y B Y † A Y C Y † C . The four complex scalar fields Y A (A = 1, 2, 3, 4) represent the eight directions X I (I = 1, · · · , 8) transverse to the M2-branes with Y A = X A + iX A+4 . (2.6) This action has N = 6 supersymmetry with the following transformation rules δY A = iω AB Ψ B , δY † A = iΨ †B ω AB , δΨ B = −γ µ ω AB D µ Y A + 2π k ω BC Y C Y † A Y A − Y A Y † A Y C + 4π k ω AC Y A Y † B Y C , δΨ †B = D µ Y † A ω AB γ µ + 2π k ω BC Y † A Y A Y † C − Y † C Y A Y † A − 4π k ω AC Y † A Y B Y † C , δA µ = − 2π k Y A Ψ †B γ µ ω AB + ω AB γ µ Ψ B Y † A , δ µ = − 2π k Ψ †B γ µ ω AB Y A + Y † A ω AB γ µ Ψ B ,(2.7) where ω AB = −ω BA = (ω AB ) * = 1 2 ǫ ABCD ω CD . Since the ABJM theory describes low energy dynamics of N stacked M2-branes, it is intriguing to consider this theory in the background of a constant transverse four-form and the dual sevenform field strengths. Interaction between the M2-branes and the background three-form gauge fields is depicted by the WZ-type coupling. In the presence of a constant transverse four-form field strength F 4 , the components of the corresponding three-form gauge field C 3 have the following transverse scalar dependence: C µνρ , C µνA , C µAB , C µAB and their complex conjugate (c.c.) are constants, C ABC , C ABC and their c.c. are linear in transverse scalars. (2.8) Here we employed the index notations of [4], where the unbarred indices are contracted with bifundamental fields while the barred ones are contracted with anti-bifundamental fields. We can set the constant components of C 3 in (2.8) to zero by using the gauge transformation of the threeform gauge field, δC 3 = dΛ 2 . In addition, one cannot construct U(N)×U(N) gauge-invariant WZ-type coupling with linear C ABC [10]. Therefore, the only gauge-invariant WZ-type coupling for this particular choice of the three-form gauge field is read from the equation (2.3) of Ref. [10], S (3) C = λ d 3 x 1 3! ǫ µνρ tr C ABC D µ Y A D ρ Y † B D ν Y C + (c.c.) ,(2.9) where λ = 2πl 3/2 P and l P is the Planck length. The dual seven-form field strength F 7 is expressed in terms of F 4 as F 7 = * F 4 + 1 2 C 3 ∧ F 4 . (2.10) According to the argument of the previous paragraph, in the presence of the constant transverse F 4 , the C 3 ∧ F 4 term in (2.10) is linear in the transverse scalar, while the * F 4 term is constant. Keeping this in mind, we notice the following transverse scalar dependence for the six-form gauge field C 6 : C µABCDE , C µABCDĒ , · · · , C µνABCD , C µνABCD , · · · are constants, C µνρABC , C µνρABC , · · · are linear in transverse scalars, C ABCDEF , C ABCDEF , · · · are quadratic in transverse scalars. (2.11) Setting the constant components of C 6 in (2.11) to zero using gauge degrees of freedom, we read the gauge-invariant WZ-type coupling from the equation (2.8) of Ref. [10], S C = − π kλ d 3 x 1 3! ǫ µνρ {tr} C µνρABC β AB C + λ 3 C ABCDĒF D µ Y A D ν Y B D ρ Y † E β CD F (2.12) + C ABCDĒF D µ Y A D ν Y † D D ρ Y † E β BC F + C ABCDĒF D µ Y † C D ν Y † D D ρ Y † E β AB F + (c.c.) , where β AB C ≡ 1 2 (Y A Y † C Y B − Y B Y † C Y A ).(6) In this paper, we consider the infinite tension limit of the M2-brane (λ → 0), which was also considered in Ref. [9], in order to turn off the coupling to gravity modes. In this limit, the three-form coupling in (2.9) and all the six-form couplings in (2.12) except the first term can be neglected. Then it is enough to take into account the following WZ-type coupling, S WZ = − π λk d 3 x 1 3! ǫ µνρ tr C µνρABC β AB C + C † µνρABC (β AB C ) † . (2.13) The six-form gauge fields which are linear in the transverse scalars are given by C µνρABC = −2λǫ µνρ T ABCD Y † D , C † µνρABC = −2λǫ µνρ T CDĀB Y D ,(2.14) where the complex-valued constant parameters T ABCD = (T CDĀB ) * are antisymmetric in the last two barred indices as well as the first two unbarred indices. Therefore, the action in (2.13) is simplified as S WZ = 4π k d 3 x tr T ABCD Y † C Y A Y † D Y B . (2.15) As we will see later, the quartic flux term of the N = 6 mABJM theory can be expressed by the WZ-type coupling (2.15). In addition, different choice of constant flux can be taken into account in M-theory. If the masses of the fermionic and bosonic fields are appropriately chosen, the supersymmetry is partially preserved. Supersymmetry-preserving Mass-deformations In this section we discuss possible mass deformations of the ABJM theory in the presence of the constant flux term (2.15), which preserve some amount of supersymmetry. We start by introducing general gauge-invariant mass terms for scalar and fermion fields in addition to the quartic WZ-type coupling (2.15), L m bos = −tr M B A Y A Y † B with M B A = (M A B ) * , L m ferm = −itr µ B A Ψ †A Ψ B with µ B A = (µ A B ) * ,(3.16) where M B A and µ B A are constant mass matrices. Then the total Lagrangian is written as L tot = L ABJM + L WZ + L m bos + L m ferm . (3.17) The corresponding supersymmetry transformation rules in (2.7) for the scalar and gauge fields are unaffected by the mass-deformation while those for the fermionic fields are modified by δ ′ Ψ A = µ B A ω BC Y C , δ ′ Ψ †A = µ A B ω BC Y † C . (3.18) From the invariance of the total Lagrangian (3.17) under the total supersymmetry transformation δ + δ ′ , we fix the values of T ABCD , M B A , and µ B A according to the number of supersymmetry. Since δL ABJM = δ ′ L WZ = δ ′ L m bos = 0, we need to verify only the following invariance, δ ′ L ABJM + δ(L WZ + L m bos + L m ferm ) + δ ′ L m ferm = 0 (3.19) up to total derivative. Using the supersymmetry transformation rules in (2.7) and (3.18), one can verify (3.19) under the conditions, µ A A = 0, (3.20) µ B A µ C B ω CD − M B D ω AB = 0, (3.21) µ B A ω CD − µ B C ω AD − µ E A δ B C ω ED + µ E C δ B A ω ED − 2T ACBĒ ω ED = 0. (3.22) In order to check the validity of this general setup, we apply it to the well-known maximal supersymmetry preserving case [7,8]. In this case SU(4) R-symmetry of the ABJM theory is broken to SU(2)×SU (2) This result exactly matches the known result for the case of maximally supersymmetric mABJM theory [7,8] and the choice (3.23) is unique up to field redefinitions [18]. In the subsequent two subsections, we consider the cases with flat directions in bosonic potentials, where some of scalar fields and corresponding superpartners are massless. In those cases some of the supersymmetries are necessarily broken. The models with N = 2 and N = 4 supersymmetries are constructed. N = 2 Let us consider a bosonic potential which is flat along only one complex scalar field. By supersymmetry, the corresponding single complex fermion field should be massless while the other three fermion fields remains to be massive. Without loss of generality, we choose the fermionic mass matrix of the three massive fermionic fields as µ B A = diag(0, m 2 , m 3 , m 4 ),(3.M B A = diag(m 2 4 , m 2 3 , m 2 2 , 0),(3.25) and then nonvanishing components of T ABCD are determined by (3.22) as T 1212 = −T 3434 = m 2 2 , T 1313 = −T 2424 = m 3 2 , T 1414 = −T 2323 = m 4 2 . (3.26) One may also choose different nonvanishing components, for instance, ω 12 or ω 13 , however, the results are equivalent to the aforementioned case of the nonvanishing ω 14 , up to field redefinition. N = 4 In order to obtain the mass-deformed theory with N = 4 supersymmetry, we have to turn on the mass term for two complex scalar fields. Then two complex fermionic fields become massive while the other two are massless. This implements an appropriate choice for the corresponding fermionic mass matrix Like the N = 2 case in subsection 3.1, this choice is unique up to field redefinition. In the original ABJM theory it was conjectured that the N = 6 supersymmetry is enhanced to N = 8 at Chern-Simons levels k = 1, 2 [1]. The existence of such additional N = 2 supersymmetries was verified in terms of the monopole operators [19][20][21][22]. For k > 2, the supersymmetry enhancement is not possible due to orbifolding. On the other hand, in order to implement the MP Higgsing procedure one has to move the M2-branes away from the orbifold singularity and this leads to an enhancement of the supersymmetry [4]. For instance, after the MP Higgsing procedure, the N = 6 supersymmetry of the ABJM theory is enhanced to the N = 8 supersymmetry of the three-dimensional SYM theory. The latter theory flows to the supersymmetry enhanced ABJM theory on flat transverse space (k = 1) at the IR fixed point [23]. However, as we shall show in the next section, there is no supersymmetry enhancement after the MP Higgsing procedure in the N = 2, 4 mABJM theories. This implies the absence of the supersymmetry enhancement in the N = 2, 4 mABJM theories, unlike the N = 6 mABJM theory. µ B A = diag(0, 0, m, −m). Classification The dimensional reduction of the ABJM theory with U(N)×U(N) gauge symmetry [1] via the MP Higgsing procedure [2] leads to the three-dimensional N = 8 SYM theory with U(N) gauge symmetry [3,4]. In Ref. [4] we have shown that the Higgsing of the ABJM theory deformed by WZ-type couplings of constant fluxes results in effective theories of D2-branes in the background of constant RR fluxes. By supersymmetric completion, for few choices of constant fluxes, we obtained N = 2, 4 mSYM theories. In the pervious section we have found the N = 2, 4 mABJM theories. Since these theories possess bosonic potentials with flat direction, the MP Higgsing procedure can be carried out for these cases. The resultant theories are compared with the aforementioned N = 2, 4 mSYM theories as discussed in Ref. [4]. The N = 1 mSYM theory in Ref. [4] cannot be obtained from the MP Higgsing procedure of mABJM theory due to the following reason. In order to apply this procedure the bosonic potential is required to involve at least one massless complex scalar field. After the Higgsing, this complex field turns to one dynamical real massless scalar field and one would-be Goldstone boson. In fact the N = 1 mSYM theory of Ref. [4] does not possess any massless scalar field. MP Higgsing of the N = 2, 4 mABJM Theories To pursue the MP Higgsing procedure [2] we proceed by introducing vacuum expectation value v for the massless scalar Y 4 along a transverse direction Y A = v 2 T 0 δ A4 +X A + iX A+4 ,(4.30) whereX I 's (I = 1, 2, · · · , 8) are Hermitian scalar fields. Correspondingly we introduce Hermitian fermionic fieldsψ r (r = 1, 2, · · · , 8) as Ψ A =ψ A + iψ A+4 .(4.31) When the vacuum expectation value v is turned on, in the MP Higgsing procedure, the U(N) × U(N) gauge symmetry is broken to U(N) and the Hermitian scalar and fermionic fields transform in adjoint representation of the unbroken U(N). Then taking double scaling limit of the large v and large Chern-Simons level k with finite v/k, the Yang-Mills coupling g is identified as g = 2πv/k and the matter fields are rescaled asφ →φ/g for dimensional reason. The detailed procedures are explained in Ref. [4]. Application of the Higgsing procedure to the total Lagrangian (3.17) results iñ L N =2,4 YM =L N =8 YM + 1 g 2 tr iT ijkX i [X j ,X k ] −M ijX iX j − iμ rsψrψs ,(4.32) where i, j, k = 1, 2, · · · , 7, and L N =8 YM = 1 g 2 tr − 1 2F µνF µν −D µX iDµX i + iψ r γ µD µψr + 1 2 [X i ,X j ] 2 − Γ rs iψ r [X i ,ψ s ] . (4.33) The cubic interaction term in (4.32) is the result of the MP Higgsing of the WZ-type coupling (2.15) and the antisymmetric tensorsT ijk are related to the constant four-form tensor T ABCD in (2.14) as follows:T These results can be compared with the corresponding mSYM theories in Ref. [4] with suitable field redefinitions and parameter choices. More precisely, the N = 2 mSYM theory we obtained in this paper is equivalent to that of Ref. [4] if we make the following field redefinitions and identifications of the mass parameters of the two theories: ab4 =T 4a+4b+4 = 2i 3 T a4b4 − T b4ā4 ,T a4b+4 = − 2 3 T a4b4 + T b4ā4 , T abc = −i T abc4 − T c4āb ,T a+4b+4c+4 = T abc4 + T c4āb , T abc+4 = 1 3 2T a4bc − T c4āb − 2T abc4 + T bcā4 , T ab+4c+4 = i 3 2T c4āb − T a4bc − 2T abc4 + T bcā4 ,X 1 →X 1 ,X 2 →X 3 ,X 3 →X 6 ,X 4 →X 7 ,X 5 →X 2 ,X 6 →X 4 ,X 7 →X 5 , m 2 → µ 3 = µ 4 , m 3 → µ 5 = µ 6 , m 4 → µ 7 = µ 8 , with µ 3 + µ 5 + µ 7 = 0, (4.37) where µ i 's are the mass parameters used in Ref. [4]. We call this theory 'D = 3 N = 2 mSYM I'. Actually, in the case of Ref. [4] one can make other two more choices of mass parameters satisfying all the constraints imposed by supersymmetry. We call these theories 'D = 3 N = 2 mSYM II & III'. However, these choices cannot be related with the Higgsing of the N = 2 mABJM theory by field redefinition. The reason is the fact that in the N = 2 mABJM theory we have only three mass parameters of the three massive complex fields while in the case of Ref. [4] we have six mass parameters of the six massive real fields. Therefore, in the latter case there are more freedoms in choosing the mass parameters. See the next subsection for the details. The comparison for the N = 4 theories obtained here and Ref. [4] can be made by setting m 2 = 0 → µ 3 = µ 4 = 0 and using the same field redefinitions and parameter choices as in (4.37). In this case other possible choices of mass parameters in Ref. [4] are also identical to the choice in (4.36) up to field redefinitions. The dimensional reduction of the four dimensional N = 2 * mSYM theory [12] also gives this N = 4 mSYM theory. (For the details see Appendix A.) It is also important to recall that we started with a general setting of mass deformation in ABJM theory to obtain the N = 4 mABJM theory. In these regards, the N = 4 mABJM and mSYM theories discussed in this paper seem unique. Classification of the N = 2 mSYM theories In Ref. [4] we have shown that introduction of the mass deformation to the N = 8 SYM theory preserves N = 1, 2, 4 supersymmetries depending on the choices of the fermionic and bosonic mass parameters and components of the antisymmetric tensorT ijk . In this subsection we fully classify N = 2 mSYM theories according to mass parameter choices. The N = 1 mSYM theory contains one massless gauge boson and seven massive scalar fields. Together with their superpartners which are one massless and seven massive fermionic fields, these sets of fields form one N = 1 vector multiplet and seven massive matter multiplets. Since all the massive scalar fields belong to different multiplets, they are allowed to have different masses of which the parameters are unrestricted unlike the higher supersymmetry cases. In this reason, there is no candidate in the mABJM theory to be linked with this N = 1 mSYM theory. In the case of the N = 2 mSYM theory, the supersymmetry invariance of the action requires µ rs = diag(0, 0, µ 3 , µ 4 , µ 5 , µ 6 , µ 7 , µ 8 ),M ij = diag(µ 2 8 , µ 2 7 , µ 2 6 , µ 2 5 , µ 2 4 , µ 2 3 , 0), T 145 = 1 3 (µ 3 + µ 6 + µ 7 ),T 246 = 1 3 (µ 3 + µ 5 + µ 7 ),T 347 = 1 3 (µ 5 + µ 6 ), T 127 = − 1 3 (µ 7 + µ 8 ),T 136 = − 1 3 (µ 4 + µ 5 + µ 7 ),T 235 = − 1 3 (µ 3 + µ 5 + µ 8 ), T 567 = − 1 3 (µ 3 + µ 4 ),(4.38) where the mass parameters are constrained as, µ 2 4 = µ 2 3 , µ 2 6 = µ 2 5 , µ 2 8 = µ 2 7 , µ 3 + µ 4 + µ 5 + µ 6 + µ 7 + µ 8 = 0. (4.39) There are three independent mass parameter choices satisfying the constraints in (4.39), case I : µ 3 = µ 4 , µ 5 = µ 6 , µ 7 = µ 8 with µ 3 + µ 5 + µ 7 = 0, T 145 =T 246 =T 136 =T 235 = 0,T 347 = 2 3 µ 5 ,T 127 = − 2 3 µ 7 ,T 567 = − 2 3 µ 3 , case II : µ 4 = −µ 3 , µ 6 = −µ 5 , µ 8 = −µ 7 , T 145 = 1 3 (µ 3 − µ 5 + µ 7 ),T 246 = 1 3 (µ 3 + µ 5 + µ 7 ),T 136 = 1 3 (µ 3 − µ 5 − µ 7 ) , T 235 = 1 3 (−µ 3 − µ 5 + µ 7 ),T 347 =T 127 =T 567 = 0, case III : µ 4 = −µ 3 , µ 6 = −µ 8 = −µ 7 = µ 5 , T 145 =T 246 =T 136 = −T 235 = 1 3 µ 3 ,T 347 =T 127 = 2 3 µ 5 ,T 567 = 0. (4.40) As discussed previously the case I is identical to the Higgsed N = 2 mABJM theory of the previous subsection through the field redefinitions in (4.37). As shown in appendix A, the case II is obtained as a result of the dimensional reduction of the four-dimensional N = 1 * mSYM theory [12]. The case III is a N = 2 mSYM theory which can be connected to neither N = 2 mABJM theory through the MP Higgsing nor the N = 1 * mSYM theory through dimensional reduction. The reason why the cases II and III are not related with the Higgsed N = 2 mABJM theory of subsection 3.1 is the following. For the latter case, any set of two fermionic fields belonging to the same supermultiplet is inherited from the real and imaginary components of a complex fermionic field in the original mABJM theory. Therefore, they have the same masses. For the former cases, the fermionic fields in the same multiplet have either the same or opposite signs for their mass parameters as indicated in (4.40). For convenience we summarize classification of the mSYM theories in the diagram of Fig. 1. Conclusion In this paper we classified some parity-preserving three-dimensional supersymmetric mass-deformed gauge theories. In the ABJM theory, we introduced a generic WZ-type coupling to constant fourform and dual seven-form field strengths in the limit of infinite M2-brane tension. We showed, with appropriate choice of the fermionic and bosonic mass terms, such deformed ABJM theory possesses N = 2, 4, 6 supersymmetries. In Ref. [4] we already constructed three distinct mSYM theories in three dimensions, which are one N = 1, three N = 2, and one N = 4 mSYM theories. Here we verified that one of the three N = 2 theories and the N = 4 theory are obtained through the MP Higgsing of the N = 2 and N = 4 mABJM theories, respectively. One of the remaining N = 2 theories is obtained by dimensional reduction of the four-dimensional N = 1 * theory, while the N = 4 mSYM theory is also obtained by the dimensional reduction to the N = 2 * theory. The third N = 2 and the N = 1 mSYM theories are not connected by the MP Higgsing of the mABJM theory or the dimensional reduction of the four-dimensional mSYM theory. We may extend our analysis in this paper to the cases of the parity-violating three-dimensional gauge theories, such as the N = 3 level-deformed ABJM theory developed by Gaiotto and Tomasiello (GT) [24] (see also Ref. [25]). As the ABJM theory does, the GT theory allows the supersymmetry-preserving mass-deformation [26] and the circle compactification via the MP Higgsing procedure [27]. Utilizing these properties, one can construct the less supersymmetric mass-deformed GT theories with flat directions which implement the MP Higgsing procedure. This analysis may shed some light on M-theory brane configuration of the GT theory. Holographic dual of the N = 6 mABJM theory is proposed in Ref. [28], which is the Z k -quotient of the Lin-Lunin-Maldacena (LLM) geometry [29] (see also Ref. [30]). The proposal of the dual gravity gets much insights from the structure of the vacuum space of the gauge theory. The dual gravity theories are not yet understood for the partially supersymmetric mABJM theories. The N = 4 mABJM theory does not contain any Higgs vacuum solution and does not seem to have a dual gravity theory related to the LLM geometry. On the other hand, after the MP Higgsing and the dimensional uplift, the resulting N = 2 * mSYM theory turns out to be dual to the Pilch-Warner geometry in type IIB supergravity [31]. It is interesting to figure out the M-theory uplifting of this geometry and to identify the dual geometry of the N = 4 mABJM theory. The N = 2 mABJM theory has Higgs vacuum solutions but it is still unclear how to modify the LLM geometry to obtain the corresponding dual gravity. A Four-dimensional Mass-deformed SYM Theories The four-dimensional N = 1 * theory by Polchinski and Strassler [12] is constructed by introducing a mass-deformation to the N = 4 SYM theory. The action for the latter is given bỹ L =tr − 1 2 F αβ F αβ −D αΦ aDαΦa +g 2 2 [Φ a ,Φ b ] 2 + iψ pγ αD α ψ p −g ψ p ∆ pq a 1 + γ 5 2 +∆ pq a 1 − γ 5 2 [Φ a , ψ q ] , (A.41) where α, β = 0, ..., 3, a, b = 1, ..., 6, p, q = 1, ..., 4, ψ p 's are Majorana fermions andΦ a 's are Hermitian scalar fields. ∆ pq a = g p ∆ a g q and∆ pq a = g * p ∆ a g * q are constants. ∆ a are the gamma matrices of the six-dimensional Euclidean space, and g p , g * p are the eigenvectors of Γ * = −i∆ 1 ...∆ 6 with eigenvalues +1, -1, respectively. The covariant derivative is given byD α = ∂ α + ig[A α , .]. The Clifford algebra for the gamma matrices is given by: {γ α ,γ β } = −2η αβ with the signature η αβ = diag (−1, 1, 1, ...). The N = 4 supersymmetry transformation rules are δ ǫ A α = iǭ pγα ψ p , δ ǫΦa = iǭ p ∆ pq a 1 + γ 5 2 +∆ pq a 1 − γ 5 2 ψ q , (A.42) δ ǫ ψ p = iF αβ Σ αβ ǫ p +γ αD αΦa ∆ pq a 1 + γ 5 2 +∆ pq a 1 − γ 5 2 ǫ q −g[Φ a ,Φ b ] ∆ pq ab 1 + γ 5 2 +∆ pq ab 1 − γ 5 2 ǫ q , where the supersymmetry parameters ǫ p 's are Majorana fermions, and ∆ pq ab = g * p Σ ab g q ,∆ ab pq = g p Σ ab g * q with Σ ab = − i 4 [∆ a , ∆ b ]. After some algebra we obtain ∆ pq ab = i 4 ∆ po a ∆ oq b −∆ po b ∆ oq a , ∆ pq ab = i 4 ∆ po a∆ oq b − ∆ po b∆ oq a , (A.43) where the components of ∆ a 's are given by ∆ pq 1 −∆ pq 1 = 2i δ p1 δ q4 − δ p4 δ q1 + δ p2 δ q3 − δ p3 δ q2 , ∆ pq 1 +∆ pq 1 = 0, ∆ pq 2 −∆ pq 2 = 2i δ p1 δ q2 − δ p2 δ q1 + δ p3 δ q4 − δ p4 δ q3 , ∆ pq 2 +∆ pq 2 = 0, ∆ pq 3 −∆ pq 3 = 2i δ p1 δ q3 − δ p3 δ q1 − δ p2 δ q4 + δ p4 δ q2 , ∆ pq 3 +∆ pq 3 = 0, ∆ pq 4 +∆ pq 4 = −2 δ p1 δ q4 − δ p4 δ q1 − δ p2 δ q3 + δ p3 δ q2 , ∆ pq 4 −∆ pq 4 = 0, ∆ pq 5 +∆ pq 5 = 2 δ p1 δ q2 − δ p2 δ q1 − δ p3 δ q4 + δ p4 δ q3 , ∆ pq 5 −∆ pq 5 = 0, ∆ pq 6 +∆ pq 6 = −2 δ p1 δ q3 − δ p3 δ q1 + δ p2 δ q4 − δ p4 δ q2 , ∆ pq 6 −∆ pq 6 = 0. (A.44) In the N = 1 * theory, without loss of generality we can choose ǫ = ǫ 4 as the unbroken supersymmetry parameter with the other supersymmetry parameters set to zero. Then the supersymmetry transformation rules in (A.42) are reduced to δ ǫ A α = iǭγ α λ, δ ǫΦa = iǭ ∆ 4t a 1 + γ 5 2 +∆ 4t a 1 − γ 5 2 ψ t , δ ǫ ψ t =γ αD αΦa ∆ t4 a 1 + γ 5 2 +∆ t4 a 1 − γ 5 2 ǫ −g[Φ a ,Φ b ] ∆ t4 ab 1 + γ 5 2 +∆ t4 ab 1 − γ 5 2 ǫ, δ ǫ λ = iF αβ Σ αβ ǫ −g[Φ a ,Φ b ] ∆ 44 ab 1 + γ 5 2 +∆ 44 ab 1 − γ 5 2 ǫ, (A.45) where λ = ψ 4 and t = 1, 2, 3. The mass-deformation preserving the N = 1 supersymmetry is When µ 1 = µ 2 and µ 3 = 0, we easily notice that the supersymmetry is enhanced to N = 2. This gives the N = 2 * theory discussed in Ref [12]. L µ = tr − iµ pqψp ψ q − M abΦaΦb + igT abcΦa [Φ b ,Φ c ] , A.1 Reduction to three dimensions In order to reduce the N = 1 * theory to three dimensions we assume that the fields do not depend on the compactified direction. For the bosonic part, by introducing V 1 2 A α = (A µ , φ) with V the volume of the compactfied direction and µ, ν = 0, 1, 2, we obtain V F αβ F αβ = F µν F µν + 2F 3µ F 3µ = F µν F µν + 2D µ φD µ φ, (A.50) and, by setting V 1 2Φ a = Φ a and V − 1 2g = g, we have VD αΦ aDαΦa = D µ Φ a D µ Φ a − g 2 [φ, Φ a ] 2 , V M abΦaΦb = M ab Φ a Φ b , iVgT abcΦa [Φ b ,Φ c ] = igT abc Φ a [Φ b , Φ c ], Vg 2 [Φ a ,Φ b ] 2 = g 2 [Φ a , Φ b ] 2 , (A.51) where the covariant derivative is given by D µ = ∂ µ − ig[A µ , .]. Using the relation d 4 xL bos = d 3 xL bos and substituting the obtained results into the bosonic part of the four-dimensional action in (A.41) and (A.46), we write the bosonic part of the Lagrangian density in three dimension as L bos = tr − 1 2 F µν F µν − D µ φD µ φ − D µ Φ a D µ Φ a + 1 2 g 2 [φ, Φ a ] 2 + [Φ a , φ] 2 + 1 2 g 2 [Φ a , Φ b ] 2 − M ab Φ a Φ b + igT abc Φ a [Φ b , Φ c ] = −tr 1 2 F µν F µν + D µX i D µX i − 1 2 g 2 [X i ,X j ] 2 + M ijX iX j − igT ijkX i [X j ,X k ] , ×U(1) due to the mass matrix µ B A = diag(m, m, −m, −m) with a mass parameter m. Then we determine the bosonic mass matrix and the nonvanishing components of the constant four-form tensor from the conditions (3.21) -(3.22) as M B A = m 2 δ B A , T 1212 = −m, T 3434 = m. (3.23) 24) where m A 's (A = 2, 3, 4) are real mass parameters. Then we notice that m 2 + m 3 + m 4 = 0 due to the condition(3.20). In order to satisfy the conditions in (3.21) and (3.22), we should keep only one complex component of ω AB and its complex conjugate nonvanishing. To be specific, we choose nonvanishing ω 14 and then ω 23 is also nonvanishing by the reality condition of ω AB . Substitution of these into (3.21) determine M B A as in (3.21) -(3.22) are satisfied only when we keep two nonvanishing complex supersymmetric parameters and their complex conjugates. One possible choice is nonvanishing ω 13 and ω 14 and then ω 24 and ω 23 are also nonvanishing. With this choice we read the bosonic mass matrix from (3.21), M B A = diag(m 2 , m 2 , 0, 0), (3.28) and the following nonvanishing components of T ABCD from (3.22) T 1313 = −T 1414 = T 2323 = −T 2424 = m 2 . (3.29) and fermionic mass terms in (4.32) are obtained from the MP Higgsing of the mass terms (3.16). For the N = 2 theory of subsection 3.1 we read the nonvanishing components ofT ijk as well as the fermionic and bosonic mass matrices from (3.24)-(3.26), µ rs = diag(0, m 2 , m 3 , m 4 , 0, m 2 , m 3 , m 4 ),M ij = diag(the N = 4 theory of subsection 3.2, we have those quantities from (3.27)-(3.29), µ rs = diag(0, 0, m, −m, 0, 0, m, −m),M ij = diag(m 2 , m 2 , 0, 0, m 2 , m 2 , 0), T 145 = − 2 3 m,T 246 = − 2 3 m. (4.36) Figure 1 : 1Classification of N = 1, 2, 4 supersymmetric mass-deformed gauge theories in three dimensions and their relationship to the N = 1 * , 2 * mSYM in four-dimensions. pq = diag(µ 1 , µ 2 , µ 3 , 0), M ab = diag( = (Φ a , φ) for i = 1, ..., 7 are the seven transverse scalar fields and M 7i = T 7ij = 0. For the fermionic part, we split the four-dimensional gamma matrices as followsγ 0 = σ 3 ⊗ iσ 2 ,γ 1 = σ 3 ⊗ σ 1 ,γ 2 = σ 3 ⊗ σ 3 ,γ 3 = σ 1 ⊗ I, (A.53)whereas the three-dimensional gamma matrices are given by γ 0 = iσ 2 , γ 1 = σ 1 , γ 2 = σ 3 . r 's form the basis of R 2 and ψ r p 's are Majorana spinors in three dimensions. With γ 5 = −iγ 0γ1γ2γ3 , their chiral components are written in terms of the three-dimensional Majorana spinors as σ 2 ⊗ I 2 e r ⊗ ψ r p . (A.56)Finally, the covariant derivatives are given byV 1 2D µ ψ p = e r ⊗ D µ ψ r p , V 1 2D 3 ψ p = ige r ⊗ [φ, ψ r p ], (A.57) (e r ⊗ ψ r p ) † σ 3 ⊗ iσ 2 .(A.58) Acknowledgementswhereψ r p = ψ r † p γ 0 is the Dirac conjugation in three dimensions, γ 0 rs = e r † γ 0 e s = ie r † σ 2 e s , and we have used e r † e s = δ rs . The fermionic mass term and the Yukawa-type interaction terms arewhere γ 1 rs = e r † γ 1 e s = e r † σ 1 e s and γ 2 rs = e r † γ 2 e s = e r † σ 3 e s . 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Recently, in a paper by Jentzen and Kloeden [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649-667], a new method for simulating nearly linear stochastic partial differential equations (SPDEs) with additive noise has been introduced. The key idea was to use suitable linear functionals of the noise process in the numerical scheme which allow a higher approximation order to be obtained. Following this approach, a new simplified version of the scheme in the above named reference is proposed and analyzed in this article. The main advantage of the convergence result given here is the higher convergence order for nonlinear parabolic SPDEs with additive noise, although the used numerical scheme is very simple to simulate and implement.

10.1214/10-aap711

1210.8320

b4a351fb9510aed2edd6af367e96c4e3935839a9

EFFICIENT SIMULATION OF NONLINEAR PARABOLIC SPDES WITH ADDITIVE NOISE Oct 2012. 2011 Arnulf Jentzen Princeton University Johann Wolfgang Goethe University and Johann Wolfgang Goethe University Peter Kloeden Princeton University Johann Wolfgang Goethe University and Johann Wolfgang Goethe University Georg Winkel Princeton University Johann Wolfgang Goethe University and Johann Wolfgang Goethe University EFFICIENT SIMULATION OF NONLINEAR PARABOLIC SPDES WITH ADDITIVE NOISE The Annals of Applied Probability 213Oct 2012. 201110.1214/10-AAP711 Recently, in a paper by Jentzen and Kloeden [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 649-667], a new method for simulating nearly linear stochastic partial differential equations (SPDEs) with additive noise has been introduced. The key idea was to use suitable linear functionals of the noise process in the numerical scheme which allow a higher approximation order to be obtained. Following this approach, a new simplified version of the scheme in the above named reference is proposed and analyzed in this article. The main advantage of the convergence result given here is the higher convergence order for nonlinear parabolic SPDEs with additive noise, although the used numerical scheme is very simple to simulate and implement. 1. Introduction. In this article, the numerical approximation of nonlinear parabolic stochastic partial differential equations (SPDEs) is considered. Following the idea in [10] for somewhat linear SPDEs, a new numerical method for simulating nonlinear SPDEs with additive noise is proposed and analyzed in this article. The main advantage of the convergence result in this article is the higher convergence order for nonlinear parabolic SPDEs with additive noise in comparison to convergence results of classical schemes such as the linear implicit Euler scheme. Nevertheless, the here presented scheme is very simple to simulate and implement. More precisely, let T ∈ (0, ∞) be a real number, let (Ω, F, P) be a probability space and let H = L 2 ((0, 1), R) be the R-Hilbert space of equiv- [13]). It is a classical result (see, e.g., Proposition 2.1.5 in [13]) that the covariance operator Q : H → H of the Wiener process W Q : [0, T ] × Ω → H has an orthonormal basis g j ∈ H, j ∈ N, of eigenfunctions with summable eigenvalues µ j ∈ [0, ∞), j ∈ N. In order to have a more concrete example, we consider the choice g j (x) = √ 2 sin(jπx) and µ j = cj −(r+1) for all x ∈ (0, 1) and all j ∈ N with some c ∈ [0, ∞) and some arbitrarily small r ∈ (0, ∞) in the following and refer to Section 2 for our general setting. Then we consider the SPDE dX t = ∂ 2 ∂x 2 X t + f (x, X t ) dt + dW Q t ,(1) X t (0) = X t (1) = 0, Then our goal is to solve the strong approximation problem of the SPDE (1). More precisely, we want to compute a F/B(H)-measurable numerical approximation Y : Ω → H such that X 0 = ξ,E 1 0 |X T (x) − Y (x)| 2 dx 1/2 < ε (3) holds for a given precision ε > 0 with the least possible computational effort (number of computational operations and independent standard normal random variables needed to compute Y : Ω → H). A computational operation is here an arithmetical operation (addition, subtraction, multiplication, division), a trigonometrical operation (sine, cosine) or an evaluation of f : (0, 1) × R → R or the exponential function. In order to be able to calculate such a numerical approximation on a computer, both the time interval [0, T ] and the infinite-dimensional R-Hilbert space H = L 2 ((0, 1), R) have to be discretized. While for temporal discretizations the linear implicit Euler scheme is often used, spatial discretizations are EFFICIENT SIMULATION OF SPDES 3 usually achieved with finite elements, finite differences and spectral Galerkin methods. For instance, the linear implicit Euler scheme combined with spectral Galerkin methods which we denote by F/B(H)-measurable mappings Z N n : Ω → H, n ∈ {0, 1, . . . , N 2 }, N ∈ N := {1, 2, . . .}, is given by Z N 0 := P N (ξ) and for all x ∈ (0, 1), v ∈ H and all N ∈ N. Note that the infinite-dimensional R-Hilbert space H is projected down to the N -dimensional R-Hilbert space P N (H) for the spatial discretization and the time interval [0, T ] is divided into N 2 subintervals, that is, N 2 time steps are used, for the temporal discretization in the scheme Z N n , n ∈ {0, 1, . . . , N 2 }, above for N ∈ N. The exact solution X : [0, T ] × Ω → H of the SPDE (1) enjoys at least twice the regularity in space than in time and therefore, the quadratic number of time steps is used in the scheme (4) above (see also Walsh [15] for details). Z N n+1 := I − T N 2 A −1 (4) × Z N n + T N 2 · (P N F )(Z N n ) + P N (W Q (n+1)T /N 2 − W Q nT /N 2 ) We now review how efficiently the numerical method (4) solves the strong approximation problem (3) of the SPDE (1). Standard results in the literature (see, e.g., Theorem 2.1 in Hausenblas [7]) yield the existence of a real number C > 0 such that E 1 0 |X T (x) − Z N N 2 (x)| 2 dx 1/2 ≤ C · N −1(6) holds for all N ∈ N. Since P N (H) is N -dimensional and since N 2 time steps are used in (4), O(N 3 log(N )) computational operations and independent standard normal random variables are needed to compute Z N N 2 for N ∈ N. The log term in O(N 3 log(N )) for N ∈ N arises due to computing the nonlinearity with fast Fourier transform (aliasing errors are neglected here). Combining the computational effort O(N 3 log(N )) and the estimate (6) shows that the linear implicit Euler scheme needs about O(ε −3 ) computational operations and independent standard normal random variables to achieve a precision of size ε > 0 in the sense of (3). In fact, we have demonstrated that the linear implicit Euler scheme method (4) needs O(ε −(3+δ) ) computational operations and random variables to solve (3) for every arbitrarily small δ ∈ (0, ∞) but for simplicity we write about O(ε −3 ) computational operations and random variables here and below. Recently, in [10], a new numerical method for simulating somewhat linear SPDEs with additive noise has been introduced. The key idea in [10] is to use suitable linear functionals of the noise process in the numerical scheme which allows a higher approximation order to be obtained. In this paper, we extend this idea to the case of nonlinear SPDEs of the form (1). More precisely, we introduce the following numerical scheme which is a simplified version of the scheme considered in [10]. Let Y N n : Ω → H, n ∈ {0, 1, . . . , N }, N ∈ N, be F/B(H)-measurable mappings given by Y N 0 := P N (ξ) and Y N n+1 := e AT /N Y N n + T N · (P N F )(Y N n ) (7) + P N (n+1)T /N nT /N e A((n+1)T /N −s) dW Q s P-a. s. for every n ∈ {0, 1, . . . , N − 1} and every N ∈ N. Note that the infinitedimensional R-Hilbert space H is projected down to the N -dimensional R-Hilbert space P N (H) for the spatial discretization and the time interval [0, T ] is divided into N subintervals, that is, N time steps are used, for the temporal discretization in the scheme Y N n , n ∈ {0, 1, . . . , N }, above for N ∈ N. We now illustrate the main result of this article (Theorem 1) and show how efficiently the method (7) solves the strong approximation problem (3) of the SPDE (1). Theorem 1 shows the existence of real numbers C δ > 0, δ ∈ (0, 1), such that E 1 0 |X T (x) − Y N N (x)| 2 dx 1/2 ≤ C δ · N (δ−1)(8) holds for all N ∈ N and all arbitrarily small δ ∈ (0, 1). The stochastic integrals P N (n+1)T /N nT /N e A((n+1)T /N −s) dW Q s(9) for n ∈ {0, 1, . . . , N } and N ∈ N in (7) provide more information about the exact solution and this allows us to obtain the estimate (8) although only N time steps (instead of N 2 time steps in the case of the linear implicit Euler scheme) are used in (7). Nevertheless, since the stochastic integrals (9) in (7) depend linearly on the Wiener process W Q : [0, T ] × Ω → H, they are again normally distributed and hence easy to simulate. More precisely, since P N (H) is N -dimensional and since N time steps are used in (7), O(N 2 log(N )) computational operations and independent standard normal random variables are needed to compute Y N N for N ∈ N. The log term in EFFICIENT SIMULATION OF SPDES 5 O(N 2 log(N )) for N ∈ N also arises due to computing the nonlinearity with fast Fourier transform (aliasing errors are neglected here). Combining the computational effort O(N 2 log(N )) and the estimate (8) shows that the numerical scheme (7) needs about O(ε −2 ) computational operations and independent standard normal random variables to achieve a precision of size ε > 0 in the sense of (3). The estimates (6) and (8) are both asymptotic results since there is no information about the size of the corresponding error constants. In particular, the error constants C δ ∈ (0, ∞), δ ∈ (0, 1), in (8) could be much bigger than in (6). Therefore, from a practical point of view, one may ask whether the numerical method (7) solves the strong approximation problem (3) more efficiently than the linear implicit Euler scheme (4) for a given example of the form (1) and a given concrete ε > 0. In order to analyze this question, we compare both methods in the case of a simple reaction diffusion SPDE of the form (1) (see Section 4.1 for details) and assume that the strong approximation problem (3) should be solved with the precision ε = 1 300 . In that example, it turns out that the linear implicit Euler scheme precisely needs 2 21 = 2,097,152 independent standard normal random variables while the numerical method (7) precisely needs 2 16 = 65,536 independent standard normal random variables to achieve an approximation error of size ε = 1 300 (see Tables 1 and 2 in Section 4.1). We also emphasize that the numerical scheme (7) is very simple to implement and refer to Figure 2 for a short matlab code. Having illustrated the main result of this article, we now sketch the key idea in the proof of Theorem 1. The main difficulty was to estimate the discretization error for nonlinear F . In that case, the main problem was to establish estimates of the form N −1 n=0 (n+1)T /N nT /N e A(T −s) (F (X s ) − F (X nT /N )) ds L 2 (Ω;H) ≤ C δ · N (δ−1)(10) for all N ∈ N and all δ ∈ (0, 1) where C δ ∈ (0, ∞), δ ∈ (0, 1), are appropriate constants and where we write Y L 2 (Ω;H) : = (E[ 1 0 |Y (x)| 2 dx]) 1/2 ∈ [0, ∞] for every F/B(H)-measurable mapping Y : Ω → H for simplicity. The smoothness of the Nemytskii operator F on an appropriate subspace V ⊂ H shows that it remains to estimate N −1 n=0 (n+1)T /N nT /N e A(T −s) F ′ (X nT /N )(X s − X nT /N ) ds L 2 (Ω;H) (11) ≤ C δ · N (δ−1) for all N ∈ N and all δ ∈ (0, 1). In [10], the linear operators F ′ (v) for v ∈ H and A : D(A) ⊂ H → H are assumed to commute in some sense which is fulfilled in the case of linear F such as F (v) = v, v ∈ H, but excludes nonlinear Nemytskii operators such as F (v) = (1−v) (1+v 2 ) , v ∈ H (see Assumption 2.4 in [10] for details). Under this commutativity condition, (11) can easily be established by using the smoothing effect of the semigroup e At , t ∈ [0, T ] (see Section 5.b.i in [10]). Instead of this condition, our key assumption on the nonlinearity is an appropriate estimate on the adjoint operators of the Fréchet derivative operators of F [see (13)]. Since in our examples F is a (nonlinear) Nemytskii operator, the derivative operators F ′ (v), v ∈ V , are self-adjoint and hence, it can easily be seen that this assumption is fulfilled [see (17) in Section 4 for details]. Moreover, this assumption enables use to show (11) and hence (10) [see Section 6.1.1 and particularly estimate (31)]. We also mention that the difficulty to estimate (10) can be avoided by using a more complicated scheme with a second linear functional (see Section 6.4 in [11]). Finally, we would like to point out limitations of the here presented numerical method. The following assumption is essential to apply our algorithm. The eigenfunctions of the dominating linear operator and of the covariance operator of the driving additive noise process of the SPDE must coincide and must be known explicitly. The rest of this article is organized as follows. The basic setting and the assumptions that we use (including our key assumption on the adjoint of the Fréchet derivative of the nonlinearity) are presented in Section 2. The new numerical scheme and its convergence theorem which is the main result of this article are given in Section 3. This result is illustrated with some examples and some numerical simulations in Section 4. Although our setting in Section 2 uses the standard global Lipschitz assumption on the nonlinearity of the SPDE, we demonstrate the efficiency of our method numerically for a SPDE with a cubic nonglobally Lipschitz nonlinearity in Section 5. Proofs are postponed to the final section. 2. Setting and assumptions. Fix T ∈ (0, ∞) and let (Ω, F, P) be a probability space with a normal filtration (F t ) t∈[0,T ] which means F t+ = F t for all t ∈ [0, T ) and {A ∈ F|P[A] = 0} ⊂ F 0 (see, e.g., Definition 2.1.11 in [13]). In addition, let (V, · V ) be a separable R-Banach space and let (H, ·, · H , · H ) be a separable R-Hilbert space with V ⊂ H continuously. The following assumptions will be used. Let D((−A) r ) with v D((−A) r ) = (−A) r v H for v ∈ D((−A) r ) and r ∈ R denote the domains of fractional powers of the linear operator −A (see, e.g., Section 3.7 in [14]). Assumption 2 (Nonlinearity F ). Assume D((−A) 1/2 ) ⊂ V continuously and let F : V → V be a twice continuously Fréchet differentiable mapping with F ′ (v)w H ≤ c w H , (12) F ′ (v) L(V ) ≤ c, F ′′ (v) L (2) (V ) ≤ c, (F ′ (u)) * L(D((−A) 1/2 )) ≤ c(1 + u D((−A) 1/2 ) ) (13) for every v, w ∈ V and every u ∈ D((−A) 1/2 ) where c ∈ [0, ∞) is a given real number. By definition F ′ (v) ∈ L(V ) is a bounded linear mapping from V to V for every v ∈ V . Due to the first condition in (12), we also have that F ′ (v) ∈ L(H) is a bounded linear mapping from H to H for every v ∈ V . In that sense, the adjoint operator (F ′ (v)) * ∈ L(H) given by (F ′ (v)) * u, w H = u, F ′ (v)w H for all u, w ∈ H is well defined for every v ∈ V . Due to (13), the operator (F ′ (v)) * ∈ L(H) is also a bounded linear mapping from D( (−A) 1/2 ) to D((−A) 1/2 ) for every v ∈ D((−A) 1/2 ). Assumption 3 (Stochastic process O). Let O : [0, T ] × Ω → D((−A) γ ) be a centered and adapted stochastic process with continuous sample paths such that O t 2 − e A(t 2 −t 1 ) O t 1 is independent of F t 1 for all 0 ≤ t 1 < t 2 ≤ T and such that E sup 0≤t≤T (−A) γ O t 4 H + sup 0≤t 1 <t 2 ≤T ((t 2 − t 1 ) −4θ E[ O t 2 − O t 1 4 V ]) < ∞ holds where γ ∈ [ 1 2 , 1) and θ ∈ (0, 1 2 ] are given real numbers. Assumption 4 (Initial value ξ). Let ξ : Ω → D(A) be a F 0 /B(D(A))- measurable mapping with E[ Aξ 4 H ] < ∞. These assumptions suffice to ensure the existence of a unique solution of the SPDE (14). X t (ω) = e At ξ(ω) + t 0 e A(t−s) F (X s (ω)) ds + O t (ω)(14)for all t ∈ [0, T ] and all ω ∈ Ω. Moreover, X : [0, T ] × Ω → D((−A) γ ) satisfies E[sup 0≤t≤T (−A) γ X t 4 H ] < ∞. The proof of Lemma 1 is given in Section 6. Some examples satisfying Assumptions 1-4 are presented in Section 4. Numerical scheme and main result. For numerical approximations of the SPDE (14), we have to discretize both the time interval [0, T ] and the R-Hilbert space H. To this end, we use projections P N : H → H given by P N (v) := N n=1 e n , v H e n for every v ∈ H, N ∈ N and finite-dimensional R-Hilbert spaces H N ⊂ H given by H N := P N (H) for every N ∈ N. Fi- nally, we define F/B(H N )-measurable mappings Y N,M m : Ω → H N for m ∈ {0, 1, . . . , M } and N, M ∈ N by Y N,M 0 (ω) := P N (ξ(ω)) + P N (O 0 (ω)) and by Y N,M m+1 (ω) := e AT /M Y N,M m (ω) + T M · (P N F )(Y N,M m (ω)) (15) + P N (O (m+1)T /M (ω) − e AT /M O mT /M (ω)) for every m ∈ {0, 1, . . . , M − 1}, N, M ∈ N and every ω ∈ Ω. In many examples, this scheme is as easy to simulate as the classical linear implicit Euler scheme. We refer to Section 4 for a detailed description of the implementation of our numerical scheme including a short matlab code. Theorem 1. Let Assumptions 1-4 be fulfilled. Then there is a real number C > 0 such that (E[ X mT /M − Y N,M m 2 H ]) 1/2 ≤ C 1 (λ N ) γ + (1 + log(M )) M 2θ (16) holds for every m ∈ {0, 1, . . . , M } and every N, M ∈ N where (λ N ) N ∈N ⊂ (0, ∞) is given in Assumption 1 where γ ∈ [ 1 2 , 1) and θ ∈ (0, 1 2 ] are given in Assumption 3 where X : [0, T ] × Ω → D((−A) γ ) is the solution of the SPDE (14) and where Y N,M m : Ω → H N , m ∈ {0, 1, . . . , M }, N, M ∈ N, is given by (15). Here and below log is the natural logarithm. While the expression 1 (λ N ) γ for N ∈ N in (16) arises due to discretizing the infinite-dimensional R-Hilbert space H, the expression (1+log(M )) M 2θ for M ∈ N arises due to discretizing the time interval [0, T ]. We would like to remark that the logarithmic term in (1+log(M )) M 2θ for M ∈ N can be avoided by assuming F (D((−A) 1/2 )) ⊂ D((−A) ε ) and an appropriate linear growth condition on F for some ε > 0. Although this condition is fulfilled in our examples below, we use this logarithmic term in Theorem 1 here in order to formulate Assumption 2 in our abstract setting as simple as possible. A similar result could be obtained for SPDEs of the form (14) but with a time dependent nonlinearity F . However, we omit the time dependency of the nonlinearity here for simplicity. Examples. Let H = L 2 ((0, 1), R) be the R-Hilbert space of equivalence classes of B((0, 1))/B(R)-measurable and square integrable functions from (0, 1) to R with the scalar product and the norm given by v, w H = 1 0 v(s)w(s) ds, v H = 1 0 |v(s)| 2 ds 1/2 for every v, w ∈ H. In addition, let V = C([0, 1], R) be the R-Banach space of continuous functions from [0, 1] to R equipped with the norm v V = sup 0≤x≤1 |v(x)| for every v ∈ V . Let κ ∈ (0, ∞) be a given positive real number and let (λ n ) n∈N ⊂ (0, ∞) and (e n ) n∈N ⊂ H be given by λ n := κn 2 π 2 , e n (x) := √ 2 sin(nπx) for every x ∈ (0, 1) and every n ∈ N. Hence, the linear operator A : D(A) ⊂ H → H reduces to the Laplacian with Dirichlet boundary conditions on the interval (0, 1) times the constant κ ∈ (0, ∞) (see, e.g., Section 3.8.1 in Sell and You [14]). In particular, D((−A) 1/2 ) reduces to the R-Sobolev space H 1 0 ((0, 1), R) equipped with the norm u D((−A) 1/2 ) = (−A) 1/2 u H = ∞ n=1 κn 2 π 2 | e n , u H | 2 1/2 = √ κ 1 0 |u ′ (x)| 2 dx 1/2 for all u ∈ D((−A) 1/2 ). (See Sell and You [14] for more information about this space.) Furthermore, let f : [0, 1] × R → R be a twice continuously differentiable function with the bounded partial derivatives ∂f ∂y (x, y) ≤ K, ∂ 2 f ∂x ∂y (x, y) ≤ K, ∂ 2 f ∂y 2 (x, y) ≤ K for all x ∈ [0, 1] and all y ∈ R with an arbitrary constant K ∈ [0, ∞). Then the Nemytskii operator F : V → V given by (F (v))(x) = f (x, v(x)) for every x ∈ [0, 1] and every v ∈ V satisfies Assumption 2. To see this note that F ′ (u)(v) = ∂f ∂y (x, u(x)) · v(x), F ′′ (u)(v, w) = ∂ 2 f ∂y 2 (x, u(x)) · v(x) · w(x) holds for all u, v, w ∈ V . Therefore, we have In view of the above choice, the SPDE (14) reduces to (F ′ (u)) * v 2 D((−A) 1/2 ) = (−A) 1/2 F ′ (u)v 2 H = κ 1 0 ∂ ∂x ∂f ∂y (x, u(x)) · v(x) 2 dx = κ 1 0 ∂ ∂x ∂f ∂y (x, u(x)) v(x) + ∂f ∂y (x, u(x)) · v ′ (x) 2 dx ≤ 2κ 1 0 ∂ ∂x ∂f ∂y (x, u(x)) v(x) 2 dx + 2κ 1 0 ∂f ∂y (x, u(x)) · v ′ (x) 2 dx and (F ′ (u)) * v 2 D((−A) 1/2 ) ≤ 2κ v 2 V 1 0 ∂ ∂x ∂f ∂y (x, u(x)) 2 dx + 2κK 2 1 0 |v ′ (x)| 2 dx ≤ 4 v 2 D((−A) 1/2 ) 1 0 ∂ 2 f ∂x ∂y (x, u(x)) 2 dx + 4 v 2 D((−A) 1/2 ) 1 0 ∂ 2 f ∂y 2 (x, u(x)) · u ′ (x) 2 dx + 2K 2 v 2 D((−A) 1/2 ) for all u, v ∈ D((−A) 1/2 ). Hence, we obtain (F ′ (u)) * v 2 D((−A) 1/2 ) ≤ 4K 2 v 2 D((−A) 1/2 ) + 4K 2 v 2 D((−A) 1/2 ) 1 0 |u ′ (x)| 2 dx + 2K 2 v 2 D((−A) 1/2 ) = 6K 2 v 2 D((−A) 1/2 ) + 4K 2 κ −1 v 2 D((−A) 1/2 ) u 2 D((−A) 1/2 ) EFFICIENT SIMULATION OF SPDES 11 and (F ′ (u)) * v D((−A) 1/2 ) ≤ 6K 2 v 2 D((−A) 1/2 ) + 4K 2 κ −1 v 2 D((−A) 1/2 ) u 2 D((−A) 1/2 ) ≤ √ 6K v D((−A) 1/2 ) + 2Kκ −1/2 v D((−A) 1/2 ) u D((−A) 1/2 ) (17) ≤ K v D((−A) 1/2 ) (3 + 2κ −1/2 u D((−A) 1/2 ) ) ≤ (3 + 2κ −1/2 )K v D((−A) 1/2 ) (1 + u D((−A) 1/2 ) ) for all u, v ∈ D((−A) 1/2 ). This shows that F indeed satisfies Assumption 2 with c = 3K. Let (b n ) n∈N ⊂ R be a sequence of real numbers with ∞ n=1 n ε |b n | 2 < ∞ for some arbitrarily small ε ∈ (0, ∞).dX t = κ ∂ 2 ∂x 2 X t + f (x, X t ) dt + B dW t , X t (0) = X t (1) = 0,(18)X 0 (x) = sin(πx) √ 2 + 3 √ 2 5 sin(3πx) for x ∈ [0, 1] and t ∈ [0, T ] where the linear operator B : H → H is given by Bv = ∞ n=1 b n e n , v e n for all v ∈ H and where (W t ) t∈[0,T ] is a cylindrical I-Wiener process on H. Since Assumption 3 is fulfilled for θ = 1 2 and γ = 1 2 , Theorem 1 shows the existence of a real number C > 0, such that E 1 0 |X nT /M (x) − Y N,M n (x)| 2 dx 1/2 ≤ C 1 N + (1 + log(M )) M(19) holds for all n ∈ {0, 1, . . . , M } and all N, M ∈ N. While the expression 1 N for N ∈ N in (19) corresponds to the spatial discretization error, the expression (19) corresponds to the temporal discretization error. Since these error terms are nearly of the same size, we choose M = N and consider the numerical approximations Y N,N n : Ω → H N , n ∈ {0, 1, . . . , N }, N ∈ N, in the following. Due to (19), we obtain the existence of real numbers C δ > 0, δ ∈ (0, 1), such that (1+log(M )) M for M ∈ N inE 1 0 |X T (x) − Y N,N N (x)| 2 dx 1/2 ≤ C δ · N (δ−1)(20) holds for all N ∈ N and all arbitrarily small δ ∈ (0, 1). In order to describe the implementation of the numerical scheme (15) = 1 2 , Y N,N 2,0 = 0, Y N,N 3,0 = 3 5 , Y N,N 4,0 = Y N,N 5,0 = · · · = 0 and Y N,N 1,n+1 = e −κπ 2 T /N Y N,N 1,n + T N e 1 , F (Y N,N n ) H + 1 − e −2κπ 2 T /N b 1 · π √ 2κ χ N 1,n , Y N,N 2,n+1 = e −κπ 2 2 2 T /N Y N,N 2,n + T N e 2 , F (Y N,N n ) H + 1 − e −2κ2 2 π 2 T /N b 2 · 2π √ 2κ χ N 2,n ,(21) . . . . . . . We remark that the log term in the computational effort O(N 2 log(N )) for N ∈ N arises if one computes the nonlinearity in (21) with fast Fourier transform (see Figure 2 for details). Y N,N N,n+1 = e −κπ 2 N 2 T /N Y N,N N,n + T N e N , F (Y N,N n ) H + 1 − e −2κN 2 π 2 T /N b N · N π √ 2κ χ N N, In order to compare the new numerical scheme (21) with classical schemes, we consider the well-known linear implicit Euler scheme combined with spectral Galerkin methods applied to the SPDE (18). The linear implicit Euler scheme is denoted by F/B(H N )-measurable mappings Z N n : Ω → H N , n ∈ {0, 1, . . . , N 2 }, N ∈ N, given by Z N 0 (ω) := P N (ξ(ω)) + P N (O 0 (ω)) and Z N n+1 (ω) := I − T N 2 A −1 Z N n (ω) + T N 2 (P N F )(Z N n (ω)) (22) + P N (B(W (n+1)T /N 2 (ω) − W nT /N 2 (ω))) for every n ∈ {0, 1, . . . , N 2 − 1} and every N ∈ N. It has been shown in the literature (see, e.g., Walsh [15], Gyöngy [4] and Hausenblas [7]) that the linear implicit Euler scheme (22) and other classical numerical schemes such as the linear implicit Crank-Nicolson scheme combined with finite elements, finite differences and spectral Galerkin methods converge with order dX t = 1 100 ∂ 2 ∂x 2 X t + 5 (1 − X t ) (1 + X 2 t ) dt + B dW t , X t (0) = X t (1) = 0,(23)X 0 (x) = sin(πx) √ 2 + 3 √ 2 5 sin(3πx) for x ∈ [0, 1] and t ∈ [0, 1]. In Figure 1 (see also Tables 1 and 2), we plot the root mean square discretization error of the numerical scheme (21) versus N 2 log(N ) (up to a constant the computational effort) and the root mean square discretization error of the linear implicit Euler scheme (22) versus N 3 log(N ) (up to a constant the computational effort) for different N ∈ N. The "expectations" are based on 40 independent random realizations and the unknown "exact" solution is approximated with a very high accuracy there. The short matlab code in Figure 2 shows that the solution of SPDE (23) can be simulated quite easily with the numerical scheme (21). Figure 3 is the result of the matlab code in Figure 2. It shows the solution of the stochastic reaction diffusion equation (23) at time t = T = 1 for one sample path ω ∈ Ω approximated with the numerical method (21). E 1 0 |X T (x) − Y N,N N (x)| 2 dx 1/2(24)E 1 0 |X T (x) − Z N N 2 (x)| 2 dx 1/2(25) 4.2. A stochastic partial differential equation with a spatially dependent f . This time let κ = 1 50 , T = 1, b n = n −0.6 5 for all n ∈ N and consider f : [0, 1] × R → R given by f (x, y) = (3.8x 2 − 2)y for all x ∈ [0, 1], y ∈ R to obtain the SPDE dX t = 1 50 ∂ 2 ∂x 2 X t + (3.8x 2 − 2)X t dt + B dW t ,X t (0) = X t (1) = 0,(26)X 0 (x) = sin(πx) √ 2 − 3 √ 2 5 sin(3πx) for x ∈ [0, 1] and t ∈ [0, T ]. Here too, the numerical approximation (21) converges to the exact solution with order 1 2 − with respect to up to a constant the computational effort (see Figure 4). Finally, in Figure 5 we illustrate how the two different f from examples (23) and (26) affect the evolution of the respective solution X t (ω, x), x ∈ [0, 1], for t ∈ {0, 1 10 , 3 10 , 6 10 , 1} and one sample path ω ∈ Ω. 5. A further numerical example. Although our setting in Section 2 uses the standard global Lipschitz assumption on the nonlinearity of the SPDE, we demonstrate the efficiency of our method numerically for a SPDE with a cubic nonglobally Lipschitz nonlinearity in this section. More formally, we consider the SPDE dX t = 1 10 ∂ 2 ∂x 2 1 + ∂ 2 ∂x 2 2 X t + X t − X 3 t dt + dW Q t ,(27) with X t | ∂(0,1) 2 ≡ 0 and X 0 (x 1 , x 2 ) = sin(πx 1 ) sin(πx 2 ) for all x 1 , x 2 ∈ (0, 1) and all v ∈ H. Of course, (27) is not included in our setting in Section 2. Even worse, it has recently been shown in [9] that many numerical methods fail to converge to the solution of a stochastic differential equation with super linearly growing coefficients in the strong root mean square sense. However, convergence in the pathwise sense often holds due to Gyöngy's result [3]. Therefore, we plot in Figure 6 the pathwise difference [4][5][6] and [12], for instance. Finally, we plot the solution of SPDE (27) for t ∈ {0, 6 10 } and one random ω ∈ Ω in Figure 7. 6. Proofs. The notation Xt(ω, x1, x2), x1, x2 ∈ [0, 1], of the SPDE (27) for t ∈ {0, 6 10 } and one random ω ∈ Ω approximated with the numerical method (15). . Combining these three parts will then yield the desired assertion via Gronwall's lemma as we will see below. Z L p (Ω;W ) := (E[ Z p W ]) 1/p ∈ [0, ∞] Before we begin with the first part, we introduce a universal constant R > 0 which is needed throughout this proof. More precisely, let R ∈ (0, ∞) be a real number which satisfies F (X t ) L 2 (Ω;H) ≤ R, (X t 2 − O t 2 ) − (X t 1 − O t 1 ) L 2 (Ω;H) ≤ R|t 2 − t 1 |, 1 λ 1 + 1 (1 − γ) + T + c ≤ R, v H ≤ R v V , O t 2 − O t 1 L 4 (Ω;V ) ≤ R|t 2 − t 1 | θ , ξ L 2 (Ω;D((−A) γ ) ≤ R, O t L 4 (Ω;D((−A) γ )) ≤ R, X t L 4 (Ω;D((−A) 1/2 )) ≤ R for every t, t 1 , t 2 ∈ [0, T ] and every v ∈ V where λ 1 ∈ (0, ∞) is given in Assumption 1 where c ∈ [0, ∞) is given in Assumption 2 and where γ ∈ [ 1 2 , 1) and θ ∈ (0, 1 2 ] are given in Assumption 3. Indeed, such a real number exists due to Assumptions 1-4 and Lemma 4 in Section 6.2. 6.1.1. Temporal discretization error. Due to (14), we have X mh = e Amh ξ + mh 0 e A(mh−s) F (X s ) ds + O mh = e Amh ξ + m−1 k=0 (k+1)h kh e A(mh−s) F (X s ) ds + O mh+ 2Rh ≤ m−2 k=0 (k+1)h kh e A(mh−s) (F (X s ) − F (X kh )) ds L 2 (Ω;H) + 2Rh + m−2 k=0 (k+1)h kh e A(mh−s) F (X kh ) ds − h m−2 k=0 e A(mh−kh) F (X kh ) L 2 (Ω;H) and X mh − Y M m L 2 (Ω;H) ≤ m−2 k=0 (k+1)h kh e A(mh−s) (F (X s ) − F (X kh + O s − O kh )) ds L 2 (Ω;H) EFFICIENT SIMULATION OF SPDES 23 + m−2 k=0 (k+1)h kh e A(mh−s) (F (X kh + O s − O kh ) − F (X kh )) ds L 2 (Ω;H) + m−2 k=0 (k+1)h kh (e A(mh−s) − e A(mh−kh) )F (X kh ) ds L 2 (Ω;H) + 2Rh for every m ∈ {0, 1, . . . , M } and every M ∈ N. Hence, we obtain X mh − Y M m L 2 (Ω;H) ≤ m−2 k=0 (k+1)h kh e A(mh−s) L(H) F (X s ) − F (X kh + O s − O kh ) L 2 (Ω;H) ds + m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − O kh ) ds L 2 (Ω;H) + m−2 k=0 (k+1)h kh e A(mh−s) 1 0 F ′′ (X kh + r(O s − O kh )) × (O s − O kh , O s − O kh ) × (1 − r) dr ds L 2 (Ω;H) + m−2 k=0 (k+1)h kh (e A(mh−s) − e A(mh−kh) )F (X kh ) L 2 (Ω;H) ds + 2R 2 M −1 and X mh − Y M m L 2 (Ω;H) ≤ c m−2 k=0 (k+1)h kh X s − (X kh + O s − O kh ) L 2 (Ω;H) ds + m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds× 1 0 F ′′ (X kh + r(O s − O kh )) × (O s − O kh , O s − O kh ) L 2 (Ω;H) dr ds + m−2 k=0 (k+1)h kh e A(mh−s) − e A(mh−kh) L(H) F (X kh ) L 2 (Ω;H) ds + 2R 2 M −1 for every m ∈ {0, 1, . . . , M } and every M ∈ N. Therefore, we have X mh − Y M m L 2 (Ω;H) ≤ c m−2 k=0 (k+1)h kh (X s − O s ) − (X kh − O kh ) L 2 (Ω;H) ds + m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds+ m−2 k=0 (k+1)h kh (mh − kh − mh + s) (mh − s) F (X kh ) L 2 (Ω;H) ds + 2R 2 M −1 for every m ∈ {0, 1, . . . , M } and every M ∈ N due to Lemma 2 below (see Section 6.2). Furthermore, we have E m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 H = m−2 k,k=0 E (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds,(k+1)E m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 H = m−2 k=0 E (k+1)h kh e A(mh−s) F ′ (X kh ) × (O s − e A(s−kh) O kh ) ds 2 H + m−2 k,k=0 k =k E (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds, (k+1)h kh e A(mh−s) F ′ (Xk h )(O s − e A(s−kh) Ok h ) dsE (k+1)h kh e A(mh−s) F ′ (X kh ) × (O s − e A(s−kh) O kh ) ds,(k+1)= m−2 k=0 E (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 H + 2 m−2 k,k=0 k<k E E (k+1)h kh e A(mh−s) F ′ (X kh ) × (O s − e A(s−kh) O kh ) ds, (k+1)h kh e A(mh−s) F ′ (Xk h ) × (O s − e A(s−kh) Ok h ) ds H Fk h for every m ∈ {0, 1, . . . , M } and every M ∈ N. Hence, we obtain E m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 H = m−2 k=0 E (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 H + 2 m−2 k,k=0 k<k E (k+1)h kh e A(mh−s) F ′ (X kh ) × (O s − e A(s−kh) O kh ) ds, (k+1)h kh e A(mh−s) F ′ (Xk h ) × (E[O s − e A(s−kh) Ok h |Fk h ]) ds H and E m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) dsX mh − Y M m L 2 (Ω;H) ≤ c m−2 k=0 (k+1)h kh (X s − O s ) − (X kh − O kh ) L 2 (Ω;H) ds + m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 L 2 (Ω;H) 1/2 + m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )((e A(s−kh) − I)O kh ) L 2 (Ω;H) ds + cR m−2 k=0 (k+1)h kh O s − O kh 2 L 4 (Ω;V ) ds + m−2 k=0 (k+1)h kh R (s − kh) (mh − s) ds + 2R 2 M −1 for every m ∈ {0, 1, . . . , M } and every M ∈ N. Hence, we obtain X mh − Y M m L 2 (Ω;H) ≤ cR m−2 k=0 (k+1)h kh (s − kh) ds + m−2 k=0 (k+1)h kh R (s − kh) (mh − (k + 1)h) ds + m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 L 2 (Ω;H) 1/2 + m−2 k=0 (k+1)h kh e A(mh−s) (−A) 1/2 L(H) × (−A) −1/2 F ′ (X kh )((e A(s−kh) − I)O kh ) L 2 (Ω;H) ds + cR m−2 k=0 (k+1)h kh O s − O kh+ R m−2 k=0 h 2(m − k − 1) + 2R 2 M −1 + m−2 k=0 (k+1)h kh e A(mh−s) F ′ (X kh ) × (O s − e A(s−kh) O kh ) L 2 (Ω;H) ds 2 1/2 + m−2 k=0 (k+1)h kh (mh − s) −1/2 × (−A) −1/2 F ′ (X kh )((e A(s−kh) − I)O kh ) L 2 (Ω;H) ds + cR m−2 k=0 (k+1)h kh (R(s − kh) θ ) 2 ds for every m ∈ {0, 1, . . . , M } and every M ∈ N. This yields X mh − Y M m L 2 (Ω;H) ≤ 1 2 cRT h + 1 2 Rh m−1 k=1 1 k + 2R 2 M −1 + cR 3 m−2 k=0 (k+1)h kh (s − kh) 2θ ds + m−2 k=0 (k+1)h kh F ′ (X kh )(O s − e A(s−kh) O kh ) L 2 (Ω;H) ds 2 1/2 + m−2 k=0 (k+1)h kh (mh − (k + 1)h) −1/2 × (−A) −1/2 F ′ (X kh )((e A(s−kh) − I)O kh ) L 2 (Ω;H) ds and X mh − Y M m L 2 (Ω;H) ≤ 1 2 cRT 2 M −1 + 1 2 R 2 M −1 1 + m−1 k=2 1 k + 2R 2 M −1 + cR 3 M h (1+2θ) + √ h m−2 k=0 (k+1)h kh F ′ (X kh )(O s − e A(s−kh) O kh ) 2 L 2 (Ω;H) ds 1/2 + √ T m−2 k=0 1 (m − k − 1)h (k+1)h kh (−A) −1/2 F ′ (X kh ) × ((e A(s−kh) − I)O kh ) L 2 (Ω;H) ds for every m ∈ {0, 1, . . . , M } and every M ∈ N. Hence, we have X mh − Y M m L 2 (Ω;H) ≤ 1 2 R 4 M −1 + 1 2 R 2 M −1 1 + M 1 1 s ds + 2R 2 M −1 + cR 3 T h 2θ + √ h m−2 k=0 (k+1)h kh c 2 O s − e A(s−kh) O kh 2 L 2 (Ω;H) ds 1/2 + R m−2 k=0 1 (m − k − 1)h × (k+1)h kh (−A) −1/2 F ′ (X kh )((e A(s−kh) − I)O kh ) L 2 (Ω;H) ds and X mh − Y M m L 2 (Ω;H) ≤ 1 2 R 4 M −1 + 1 2 R 2 M −1 (1 + log(M )) + 2R 2 M −1 + R 6 M −2θ + √ T cM −1/2 m−2 k=0 (k+1)h kh O s − e A(s−kh) O kh 2 L 2 (Ω;H) ds 1/2 + m−2 k=0 R (m − k − 1)h × (k+1)h kh sup w H ≤1 | w, (−A) −1/2 F ′ (X kh ) × (e A(+ m−2 k=0 R (m − k − 1)h × (k+1)h kh sup w H ≤1 | (F ′ (X kh )) * (−A) −1/2 w, (e A(s−kh) − I)O kh H | L 2 (Ω;R) ds + R 2 M −1/2 m−2 k=0 (k+1)h kh ( O s − O kh L 2 (Ω;H) + e A(s−kh) O kh − O kh L 2 (Ω;H) ) 2 ds 1/2 and X mh − Y M m L 2 (Ω;H) ≤ 4R 6 (1 + log(M )) M 2θ + m−2 k=0 R (m − k − 1)h × (k+1)h kh sup w H ≤1 (F ′ (X kh )) * (−A) −1/2 w D((−A) 1/2 ) × (e A(s−kh) − I)O kh D((−A) −1/2 ) L 2 (Ω;R) ds + R 3 M −1/2 × m−2 k=0 (k+1)h kh ( O s − O kh L 2 (Ω;V ) + (e A(R (m − k − 1)h × (k+1)h kh c(1 + X kh D((−A) 1/2 ) ) (31) × (e A(s−kh) − I)O kh D((−A) −1/2 ) L 2 (Ω;R) ds + R 3 M −1/2 × m−2 k=0 (k+1)h kh (R(s − kh) θ + (s − kh) γ O kh L 2 (Ω;D((−A) γ )) ) 2 ds 1/2 and therefore and hence X mh − Y M m L 2 (Ω;H) ≤ 4R 6 (1 + log(M )) M 2θ + m−2 k=0 cR (m − k − 1)h (k+1)h+ 2R 5 M −1/2 m−2 k=0 (k+1)h kh h 2θ ds 1/2 + m−2 k=0 cR(1 + R) (m − k − 1)h × (k+1)h kh (−A) −(γ+1/2) (e A(≤ 4R 6 (1 + log(M )) M 2θ + 2R 5 √ T M −1/2 h θ + m−2 k=0 2R 4 (m − k − 1)h (k+1)h kh (−A) (1/2−γ) L(H) × A −1 (e A(X mh − Y M m L 2 (Ω;H) ≤ 6R 6 (1 + log(M )) M 2θ + m−2 k=0 R 5 h (m − k − 1) ≤ 6R 6 (1 + log(M )) M 2θ + R 6 M −1 M k=1 1 k (32) ≤ 6R 6 (1 + log(M )) M 2θ + R 6 M −2θ 1 +Y M m − P N (Y M m ) L 2 (Ω;H) ≤ (−A) −γ (I − P N ) L(H) (−A) γ ξ L 2 (Ω;H) + (−A) −γ (I − P N ) L(H) (−A) γ O mh L 2 (Ω;H) + Rh m−1 k=0 (−A) −γ (I − P N ) L(H) (−A) γ e A(mh−kh) L(H) ≤ (λ N ) −γ ( (−A) γ ξ L 2 (Ω;H) + (−A) γ O mh L 2 (Ω;H) ) + Rh m−1 k=0 (λ N ) −γ (−A) γ e A(mh−kh) L(H) for every m ∈ {0, 1, . . . , M } and every M ∈ N. Therefore, we have Y M m − P N (Y M m ) L 2 (Ω;H) ≤ (λ N ) −γ ( ξ L 2 (Ω;D((−A) γ )) + O mh L 2 (Ω;D((−A) γ )) ) + Rh(λ N ) −γ m−1 k=0 1 (mh − kh) γ (−A(mh − kh)) γ e A(mh−kh) L(H) ≤ 2R(λ N ) −γ + Rh (1−γ) (λ N ) −γ m−1 k=0 1 (m − k) γ sup x>0 x γ e −x ≤ 2R(λ N ) −γ + Rh (1−γ) (λ N ) −γ m k=1 1 k γ ≤ 2R(λ N ) −γ + Rh (1−γ) (λ N ) −γ 1 + m k=2 1 k γ and Y M m − P N (Y M m ) L 2 (Ω;H) ≤ R(λ N ) −γ 2 + h (1−γ) 1 + M 1 1 s γ ds = R(λ N ) −γ 2 + h (1−γ) 1 + s (1−γ) (1 − γ) s=M s=1 (33) = R(λ N ) −γ 2 + h (1−γ) 1 + M (1−γ) (1 − γ) − 1 (1 − γ) ≤ R(λ N ) −γ 2 + T (1−γ) (1 − γ) ≤ 3R 3 (λ N ) −Y N,M m+1 = e Ah (Y N,M m + h · (P N F )(Y N,M m )) + P N (O (m+1)h − e Ah O mh ) = e Ah Y N,M m + h · P N e Ah F (Y N,M m ) + P N (O (m+1)h ) − e Ah P N (O mh ) = e Ah (Y N,M m − P N (O mh )) + h · P N e Ah F (Y N,M m ) + P N (O (m+1)h ) and Y N,M m+1 = e Ah e Amh (P N (ξ)) + h m−1 k=0 P N e A(mh−kh) F (Y N,M k ) + h · P N e Ah F (Y N,M m ) + P N (O (m+1)h ) = e A(m+1)h (P N (ξ)) + h m−1 k=0 P N e A((m+1)h−kh) F (Y N,M k ) + h · P N e Ah F (Y N,M m ) + P N (O (m+1)h ) = e A(m+1)h (P N (ξ)) + h m k=0 P N e A((m+1)h−kh) F (Y N,M k ) + P N (O (m+1)h ) for every N, M ∈ N, which shows (34) by induction. In the next step, (34) yields P N (Y N m ) − Y N,M m = h m−1 k=0 P N e A(mh−kh) F (X kh ) 36 A. JENTZEN, P. KLOEDEN AND G. WINKEL − h m−1 k=0 P N e A(mh−kh) F (Y N,M k ) = h m−1 k=0 P N e A(mh−kh) (F (X kh ) − F (Y N,M k )) for every m ∈ {0, 1, . . . , M } and every N, M ∈ N. Therefore, we obtain P N (Y N m ) − Y N,M m L 2 (Ω;H) ≤ h m−1 k=0 P N e A(mh−kh) (F (X kh ) − F (Y N,M k )) L 2 (Ω;H) ≤ h m−1 k=0 ( P N e A(mh−kh) L(H) F (X kh ) − F (Y N,M k ) L 2 (Ω;H) ) (35) ≤ h m−1 k=0 F (X kh ) − F (Y N,M k ) L 2 (Ω;H) ≤ ch m−1 k=0 X kh − Y N+ Y M m − P N (Y M m ) L 2 (Ω;H) + P N (Y M m ) − Y N,M m L 2 (Ω;H) ≤ 7R 6 (1 + log(M )) M 2θ + 3R 3 1 (λ N ) γ + ch m−1 k=0 X kh − Y N,e At 2 − e At 1 L(H) ≤ (t 2 − t 1 ) t 1 for every t 1 , t 2 ∈ (0, T ] with t 1 ≤ t 2 . Proof. By definition, we have e At 2 − e At 1 L(H) = (e A(t 2 −t 1 ) − I)e At 1 L(H) ≤ A −1 (e A(t 2 −t 1 ) − I) L(H) Ae At 1 L(H) = (A(t 2 − t 1 )) −1 (e A(t 2 −t 1 ) − I) L(H) × At 1 e At 1 L(H) (t 2 − t 1 ) t 1 ≤ sup x∈(0,∞) (1 − e −x ) x sup x∈(0,∞) xe −x (t 2 − t 1 ) t 1 ≤ (t 2 − t 1 ) t 1 for every t 1 , t 2 ∈ (0, T ] with t 1 < t 2 . ≤ R(t 2 − t 1 ) + cR(t 2 − t 1 ) t 1 0 (t 1 − s) (θ−1) ds + 2R(t 2 − t 1 ) = R(t 2 − t 1 ) + cR(t 2 − t 1 ) t 1 0 s (θ−1) ds + 2R(t 2 − t 1 ) ≤ (R + cR(T + 1)θ −1 + 2R)(t 2 − t 1 ) for every 0 ≤ t 1 < t 2 ≤ T . Combining this and (40) shows the assertion. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2011, Vol. 21, No. 3, 908-950. This reprint differs from the original in pagination and typographic detail. 1 2 A. JENTZEN, P. KLOEDEN AND G. WINKEL alence classes of square integrable functions from (0, 1) to R. Moreover, let f : [0, 1] × R → R be a smooth function with bounded partial derivatives, let ξ : [0, 1] → R with ξ(0) = ξ(1) = 0 be a smooth function and let W Q : [0, T ] × Ω → H be a standard Q-Wiener process with a trace class operator Q : H → H (see, e.g., Definition 2.1.9 in for x ∈ (0, 1) and t ∈ [0, T ]. Under the assumptions above, the SPDE (1) has a unique mild solution. Specifically, there exists an up to indistinguishability unique stochastic process X : [0, T ] × Ω → H with continuous sample paths which satisfies X t = e At ξ +t 0 e A(t−s) F (X s ) ds + t ∈ [0, T ] where A : D(A) ⊂ H → H is the Laplacian with Dirichlet boundary conditions on (0, 1) and where F : H → H is the Nemytskii operator (F (v))(x) := f (x, v(x)) for all x ∈ (0, 1) and all v ∈ H. for every n ∈ {0, 1, . . . , N 2 − 1} and every N ∈ N where the bounded linear operators P N : H → H, N ∈ N, are given by (P N (v)) Assumption 1 ( 1Linear operator A). Let (λ n ) n∈N ⊂ (0, ∞) be an increasing sequence of real numbers and let (e n ) n∈N ⊂ H be an orthonormal basis of H. Assume that the linear operator A : D(A) ⊂ H → H is given by Av = ∞ n=1 −λ n e n , v H e n for all v ∈ D(A) with D(A) = {w ∈ H| ∞ n=1 |λ n | 2 | e n , w H | 2 < ∞}. Lemma 1 ( 1Existence of the solution). Let Assumptions 1-4 be fulfilled. Then there exists a unique adapted stochastic process X : [0, T ]×Ω → D((−A) γ ) with continuous sample paths which fulfills Lemma 4.3 in [1] then gives the existence of an up to indistinguishability unique stochastic process O : [0, T ] × Ω → V which satisfies Assumption 3 for θ = t ∈ [0, T ] where the β n : [0, T ] × Ω → R, n ∈ N, are independent standard Brownian motions with respect to a given normal filtration (F t ) t∈[0,T ] .Moreover, the mapping ξ : Ω → V given by ω ∈ Ω and all x ∈ (0, 1) obviously satisfies Assumption 4. in this example, we use the F/B(R)-measurable mappings Y N,M n,m : Ω → R given by Y N,M n,m (ω) := e n , Y N,M m (ω) H for all n ∈ {1, 2, . . . , N }, m ∈ {0, 1, . . . , M } and all N, M ∈ N. The numerical scheme (15) for the SPDE (18) with M = N then reduces to Y N,N 1,0 n for all n ∈ {0, 1, . . . , N − 1} and all N ∈ N where the F/B(R)-measurable mappings χ N n,m : Ω → R for n ∈ {1, 2, . . . , N }, m ∈ {0, 1, . . . , N − 1} and N ∈ N are independent standard normal random variables. Since O(N 2 log(N )) computational operations and independent standard normal random variables (computational effort) are needed to compute the numerical solution Y N,N N given by (21) for N ∈ N, it follows that Y N,respect to the computational effort to the exact solution X : [0, T ] × Ω → D((−A) 1/2 ) of the SPDE (18) in the sense of (20) . A stochastic reaction diffusion equation. In this example, we set κ = 1 100 , T = 1, b n = n −0.55 3.5 for all n ∈ N and consider f : [0, 1] × R → R given by f (x, y) = 5 (1−y) (1+y 2 ) for all x ∈ [0, 1], y ∈ R. The SPDE (18) then reduces to Fig. 1 . 1Root mean square approximation error (24) of the numerical scheme (21) and root mean square approximation error (25) of the linear implicit Euler scheme (22) applied to SPDE (23) versus up to a constant the computational effort. plot( (0:N+1)/(N+1), [0,dst(Y)*sqrt(2),0], 'k', 'Linewidth', 2 ); Fig. 2. matlab code for the numerical scheme (21) applied to the SPDE (23). Fig. 3 . 3Result of the matlab code inFigure 2: Solution of the stochastic reaction diffusion equation (23) at t = T = 1 for one sample path ω ∈ Ω approximated with the numerical method (21). Fig. 4 . 4Root mean square approximation error (24) of the numerical scheme (21) applied to SPDE (26) versus up to a constant the computational effort. ( for t, x 1 , x 2 ∈ [0, 1] on the R-Hilbert space H = L 2 ((0, 1) 2 , R) of equivalence classes of B((0, 1) 2 )/B(R)-measurable and square integrable functions from (0, 1) 2 to R here where (W Q t ) t∈[0,1] is a cylindrical Q-Wiener process on H with the covariance operator Q : H → H given by Qvnπy 1 ) sin(mπy 2 )v(y 1 , y 2 ) dy 1 dy 2 |X1/ 2 Fig. 5 . 25T (ω, x 1 , x 2 ) − Y N,N N (ω, x 1 , x 2 )| 2 dx 1 dx 2 Solution Xt(ω, x), x ∈ [0,1], of the stochastic reaction diffusion equation (23) and of the SPDE (26) for t ∈ {0,1 10 , 3 10 , 6 10 , 1} and one sample path ω ∈ Ω approximated with the numerical method (21).of the exact solution X T (ω) and of the numerical approximation Y N,N N (ω) [see(15)] applied to the SPDE (27) versus up to a constant the computational effort N 3 log(N ) for N ∈ {2 2 , 2 3 , . . . , 2 7 } and one random ω ∈ Ω. It turns out that the method (15) converges with order 1 3 − with respect to the computational effort. The linear implicit Euler scheme is known to converge Fig. 6 . 6Pathwise approximation error of the numerical scheme (15) applied to SPDE (27) versus up to a constant the computational effort for one random ω ∈ Ω. in the pathwise sense with order 1 4 − with respect to the computational effort to the solution of the SPDE (27). Further pathwise approximation results for the SPDE (27) and other SPDEs with nonglobally Lipschitz coefficients can be found in is used throughout this section for an R-Banach space (W, · W ), a F/B(W )measurable mapping Z : Ω → W and a real number p ∈ [1, ∞). 6. 1 . 1Proof of Theorem 1. The F/B(H)-measurable mappings Y M m : Ω → H for m ∈ {0, 1, . . . , M } and M ∈ N given by Y M m (ω) := e Amh ξ(ω) m ∈ {0, 1, . . . , M }, ω ∈ Ω and M ∈ N are used throughout this proof. Here and below h is the time stepsize h = h M = T M with M ∈ N.This proof is divided into three parts. In the first part (see Section 6.1.1), we Fig. 7 . 7Solution m ∈ {0, 1, . . . , M } and every M ∈ N which corresponds to the temporal discretization error. In the second part (see Section 6.1.2), we estimate Y M m − P N (Y M m ) L 2 (Ω;H) for every m ∈ {0, 1, . . . , M } and every N, M ∈ N which corresponds to the spatial discretization error. Finally, we estimate P N (Y M m ) − Y N,M m L 2 (Ω;H) for every m ∈ {0, 1, . . . , M } and every N, M ∈ N in the third part (see Section 6.1.3) e for every m ∈ {0, 1, . . . , M } and every M ∈ N. From (28), we haveX mh − Y e A(mh−s) F (X s ) ds − h m−1 k=0 e A(mh−kh) F (X kh ) L 2 (Ω;H) (mh−s) F (X s ) L 2 (Ω;H) ds + h e Ah F (X max(m−1,0)h ) L 2 (Ω;H) F (X s ) L 2 (Ω;H) ds + h e Ah L(H) F (X max(m−1,0)h ) L2 (Ω;H) for every m ∈ {0, 1, . . . , M } and every M ∈ N. Therefore, we obtain X mh − Y A(mh−kh) F (X kh ) L 2 (Ω;H) e A(mh−s) F ′ (X kh )((e A(s−kh) − I)O kh ) ds e A(mh−s) F ′ (X kh )((e A(s−kh) − I)O kh ) L 2 (Ω;H) Hee for every m ∈ {0, 1, . . . , M } and every M ∈ N. A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) e h kh e A(mh−s) F ′ (Xk h ) × (O s − e A(s−kh) Ok h ) A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds2 H 26 A. JENTZEN, P. KLOEDEN AND G. WINKEL e A(mh−s) F ′ (X kh )(O s − e A(s−kh) O kh ) ds 2 H EFFICIENT SIMULATION OF SPDES 27 for every m ∈ {0, 1, . . . , M } and every M ∈ N due to Assumption 3. Combining (29) and (30) then shows s−kh) − I)O kh H | L 2 (Ω;R) ds for every m ∈ {0, 1, . . . , M } and every M ∈ N. This yields X mh − Y M m L 2 (Ω;H) 2 + 2R 2 + R 6 (1 + log(M )) M 2θ 30 A. JENTZEN, P. KLOEDEN AND G. WINKEL s−kh) − I)O kh L 2 (Ω;H) ) 2 ds 1/2 for every m ∈ {0, 1, . . . , M } and every M ∈ N. Using now condition (13) in Assumption 2 shows X mh − Y M m L 2 (Ω;H) ≤ 4R 6 (1 + log(M )) kh 1 +( 1X kh D((−A) 1/2 ) L 4 (Ω;R) × (e A(s−kh) − I)O kh L 4 (Ω;D((−A) Rh θ + Rh θ T (γ−θ) ) 2 ds 1/2 for every m ∈ {0, 1, . . . , M } and every M ∈ N. Hence, we obtain X mh − Y A(s−kh) − I)O kh L 4 (Ω;D((−A) −1/2 )) ds + R 3 M −1 s−kh) − I) L(H) × O kh L 4 (Ω;D((−A) γ )) ds for every m ∈ {0, 1, . . . , M } and every M ∈ N. This yields X mh − Y M m L 2 (Ω;H) ≤ 4R 6 (1 + log(M )) M 2θ + 2R 5 M −1/2 (M h (1+2θ) ) A) −(γ+1/2) (e A(s−kh) − I) L(H) ds and hence X mh − Y M m L 2 (Ω;H) s−kh) − I) L(H) ds for every m ∈ {0, 1, . . . , M } and every M ∈ N. Therefore, we have X mh − Y e( for every m ∈ {0, 1, . . . , M } and every M ∈ N.6.1.2. Spatial discretization error. Due to (28), we obtainY M m − P N (Y M m ) L 2 (Ω;H) = e Amh (ξ − P N (A(mh−kh) − P N e A(mh−kh) )F (X kh ) + O mh − P N (O mh ) L 2 (Ω;H) ≤ e Amh (ξ − P N (ξ)) L 2 (Ω;H) + O mh − P N (O mh ) L 2 (Ω;H) A(mh−kh) − P N e A(mh−kh) )F (X kh ) L 2 (Ω;H) and Y M m − P N (Y M m ) L 2 (Ω;H) ≤ ξ − P N (ξ) L 2 (Ω;H) + O mh − P N (O mh ) L 2 (Ω;A(mh−kh) − P N e A(mh−kh) L(H) F (X kh ) L 2 (Ω;H) ≤ (I − P N )ξ L 2 (Ω;H) + (I − P N )O mh L 2 (Ω;H) I − P N )e A(mh−kh) L(H) PP γ for every m ∈ {0, 1, . . . , M } and every M ∈ N. N e A(mh−kh) F (Y N,M k ) + P N (O mh ) (34) for every m ∈ {0, 1, . . . , M } and every N, M ∈ N. Indeed, in the case m N e A(0−kh) F (Y N,M k ) + P N (O 0 ) for every N, M ∈ N. Moreover, if (34) holds for one m ∈ {0, 1, . . . , M − 1}, then we obtain m ∈ {0, 1, . . . , M } and every N, M ∈ N. Combining (32), (33) and (35) finally yields X mh − Y N,M m L 2 (Ω;H) ≤ X mh − Y M m L 2 (Ω;H) m ∈ {0, 1, . . . , M } and every N, M ∈ N. Hence, Gronwall's lemma yieldsX mh − Y N,M m L 2 (Ω;H) ≤ 7R 6 (1 + log(M )) M 2θ + 3R 3 1 (λ N ) γ e cT ≤ 7R 6 (1 + log(M )) M 2θ + 7R 6 1 (λ N ) γ e cT(36) = (e cT 7R 6 ) (1 + log(M )) M 2θ + 1 (λ N ) γ for every m ∈ {0, 1, . . . , M } and every N, M ∈ N, which shows the assertion. Lemma 2 . 2Let Assumptions 1-4 be fulfilled. Then we have Lemma 3 . 3Let Assumptions 1-4 be fulfilled. Then we obtainsup 0≤t 1 <t 2 ≤T X t 2 − X t 1 L 2 (Ω;H) (t 2 − t 1 ) θ < ∞, where θ ∈ (0, 1 2 ] isgiven in Assumption 3 and where X : Ω×[0, T ] → D((−A) γ ) is the solution of the SPDE (14). Proof. First, let R ∈ [0, ∞) be the real number given by R := ξ L 2 (Ω;D(A)) + sup t∈[0,T ] F (X t ) L 2 (Ω;H) + sup 0≤t 1 <t 2 ≤T O t 2 −e 2O t 1 L 2 (Ω;H) (t 2 − t 1 ) θ ,which is finite due to Assumptions 1-4. Then we havee At 2 ξ − e At 1 ξ L 2 (Ω;H) = e At 1 (e A(t 2 −t 1 ) ξ − ξ) L 2 (Ω;H) ≤ e A(t 2 −t 1 ) ξ − ξ L 2 (Ω;H) (38) ≤ A −1 (e A(t 2 −t 1 ) − I) L(H) ξ L 2 (Ω;D(A)) ≤ R(t 2 − t 1 ) 40 A. JENTZEN, P. KLOEDEN AND G. WINKEL for every 0 ≤ t 1 < t 2 ≤ T . Moreover, (t 2 −s) − e A(t 1 −s) L(H) F (X s ) L 2 (Ω;H) ds ≤ R(t 2 − t 1 ) + R t 1 0 e A(t 2 −s) − e A(t 1 −s) (1−θ) L(H) e A(t 2 −s) − e A(t 1 −s) θ L(H) ds for every 0 ≤ t 1 < t 2 ≤ T . A(t 1 −s) F (X s ) ds L 2 (Ω;H) ≤ R(t 2 − t 1 ) + 2R(t 2 − t 1 ) θ) (t 2 − t 1 ) θfor every 0 ≤ t 1 < t 2 ≤ T . Combining (38), (39) and Assumption 3 yields the assertion. Lemma 4 .e 4Let Assumptions 1-4 be fulfilled. Then we obtain sup 0≤t 1 <t 2 ≤T(X t 2 − O t 2 ) − (X t 1 − O t 1 ) L 2 (Ω;H) (t 2 − t 1 ) < ∞, EFFICIENT SIMULATION OF SPDES 41 where O : [0, T ] × Ω → D((−A) γ )is given in Assumption 3 and where X :[0, T ] × Ω → D((−A) γ ) is the solution of the SPDE(14).Proof. First, let R ∈ [0, ∞) be the real number given byR := ξ L 2 (Ω;D(A)) + sup t∈[0,T ] F (X t ) L 2 (Ω;H) + sup 0≤t 1 <t 2 ≤T X t 2 − X t 1 L 2 (Ω;H) (t 2 − t 1 ) θ ,which exists due to Lemma 3. Then we havee At 2 ξ − e At 1 ξ L 2 (Ω;H) = e At 1 (e A(t 2 −t 1 ) ξ − ξ) L 2 (Ω;H) ≤ e A(t 2 −t 1 ) ξ − ξ L 2 (Ω;H) (40) ≤ R(t 2 − t 1 )for every 0 ≤ t 1 < t 2 ≤ T . Moreover, A(t 2 −s) F (X s ) ds (e A(t 2 eee 2A(t 2 −s) − e A(t 1 −s) )(F (X s ) − F (X t 1 )) ds −s) − e A(t 1 −s) )F (X t 1 ) ds A(t 2 −s) − e A(t 1 −s) L(H) F (X s ) − F (X t 1 ) L 2 (Ω;H) ds + R t 1 0 (e A(t 2 −s) − e A(t 1 −s) ) ds L(H) 42 A. JENTZEN, P. KLOEDEN AND G. WINKEL for every 0 ≤ t 1 < t 2 ≤ T . Hence, A(t 2 −s) − e A(t 1 −s) L(H) X s − X t 1 L 2 (Ω;H) A(t 1 −s) F (X s ) ds L 2 (Ω;H) Table 1 1Root mean square approximation error (24) of Y N,N N given by (21) applied to the SPDE (23) for N ∈ {2 2 , 2 3 , . . . , 2 11 }Independent standard Computational Root mean square Numerical normal random effort N 2 log(N ) approximation scheme (21) variables N 2 (up to a constant) error (24) Y 2 2 ,2 2 2 2 16 22 0.1864 Y 2 3 ,2 3 2 3 64 133 0.0914 Y 2 4 ,2 4 2 4 256 710 0.0417 Y 2 5 ,2 5 2 5 1024 3549 0.0191 Y 2 6 ,2 6 2 6 4096 17,035 0.0091 Y 2 7 ,2 7 2 7 16,384 79,496 0.0045 Y 2 8 ,2 8 2 8 65,536 363,408 0.0022 Y 2 9 ,2 9 2 9 262,144 1,635,339 0.0011 Y 2 10 ,2 10 2 10 1,048,576 7,268,174 0.0005 Y 2 11 ,2 11 2 11 4,194,304 31,979,969 0.0003 Table 2 2Root mean square approximation error (25) of Z N N 2 given by (22) applied to the SPDE (23) for N ∈ {2 1 , 2 2 , . . . , 2 7 } Linear implicit Independent standard Computational Root mean square Euler normal random effort N 3 log(N ) approximation scheme (22) variables N 3 (up to a constant) error (25) Z 2 1 2 2 8 6 0.3066 Z 2 2 2 4 64 88 0.1715 Z 2 3 2 6 512 1064 0.0837 Z 2 4 2 8 4096 11,356 0.0353 Z 2 5 2 10 32,768 113,565 0.0135 Z 2 6 2 12 262,144 1,090,226 0.0058 Z 2 7 2 14 2,097,152 10,175,444 0.0027 Acknowledgments. We strongly thank the anonymous referee for his careful reading and his very valuable advice.Proof of Lemma 1. A standard application of Banach's fix point theorem (see, e.g., Section 7.1 in[2]) yields the existence of a unique adapted stochastic process X : [0, T ] × Ω → V with continuous sample paths which fulfills(14). Moreover, we havefor every t ∈ [0, T ] and every ω ∈ Ω, sinceholds for every t ∈ [0, T ] and every ω ∈ Ω. Assumptions 3, 4 and (37) hence imply X t (ω) ∈ D((−A) γ ) for every t ∈ [0, T ] and every ω ∈ Ω. Furthermore, we havefor every t ∈ [0, T ]. This yieldsfor every t ∈ [0, T ]. Hence, Lemma 7.1.1 in[8]showsand therefore sup 0≤t≤T x (n(1−γ)) Γ(n(1 − γ) + 1)for every x ∈ [0, ∞) where Γ : (0, ∞) → (0, ∞) is the Gamma function. Galerkin approximations for the stochastic burgers equation. D Blömker, A Jentzen, Institute for Mathematics, Univ. AugsburgPreprintBlömker, D. and Jentzen, A. (2009). Galerkin approximations for the stochastic burgers equation. Preprint, Institute for Mathematics, Univ. Augsburg. Avail- able at http://opus.bibliothek.uni-augsburg.de/volltexte/2009/1444/. G Da Prato, J Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. CambridgeCambridge Univ. Press1207136Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge. MR1207136 A note on Euler's approximations. Potential Anal. I Gyöngy, 8Gyöngy, I. (1998). A note on Euler's approximations. Potential Anal. 8 205-216. MR1625576 Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise. I Gyöngy, II. Potential Anal. 111699161Gyöngy, I. (1999). Lattice approximations for stochastic quasi-linear parabolic par- tial differential equations driven by space-time white noise. II. Potential Anal. 11 1-37. MR1699161 On discretization schemes for stochastic evolution equations. I Gyöngy, A Millet, Potential Anal. 232139212Gyöngy, I. and Millet, A. (2005). On discretization schemes for stochastic evolu- tion equations. Potential Anal. 23 99-134. MR2139212 Rate of convergence of space-time approximations for stochastic evolution equations. Potential Anal. I Gyöngy, A Millet, 30Gyöngy, I. and Millet, A. (2009). Rate of convergence of space-time approxima- tions for stochastic evolution equations. Potential Anal. 30 29-64. MR2465711 Approximation for semilinear stochastic evolution equations. Potential Anal. E Hausenblas, 181953619Hausenblas, E. (2003). Approximation for semilinear stochastic evolution equations. Potential Anal. 18 141-186. MR1953619 Geometric Theory of Semilinear Parabolic Equations. D Henry, Lecture Notes in Math. 840610244SpringerHenry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math. 840. Springer, Berlin. MR610244 Strong and weak divergence in finite time of Euler's method for SDEs with non-globally Lipschitz continuous coefficients. M Hutzenthaler, A Jentzen, P E Kloeden, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467Hutzenthaler, M., Jentzen, A. and Kloeden, P. E. (2011). Strong and weak divergence in finite time of Euler's method for SDEs with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 1563-1576. Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise. A Jentzen, P E Kloeden, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465Jentzen, A. and Kloeden, P. E. (2009). Overcoming the order barrier in the nu- merical approximation of stochastic partial differential equations with additive space-time noise. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 649-667. MR2471778 Taylor expansions of solutions of stochastic partial differential equations with additive noise. A Jentzen, P E Kloeden, Ann. Probab. 38Jentzen, A. and Kloeden, P. E. (2010). Taylor expansions of solutions of stochastic partial differential equations with additive noise. Ann. Probab. 38 532-569. Numerical approximation for a white noise driven SPDE with locally bounded drift. Potential Anal. R Pettersson, M Signahl, 222135265Pettersson, R. and Signahl, M. (2005). Numerical approximation for a white noise driven SPDE with locally bounded drift. Potential Anal. 22 375-393. MR2135265 A Concise Course on Stochastic Partial Differential Equations. C Prévôt, M Röckner, Lecture Notes in Math. 2329435SpringerPrévôt, C. and Röckner, M. (2007). A Concise Course on Stochastic Partial Dif- ferential Equations. Lecture Notes in Math. 1905. Springer, Berlin. MR2329435 Dynamics of Evolutionary Equations. G R Sell, Y You, Applied Mathematical Sciences. 143SpringerSell, G. R. and You, Y. (2002). Dynamics of Evolutionary Equations. Applied Mathematical Sciences 143. Springer, New York. MR1873467 Finite element methods for parabolic stochastic PDE's. Potential Anal. J B Walsh, 232136207Walsh, J. B. (2005). Finite element methods for parabolic stochastic PDE's. Poten- tial Anal. 23 1-43. MR2136207 A Jentzen, 08544-1000 USA E-mail: [email protected] P. Kloeden G. Winkel Institute of Mathematics Johann Wolfgang Goethe University D-60054. Washington Road Princeton, New Jersey; Frankfurt am Main Germany E-mailProgram in Applied and Computational Mathematics Princeton University Fine HallA. Jentzen Program in Applied and Computational Mathematics Princeton University Fine Hall, Washington Road Princeton, New Jersey 08544-1000 USA E-mail: [email protected] P. Kloeden G. Winkel Institute of Mathematics Johann Wolfgang Goethe University D-60054 Frankfurt am Main Germany E-mail: [email protected] [email protected]

Recent developments on financial markets have revealed the limits of Brownian motion pricing models when they are applied to actual markets. Lévy processes, that admit jumps over time, have been found more useful for applications. Thus, we suggest a Lévy model based on Forward-Backward Stochastic Differential Equations (FBSDEs) for option pricing in a Lévy-type market. We show the existence and uniqueness of a solution to FBSDEs driven by a Lévy process. This result is important from the mathematical point of view, and also, provides a much more realistic approach to option pricing.

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Forward-Backward SDEs driven by Lévy Processes and Application to Option Pricing Sep 2013. June 2012 R S Pereira Centro de Matemática da Universidade do Porto Rua do Campo Alegre 6874169-007PortoPortugal E Shamarova Centro de Matemática da Universidade do Porto Rua do Campo Alegre 6874169-007PortoPortugal Forward-Backward SDEs driven by Lévy Processes and Application to Option Pricing Global and Stochastic Analysis 21Sep 2013. June 2012Received by the Editorial Board on May 2, 2012Forward-backward stochastic differential equationsFBSDEsLévy processesPartial integro-differential equationOption pricing 2010 Mathematics subject classification: 60J75 60H10 60H30 35R09 91G80 Recent developments on financial markets have revealed the limits of Brownian motion pricing models when they are applied to actual markets. Lévy processes, that admit jumps over time, have been found more useful for applications. Thus, we suggest a Lévy model based on Forward-Backward Stochastic Differential Equations (FBSDEs) for option pricing in a Lévy-type market. We show the existence and uniqueness of a solution to FBSDEs driven by a Lévy process. This result is important from the mathematical point of view, and also, provides a much more realistic approach to option pricing. Introduction Since the seminal contribution made by F. Black and M. Scholes [2], several methodologies to value contingent assets have been developed. From Plain Vanilla options to complex instruments such as Collateral Debt Obligations or Baskets of Credit Default Swaps, there are models to price virtually any type of contingent asset. There are, indeed, successful attempts on providing general models which in theory could price any kind of contingent claim (see, for example [3]), given a payoff function. The idea behind these models is quite standard: A portfolio replicating the payoff function of the asset is devised and, under non-arbitrage conditions, the price of the asset at a certain instant of time is the price of this portfolio at that time. However, in spite of all this diversity and sophistication, there is an assumption that is, up to recent times, rarely questioned. Specifically, we refer to the assumption that stock prices are continuous diffusion processes, presupposing thereby that the returns have normal distributions at any time. However, it is well known today that empirical distributions of stock prices returns tend to deviate from normal distributions, either due to skewness, kurtosis or even the existence of discontinuities (Eberlein et al. give evidence of this phenomenae in [4]). The recent developments have shown that the reliance on normal distribution can bring costly surprises, especially when extreme and disruptive events occur with a much higher frequency than the one estimated by models. As such, we believe that no matter which historical status the normal distribution has acquired throughout the years, strong efforts should be undertaken in order to develop alternative models that incorporate assumptions adequated to the observed evidence on financial markets, such as asymmetry or skewness. We do not pretend that some definitive model can actually be developed, especially when market participants' main activities are currently shifting due to the conditions imposed on financial markets. Indeed, the recently introduced regulations on financial markets severely restraining the use of own's capital for trading purposes will force the financial players to find new ways of driving a profit. This adds another layer of uncertainty about the assumptions imposed on a model. We believe, however, that in spite of the inherent inability to prove that any present model can account for future market conditions, it is worth to attempt to correctly price financial claims in the present and near future market conditions, which, as it is clear now, is fundamental to the stability of markets. Taking the above considerations into account, we propose to replicate contingent claims in Lévy-type markets, i.e. in markets with the stock-price dynamics described as S t = S 0 e Xt , where X t is a Lévy-type stochastic integral defined in [1]. This allows the relaxation of conditions posed on the pricing process such as symmetry, non-skewness or continuity, imposed by the Brownian framework. The self-similiarity of the pricing process, appearing due to a Brownian motion, is also ruled out from the assumptions. We base our model on the study of Forward-Backward Stochastic Differential Equations (FBSDEs) driven by a Lévy process. FBSDEs combine equations with the ini-tial and final conditions which allows one to search for a replicating portfolio. Specifically, we are concerned with the following fully coupled FBSDEs: X t = x + t 0 f (s, X s , Y s , Z s ) ds + ∞ i=1 t 0 σ i (s, X s− , Y s− ) dH (i) s , Y t = h(X T ) + T t g(s, X s , Y s , Z s ) ds − ∞ i=1 T t Z i s dH (i) s ,(1.1) where the stochastic integrals are written with respect to the orthogonalized Teugels martingales {H (i) t } ∞ i=1 associated with a Lévy process L t [10]. We are searching for an R P ×R Q ×(R Q ×ℓ 2 )-valued solution (X t , Y t , Z t ) on an arbitrary time interval [0, T ], which is square-integrable and adapted with respect to the filtration F t generated by L t . To the authors' knowledge, fully coupled FBSDEs of this type have not been studied before. Fully decoupled FBSDEs involving Lévy processes as drivers were studied by Otmani [6]. Backward SDEs driven by Teugels martingales were studied by Nualart and Schoutens [9]. Our method of solution to the FBSDEs could be compared to the Four Step Scheme [7]. The original four step scheme deals with FBSDEs driven by a Brownian motion, and the solution is obtained via the solution to a quasilinear PDE. Replacing the stochastic integral with respect to a Brownian motion with a stochastic integral with respect to the orthogonalized Teugels martingales leads to a partial integro-differential equation (PIDE). The solution to the PIDE is then used to obtain the solution to the FBSDEs. The organization of the paper is as follows. In Section 2, we give some preliminaires on the martingales {H (i) t }. In Section 3, under certain assumptions, we obtain the existence and uniqueness result for the associated PIDE. Our main result is Theorem 3.7, where we obtain a solution to FBSDEs (1.1) via the solution to the PIDE and prove its uniqueness. In section 4, we apply the results of Section 3 to model hedging options for a large investor in a Lévytype market. Previously, this problem was studied by Cvitanic and Ma [3] for a Brownian market model. Finally, we study conditions for the existence of replicating portfolios. Preliminaires Let (Ω, F , F t , P ) be a filtered probability space, where {F t }, t ∈ [0, T ], is the filtration generated by a real-valued Lévy process L t . Note that the Lévy measure ν of L t always satisfies the condition R (1 ∧ x 2 ) ν(dx) < ∞. We make the filtration F t P -augmented, i.e. we add all P -null sets of F to each F t . Following Nualart and Schoutens [8] we introduce the orthogonalized Teugels martingales {H (i) t } ∞ i=1 associated with L t . For this we assume that for every ε > 0, there exists a λ > 0 so that (−ε,ε) c exp(λ|x|) ν(dx) < ∞. The latter assumption guaranties that R |x| i ν(dx) < ∞ for i = 2, 3, . . . . It was shown in [10] that under the above assumptions one can introduce the power jump processes and the related Teugels maringales. Futhermore, it was shown that the strong orthogonalization procedure can be applied to the Teugels martinagles and that the orthonormalization of the Teugels martingales corresponds to the orthonormalization of the polynomials 1, x, x 2 , . . . with respect to the measure x 2 ν(dx) + a 2 δ 0 (dx), where the parameter a ∈ R is defined in Lemma 2.1 below. As in [9], by {q i (x)} we denote the system of orthonormalized polynomials such that q i−1 (x) corresponds to H (i) t . Also, we define the polynomial p i (x) = xq i−1 (x). We refer to [10] for details on the Teugels martingales and their orthogonalization procedure. In the following, Lemma 2.1 below will be usefull. H (i) t = q i−1 (0)B λ (t) + R p i (x)Ñ (t, dx), where B λ (t) = N i=1 λ i B i (t) with λ T λ = a, λ i ∈ R, {B i (t)} N i=1 are independent real-valued Brownian motions, andÑ (t, A) is the compensated Poisson random measure that corresponds to the Poisson point process ∆L t . Proof. We will use the representation below for H (i) t obtained in [9]: H (i) t = q i−1 (0)L t + 0<s tp i (∆L s ) − tE 0<s 1p (∆L s ) − tq i−1 (0)E[L 1 ], wherep i (x) = p i (x) − xq i−1 (0) , and E is the expectation with respect to P . Taking into account that L t = L c t + 0 s t ∆L s , where L c t is the continuous part of L t , we obtain: H (i) t = q i−1 (0)L c t + 0<s t p i (∆L s ) − tE 0<s 1p i (∆L s ) − tq i−1 (0)E[L 1 ] = q i−1 (0) L c t − E[L c t ] + 0<s t p i (∆L s ) − E 0<s t p i (∆L s ) = q i−1 (0)B λ (t) + R p i (x)Ñ (t, dx). In the sequence, the following lemma will be frequently applied: Lemma 2.2. It holds that R p i (x)p j (x)ν(dx) = δ ij − a 2 q i−1 (0)q j−1 (0). Proof. The proof is a straightforward corollary of the orthonormality of q i−1 (x) with respect to the measure x 2 ν(dx) + a 2 δ 0 (dx). We will need an analog of Lemma 5 from [9] which was proved in the latter article for a pure-jump L t . We obtain this result for the case when L t has both the continuous and the pure-jump parts. Lemma 2.3. Let h : Ω × [0, T ] × R → R n be a random function satisfying E T 0 |h(s, y)| 2 ν(dy) < ∞. (2.1) Then, for each t ∈ [0, T ], t<s T h(s,∆L s ) = ∞ i=1 T t R ν(dy)h(s, y)p i (x) dH (i) s + T t R h(s, y)ν(dy)ds. Proof. Note that M t = 0 s t h(s, ∆L s ) − t 0 R h(s, y)ν(dy)ds = t 0 R h(s, x)Ñ (ds, dx) (2.2) is a square integrable martingal, i.e. sup t∈[0,T ] E|M t | 2 < ∞, by (2.1). By the predictable representation theorem [8], there exist predictable processes ϕ i with E T 0 ∞ i=1 |ϕ i | 2 < ∞ and such that M t = ∞ i=1 t 0 ϕ i (s)dH (i) s . Since H (i) , H (j) t = t δ ij [10], then M, H (i) t = t 0 ϕ i (s)ds. On the other hand, by (2.2) and Lemma 2.1, M, H (i) t = t 0 R h(s, x)Ñ(ds, dx), q i−1 (0)B λ t + t 0 R p i (x)Ñ (dt, dx) t = t 0 R h(s, x)p i (x)ν(dx)ds. This implies that ϕ i (s) = R h(s, y)p i (y)ν(dy), and therefore, 0 s t h(s, ∆L s ) − t 0 R h(s, y)ν(dy)ds = ∞ i=1 t 0 R h(s, y)p i (y)ν(dy)dH (i) s . FBSDEs and the associated PIDE Problem Formulation and Assumptions Consider the FBSDEs:      X t = x + t 0 f (s, X s , Y s , Z s ) ds + t 0 σ(s, X s− , Y s− ) dH s , Y t = h(X T ) + T t g(s, X s , Y s , Z s ) ds − T t Z s dH s , t ∈ [0, T ], (3.1) where f : [0, T ] × R P × R Q × (R Q × ℓ 2 ) → R P , σ : [0, T ] × R P × R Q → R P × ℓ 2 , g : [0, T ] × R P × R Q × (R Q × ℓ 2 ) → R Q , h : R P → R Q are Borel-measurable functions. Here, the stochastic integrals t 0 σ(s, X s− , Y s− ) dH s and T t Z s dH s are shorthand notation for ∞ i=1 t 0 σ i (s, X s− , Y s− ) dH (i) s and ∞ i=1 T t Z i s dH (i) s respectively, where Z s = {Z i s } ∞ i=1 , σ = {σ i } ∞ i=1 , σ i : [0, T ]×R P ×R Q → R P . The solution to FBSDEs (3.1), when exists, will be an R P × R Q × (R Q × ℓ 2 )-valued F t -adapted triple (X t , Y t , Z t ) satisfying E T 0 |X t | 2 + |Y t | 2 + ∞ i=1 |Z i t | 2 dt < ∞, and verifying (3.1) P -a.s.. The latter includes the existence of the stochastic integrals in (3.1). Implicitly, we are assuming that X t and Y t have left limits, and that Z t is F t -predictable. So in fact, we are searching for càdlàg (X t , Y t ), which will guarantee the existence of X t− and Y t− , and predictable Z t . We associate to (3.1) the following final value problem for a PIDE:        ∂ t θ(t, x) + f k (t, x, θ(t, x), θ (1) (t, x)) ∂ k θ(t, x) + β kl (t, x, θ(t, x)) ∂ 2 kl θ(t, x) − R θ t, x + δ(t, x, θ(t, x), y) − θ(t, x) − ∂ k θ(t, x)δ k (t, x, θ(t, x), y) ν(dy) +g(t, x, θ(t, x), θ (1) (t, x)) = 0, θ(T, x) = h(x) (3.2) with θ (1) : [0, T ] × R P → R Q × ℓ 2 , θ (1) i (t, x) = R [θ t, x + δ(t, x, θ(t, x), y) − θ(t, x)]p i (y) ν(dy) + c k i (t, x, θ(t, x)) ∂ k θ(t, x). (3.3) The connection between β kl , δ, c k i and the coefficients of FBSDEs (3.1) is the following: δ(t, x, y, y ′ ) = ∞ i=1 σ i (t, x, y)p i (y ′ ), (3.4) β kl (t, x, y) = a 2 2 ∞ i=1 σ k i (t, x, y)q i−1 (0) ∞ j=1 σ l j (t, x, y)q j−1 (0) , (3.5) c k i (t, x, y) = σ k i (t, x, y) − R δ k (t, x, y, y ′ )p i (y ′ ) ν(dy ′ ). (3.6) To guarantee the existence of the above functions we will make the assumption: A0 ∞ i=1 q i−1 (0) 2 < ∞. Since σ k = {σ k i } ∞ i=1 takes values in ℓ 2 , A0 immediately guarantees the convergence of the both multipliers in (3.5). The convergence of the series in (3.4) is understood in L 2 (ν(dy ′ )) for each fixed (t, x, y). Moreover, it holds that R ∞ i=1 σ i (t, x, y)p i (y ′ ) 2 ν(dy ′ ) = σ(t, x, y) 2 R P ×ℓ 2 − a 2 ∞ i=1 σ i (t, x, y) q i−1 (0) 2 . (3.7) Indeed, applying Lemma 2.2 for each fixed N, we obtain: R N i=1 σ i (t, x, y)p i (y ′ ) 2 ν(dy ′ ) = N i,j=1 (σ i , σ j ) R p i (y ′ )p j (y ′ )ν(dy ′ ) = N i=1 |σ i | 2 − a 2 N i=1 σ i q i−1 (0) 2 . Now letting N tend to infinity, we obtain (3.7). Lemma 3.1. The following assertions hold: 1. c k i (t, x, y) = a 2 q i−1 (0) ∞ j=1 σ k j (s, x, y)q j−1 (0). 2. For each k, c k = {c k i } ∞ i=1 takes values in ℓ 2 . 3. For each (s, x, y), R δ(s, x, y, y ′ )p i (y ′ ) ν(dy ′ ) ∞ i=1 ∈ ℓ 2 . Proof. Define δ N (s, x, y, y ′ ) = N j=1 σ j (s, x, y)p j (y ′ ). By what was proved, for each (s, x, y), δ N (s, x, y, · ) → δ(s, x, y, · ) in L 2 (ν(dy ′ )), and therefore, for each i, and for each (s, x, y), R δ N (s, x, y, y ′ )p i (y ′ )ν(dy ′ ) → R δ(s, x, y, y ′ )p i (y ′ )ν(dy ′ ) as N → ∞. On the other hand, by Lemma 2.2, R δ N (s, x, y, y ′ )p i (y ′ )ν(dy ′ ) = σ i (s, x, y) − a 2 q i−1 (0) N j=1 σ j (s, x, y)q j−1 (0). Comparing the last two relations, we obtain that R δ(s, x, y, y ′ )p i (y ′ )ν(dy ′ ) = σ i (s, x, y) − a 2 q i−1 (0) ∞ j=1 σ j (s, x, y)q j−1 (0) (3.8) which proves Assertion 1. Assertion 2 is implied by Assumption A0 and Assertion 1. Finally, (3.6) implies Assertion 3. The heuristic argument behind PIDE (3.2) assumes the connection Y t = θ(t, X t ) between the solution processes X t and Y t to (3.1) via a C 1,2 -function θ. Itô's formula applied to θ(t, X t ) at points t and T leads to another BSDE which has to be the same as the given BSDE in (3.1). Thus we "guess" PIDE (3.2) by equating the drift and stochastic terms of these two BSDEs. Solvability of the PIDE We solve Problem (3.2) for a particular case when the functions f (t, x, y, z) and g(t, x, y, z) do not depend on z, and for a short time duration T . Thus, we are dealing with the following final value problem for a PIDE: ∂ t θ(t, x) = −[A(t, θ(t, ·))θ](x) + g(t, x, θ(t, x)), θ(T, x) = h(x), (3.9) where A(t, ρ(t, ·)) is a partial integro-differential operator given by [A(t, ρ(t, ·))θ](x) = f k (t, x, ρ(t, x)) ∂ k θ(t, x) + β kl (t, x, ρ(t, x)) ∂ 2 kl θ(t, x) + R θ t, x + δ(t, x, ρ(t, x), y) − θ(s, x) − ∂ k θ(t, x)δ k (t, x, ρ(t, x), y) ν(dy). (3.10) with the domain D(A(t, ρ(t, ·))) = C 2 b (R P → R Q ), the space of bounded continuous functions R P → R Q whose first and second order derivatives are also bounded. We assume the following: A1 Functions f , g, σ, and h are bounded and have bounded spatial derivatives of the first and the second order. Lemma 3.2. Let A0 and A1 be fulfilled. Then A(t, ρ(t, ·)), defined by (3.10), is a generator of a strongly continuous semigroup on C b (R P → R Q ). Proof. Note that by Assertion 1 of Lemma 3.1 and by (3.5), functions c k and β kl are bounded and Lipschitz in the spatial variables. This implies that the SDE dX k s = f k (t, X s , ρ(t, X s )) + ∞ i=1 σ k i (t, X s− , ρ(t, X s− ))dH (i) s ,(3.11) with X t = x, has a pathwise unique càdlàg adapted solution on [t, T ]. The existence and uniqueness of a solution to an SDE of type (3.11) will be proved in Paragraph 3.3. Now application of Itô's formula to ϕ(X s ), where ϕ is twice continuously differetiable, shows that the operator (3.10) is the generator of the solution to SDE (3.11), and therefore, it generates a strongly continuous semigroup on C b (R P → R Q ). The common method to deal with problems of type (3.9) is to fix a C 1,2 b -function ρ(t, x), and consider the following non-autonomous evolution equation: ∂ t θ(t, x) = −[A(t, ρ(t, ·))θ](x) − g(t, x, ρ(t, x)), θ(T, x) = h(x). (3.12) By Assumption A1 and the results of [11] and [5], there exists a backward propagator U(s, t, ρ), 0 s t T , so that θ(t, x) = [U(t, T, ρ)h](x) + T t [U(t, s, ρ)g(s, · , ρ(s, · ))](x) ds. We organize the map Φ : C b ([0, T ] × R P → R Q ) → C b ([0, T ] × R P → R Q ), ρ → θ,(3.13) and prove the existence of a fixed point. Define E = C b (R P → R Q ) and D = C 2 b (R P → R Q ).U(t, s, ρ)ϕ − U(t, s, ρ ′ )ϕ E K T sup s∈[t,T ] ρ(s, x) − ρ ′ (s, x) E ϕ D . Proof. We have: U(t, s, ρ ′ ) − U(t, s, ρ) ϕ = U(t, r, ρ ′ )U(r, s, ρ)ϕ| s r=t = s t dr U(t, r, ρ ′ ) A(r, ρ ′ (r, · )) − A(r, ρ(r, · )) U(r, s, ρ)ϕ. This implies: sup s∈[t,T ] U(t, s, ρ ′ )ϕ − U(t, s, ρ)ϕ E T sup s∈[t,T ] U(t, s, ρ ′ ) L(E) × sup r,s∈[t,T ], r s U(r, s, ρ) L(D) sup s∈[t,T ] A(s, ρ(s, · )) − A(s, ρ ′ (s, · )) L(D,E) ϕ D . (3.14) Taking into account that θ(s, · ) D = sup x∈R P |θ(s, x)| + sup x∈R P |∇θ(s, x)| + sup x∈R P |∇∇θ(s, x)|, and applying (3.5), (3.10), and Lemma 3.1, we obtain that there exists a constantK > 0 which does not depend on s, ρ, and ρ ′ , so that sup θ D 1 sup x∈R P A(s, ρ(s, x))θ − A(s, ρ ′ (s, x))θ L(D,E) K sup x∈R P |f (s, x, ρ(s, x)) − f (s, x, ρ ′ (s, x))| + σ(s, x, ρ(s, x)) − σ(s, x, ρ ′ (s, x)) R P ×ℓ 2 + sup x ′ ∈R P |∇∇θ(t, x ′ )| R |δ(s, x, ρ(t, x), y) − δ(s, x, ρ ′ (t, x), y)| 2 ν(dy) 1 2 . (3.15) By (3.7), the last summand in (3.15) is smaller than σ(s, x, ρ(s, x)) − σ(s, x, ρ ′ (s, x)) R P ×ℓ 2 up to a multiplicative constant. Therefore, modifying the constantK, if necessary, by Assumption A1, we obtain that sup θ D 1 A(s, ρ(s, · ))θ − A(s, ρ ′ (s, · ))θ E K ρ(s, · ) − ρ ′ (s, · ) E whereK does not depend on s, ρ, and ρ ′ . Now by (3.14), there exists a constant K > 0, so that sup s∈[t,T ] U(t, s, ρ)ϕ − U(t, s, ρ ′ )ϕ E K T sup s∈[t,T ] ρ(s, · ) − ρ ′ (s, · ) E ϕ D . Let us show that K does not depend on t, s, ρ, and ρ ′ . By Itô's formula, for s ∈ [t, T ] and for ϕ ∈ D, [U(t, s, ρ)ϕ](x) = E[ϕ(X s )|X t = x],(3.16) where X s is the solution to dX k s = f k (s, X s , ρ(s, X s )) + ∞ i=1 σ k i (s, X s− , ρ(s, X s− ))dH (i) s . Moreover, by the results of [11] (p. 102), U(t, s, ρ) maps D into D, and (3.16) implies that U(t, s, ρ) ∈ L(D) so that the norm U(t, s, ρ) L(D) is bounded uniformly in ρ. Next, since for each ϕ ∈ D, U(t, s, ρ)ϕ ∈ D is continuous in t and s, then it is bounded uniformly in t and s. Therefore, U(t, s, ρ) L(D) is bounded uniformly in t, s, and ρ. This implies the statement of the lemma. Proof. Consider the equation: in the space D. Since U(t, s, θ) L(D) is bounded, and g(s, x, y) is Lipschitz in y whose Lipschitz constant does not depend on s and x, the fixed point argument implies the existence of a unique solutionθ ∈ D to (3.18). Clearly, θ is also a unique solution to (3.18) in E. Henceθ = θ, and therefore, θ ∈ D. This implies that θ is the unique solution to Problem (3.9). θ(t, x) = [U(t, T, θ)h](x) + Existence and Uniqueness Theorem for the FBSDEs In Paragraph 3.2 we found some conditions under which there exists a unique solution to PIDE (3.2). However, this solution may exist under more general assumptions. Thus, we prove the existence and uniqueness of a solution to FBSDEs (3.1) assuming the existence and uniqueness of a solution to PIDE (3.2). Specifically, we will assume the following: A2 Functions f , g, and σ possess bounded first order derivatives in all spatial variables. A3 Assumption A0 is fulfilled and Final value problem (3.2) has a unique solution θ which belongs to the class C 1,2 b ([0, T ] × R P → R Q ). A4 There exists a constant K > 0 which does not depend on (t, x, y, z), such that ∞ i=1 ∂ ∂z i f (t, x, y, {z i } ∞ i=1 ) R |p i (y)| 2 ν(dy) 1 2 < K.k = {c k i } ∞ i=1 is Lipschitz in two spatial variables as an ℓ 2 -valued function. By A3, θ and ∂ k θ are Lipschitz. Therefore, the last summand in (3.3) is Lipschitz in x, and moreover, its Lipschitz constant does not depend on t by boundedness of the both multipliers. Let us prove that the map R P → R Q , x → R θ(t,x + δ(t, x, ρ(t, x), y)) p i (y)ν(dy) (3.19) is Lipschitz, wherex and t are fixed. Let x 1 , x 2 ∈ R P , and let ρ 1 = ρ(t, x 1 ) and ρ 2 = ρ(t, x 2 ), where t is fixed. We have: R [θ(t,x + δ(t, x 1 , ρ 1 , y) − θ(t,x + δ(t, x 2 , ρ 2 , y)]p i (y)ν(dy) max x∈R P |∇θ(t, x)| R |δ(t, x 1 , ρ 1 , y) − δ(t, x 2 , ρ 2 , y)| |p i (y)|ν(dy) max x∈R P |∇θ(t, x)| R |δ(t, x 1 , ρ 1 , y) − δ(t, x 2 , ρ 2 , y)| 2 ν(dy) R |p i (y)| 2 ν(dy) 1 2 K max x∈R P |∇θ(t, x)| R |p i (y)| 2 ν(dy) 1 2 σ(t, x 1 , ρ 1 ) − σ(t, x 2 , ρ 2 ) R P ×ℓ 2 . (3.20) Now the Lipschitzness of map (3.19) and the boundedness of the gradient of θ imply that the map Φ : R P → R Q × ℓ 2 , x → R θ(t, x + δ(t, x, θ(t, x), y)) − θ(t, x))p i (y) ν(dy) is also Lipschitz. Argument (3.20) implies that the Lipschitz constant of Φ has the formK R |p i (y)| 2 ν(dy) dX t = f (s, X s , θ(s, X s ), θ (1) (s, X s− ))ds + ∞ i=1 σ i (s, X s− , θ(s, X s− ))dH (i) s , X 0 = x, (3.21) where θ is the solution to (3.2) and θ (1) is defined by (3.3), has a pathwise unique càdlàg adapted solution. Proof. We will show that Ψ(X) t = x + t 0 f (s, X s , θ(s, X s ), θ (1) (s, X s− ))ds + t 0 σ(s, X s− , θ(s, X s− ))dH s is a contraction map in the Banach space S with the norm Φ 2 S = E sup t∈[0,T ] |Φ t | 2 . Take two points X s and X ′ s from S. For simplicity of notation, let σ s = σ(s, X s− , θ(s, X s− )) and σ ′ s = σ(s, X ′ s− , θ(s, X ′ s− )). To estimate the difference of the stochastic integrals with the integrands σ s and σ ′ s with respect to the · S -norm, we apply the Burkholder-Davis-Gundy inequality to the martingale t 0 (σ s − σ ′ s )dH s . We obtain that there exists a constant C > 0 such that E sup r∈[0,t] r 0 (σ s − σ ′ s )dH s 2 CE • 0 (σ s − σ ′ s )dH s t = CE • 0 (σ s − σ ′ s )dH s t + U t = CE ∞ i,j=1 t 0 (σ i − σ ′ i , σ j − σ ′ j )d H i , H j s = CE t 0 σ s − σ ′ s 2 R P ×ℓ 2 ds where [ · ] t and · t are the quadratic variation and the predictable quadratic variation, respectively. Moreover, we applied the identity H i , H j s = δ ij s and the decomposition [M] t = M t + U t for the quadratic variation of a square integrable martingale (i.e. a martingale M t with sup t |M t | 2 < ∞) into the sum of the predictable quadratic variation and a uniformly integrable martingale U t starting at zero. Next, we note that the functions x → f (s, x, θ(t, x), θ 1 (t, x)) and x → σ(t, x, θ(t, x)) are Lipschitz whose Lipschitz constants do not depend on t. This and the above stochastic integral estimate imply that there exist a constant K > 0 such that E sup s∈[0,t] |Ψ(X) s −Ψ(X ′ ) s | 2 KE t 0 |X s −X ′ s | 2 ds KE t 0 sup r∈[0,s] |X r −X ′ r | 2 ds. Iterating this n − 1 times we obtain: E sup s∈[0,t] |Ψ n (X) s − Ψ n (X) ′ s | K n t n n! E sup s∈[0,t] |X s − X ′ s | 2 . Choosing n sufficienty large so that K n T n n! < 1, we obtain that Ψ n is a contraction, and thus, Ψ is a contraction as well. By the Banach fixed point theorem, the map Ψ has a unique fixed point in the space S. Clearly, this fixed point is a unique solution to (3.21). Setting X (0) = x, and then, sucessively, X (n) = Ψ(X (n−1) ), we can choose càdlàg modifications for each X (n) . Since the X (n) 's converge to the solution X in the norm of S, X will be also càdlàg a.s.. This càdlàg solution is unique in the space S, and therefore, pathwise unique. Introduce the space S of F t -predictable R Q × ℓ 2 -valued stochastic processes with the norm Φ 2 S = E T 0 Φ s 2 R Q ×ℓ 2 ds. Now we formulate our main result. Theorem 3.7. Suppose A2, A3, and A4 hold. Let X t be the càdlàg adapted solution to (3.21). Then, the triple (X t , Y t , Z t ), where Y t = θ(t, X t ), Z t = θ (1) (t, X t− ) with θ (1) given by (3.3) , is a solution to FBSDEs (3.1). Moreover, the pair of càdlàg solution processes (X t , Y t ) is pathwise unique. The solution process Z t is unique in the space S. Proof. It suffices to prove that the triple (X t , Y t , Z t ) defined in the statement of the theorem verifies the BSDE in (3.1). Application of Itô's formula to θ(t, X t ) gives: θ(T, X t ) − θ(t, X t ) = T t ∂ s θ(s, X s− )ds + T t ∂ k θ(s, X s− )dX k s + 1 2 T t ∂ 2 kl θ(s, X s− )d[(X c ) k , (X c ) l ] s + t<s T [θ(s, X s ) − θ(s, X s− ) − ∆X k s ∂ k θ(s, X s− )], (3.22) where X c s is the continuous part of X s . Using the representation for H (i) s from Lemma 2.1 we obtain that d[(X c ) k , (X c ) l ] s = 2β kl (s, X s , θ(s, X s ))ds, where β kl is given by (3.5). The forward SDE in (3.1), the relation ∆H [9], and representation (3.4) for the function δ imply: Note that for each fixed s ∈ [0, T ] and ω ∈ Ω, the function h satisfies condition (2.1). Indeed, the mean value theorem, e.g. in the integral form, can be applied to the difference of the first two terms in (3.24). By boundedness of the partial derivatives ∂ k θ, it suffices to verify that (i) s = p i (∆L s ), obtained in∆X s = ∞ i=1 σ i (s, X s− , Y s− )∆H (i) s = δ(s, X s− , Y s− , ∆L s ).E T 0 |δ(s, X s− , θ(s, X s− ), y)| 2 ds ν(dy) < ∞. The latter holds by Assumption A0 and formula (3.7). Now Lemma 2.3 implies: t<s T h(s, ∆L s ) = ∞ i=1 T t R θ(s, X s− + δ(s, X s− , θ(s, X s− ), y)) − θ(s, X s− ) − δ k (s, X s− , θ(s, X s− ), y) ∂ k θ(s, X s− ) p i (y)ν(dy)dH (i) s + T t R θ(s, X s− + δ(s, X s− , θ(s, X s− ), y)) − θ(s, X s− ) − δ k (s, X s− , θ(s, X s− ), y) ∂ k θ(s, X s− ) ν(dy) ds. Substituting this into (3.22), replacing dX k s with the right-hand side of (3.21), and taking into acccount that Y t = θ(t, X t ) and that θ(T, X T ) = h(X T ) by (3.2), we obtain: Y t = h(X T ) − T t ∂ s θ(s, X s− ) + ∂ k θ(s, X s− )f k (s, X s− , θ(s, X s− ), θ (1) (s, X s− )) + 1 2 ∂ kl θ(s, X s− )β kl (s, X s− , θ(s, X s− )) + R θ(s, X s− + δ(s, X s− , θ(s, X s− ), y)) − θ(s, X s− ) − δ k (s, X s− , θ(s, X s− ), y) ∂ k θ(s, X s− ) ν(dy) ds − T t ∞ i=1 R θ(s, X s− + δ(s, X s− , θ(s, X s− ), y)) − θ(s, X s− ) − ∂ k θ(s, X s− )c k i (X s− , y) p i (y)ν(dy) dH (i) s . Clearly, in the first three summands under the ds-integral sign one can equivalently write X s or X s− . This is true since X s has càdlàg paths, and therefore, X s and X s− can differ only at a countable number of points. Now taking into account PIDE (3.2), we note that the integrand in the drift term is −g(s, X s− , θ(s, X s− ), θ (1) (s, X s− )) which is −g(s, X s− , Y s− , Z s ) by the definitions of Y s and Z s , or, it can be replaced by −g(s, X s , Y s , Z s ) since X s and Y s have càdlàg paths. Finally, by (3.3) and the definition of Z s , the integrand in the stochastic term is Z s . Consequently, Y t = h(X T ) + T t g(s, X s , Y s , Z s ) − T t Z s dH s , which implies that (X s , Y s , Z s ) is a solution. Let us prove the uniqueness. Let (X s , Y s , Z s ) be an arbitrary solution to (3.1). LetỸ s = θ(s, X s ), andZ s = θ (1) (s, X s− ), where θ is the solution to (3.2), and θ (1) is defined by (3.3). By the above argument, (X s ,Ỹ s ,Z s ) verifies the BSDE in (3.1). Applying Itô's product formula to |Ỹ t − Y t | 2 and taking into consideration thatỸ T = Y T , we obtain: |Ỹ t − Y t | 2 = −2 T t Ỹ s− − Y s− , d(Ỹ s − Y s ) + [Ỹ − Y ] t − [Ỹ − Y ] T . Taking the expectations in the above relation gives: E|Ỹ s − Y s | 2 + E T t Z s − Z s 2 R Q ×ℓ 2 ds = 2E T t Ỹ s − Y s , g(s, X s ,Ỹ s ,Z s ) − g(s, X s , Y s , Z s ) ds. By A2, there exists a constant C > 0 such that E|Ỹ t − Y t | 2 + E T t Z s − Z s 2 R Q ×ℓ 2 ds CE T t |Ỹ s − Y s | |Ỹ s − Y s | + Z s − Z s R Q ×ℓ 2 ds. Now using the standard estimates and applying Gronwall's inequality, we ob- tain that E|Ỹ t − Y t | 2 + cE T t Z s − Z s 2 R Q ×ℓ 2 ds = 0 for some constant c > 0. The latter relation holds for all t ∈ [0, T ]. This proves thatỸ t is a modification of Y t and that Z − Z S = 0. This implies the uniqueness result. Option Pricing with a Large Investor in Lévy-type Markets Usually, when modeling financial assets it is assumed that all investors are price takers whose individual buy and sell decisions do not influence the price of assets. Cvitanic and Ma [3] have already developed a model for hedging options in the presence of a large investor in a Brownian market. However, observation of real data suggests that patterns, like skewness, kurtosis, or the occurence of jumps are sufficiently significant (see, e.g., Eberlein and Keller [4]) to deserve to be accounted in a realistic model of option pricing. Furthermore, the graphs of the evolution of stock prices at different time-scales are sufficiently different from the self-similarity of a Brownian motion. Thus, we develop a Lévy-FBSDE option pricing model. We believe that such a model conveys a much more realistic approach to option pricing in the presence of the already mentioned empirical market characteristics. We assume the existence of a Large investor, whose wealth and strategy may induce distortions of the price process. Let M be a Lévy-type Market, i.e. a market whose stock price dynamics S t obeys the equation S t = S 0 e Xt , where X t is a Lévy-type stochastic integral [1]. The market consists of d risky assets and a money market account. For the price process P 0 (t) of the money market account, we assume that its evolution is given by the following equation dP 0 (t) = P 0 (t) r(t, W (t), Z(t)) dt, 0 t T, P 0 (0) = 1, where W (t) is the wealth process, and Z(t) is a portfolio-related process in a way that will be explained later. For the risky assets, we add the stochastic component represented by the volatility matrix σ taking values in R d × ℓ 2 . We postulate that the evolution of the d-dimensional risky asset price process P (t) = {P i (t)} d i=1 is given by the following SDE: dP i (t) = f i (t, P (t), W (t), Z(t)) dt + ∞ j=1 σ i j (t, P (t), W (t)) dH We derive the BSDE for the wealth process as in [3]. For the convinience of the reader we repeat this derivation: dW (t) = d i=1 α i (t) dP i (t) + W (t) − d i=1 α i (t)P i (t) P 0 (t) dP 0 (t), where α i (t) is the portfolio process. Substituting dP i (t) with the right-hand sides of (4.1), we obtain: dW (t) = d i=1 α i (t) f i (t, P (t), W (t), Z(t))dt + ∞ j=1 σ i j (t, P (t), W (t)) dH (j) (t) + (W (t) − d i=1 α i P i (t)) r(t, W (t), Z(t)) dt = g(t, P (t), W (t), Z(t), α(t))dt + ∞ i=1 Z i (t)dH (i) (t), (4.2) where g(t, π, w, z, a) = d i=1 a i f i (t, π, w, z) + (w − d i=1 a i π i ) r(t, w, z), a = {a i } d i=1 , π = {π i } d i=1 ; Z i (t) = d j=1 α j (t)σ j i (t, P (t), W (t)), i = 1, 2, . . . . As we are assuming the absence of risk for the money market account, the evolution of its price depends totaly on the interest rate the investor is earning. On the other hand, to describe the evolution of the risky assets we use an SDE with the stochastic term given as a sum of stochastic integrals with respect to H (j) 's. This adds explanative power to the model, as it affords the isolation of the individual contributions of each H (j) . Now, to guarantee that the stock price has positive components we will rewrite (4.1) for Q i (t) = log P i (t) using Itô's formula. For simplicity of notation, we will use the same symbols f , g, σ, and h for the coefficients of the FBSDEs which we obtain after rewriting SDE (4.1) with respect to Q(t) = {Q i (t)} d i=1 and substituting P i (t) = exp{Q i (t)}: (4.4-4.5) has a unique solution (Q(t), W (t), Z(t)) such that the pair (Q(t), W (t)) is càdlàg. Furthermore, if for some R d -valued stochastic process {α j (t)} d j=1 , α j (t) 0, relation (4.3) holds in the space S, then {α j (t)} d j=1 is a replicating portfolio. Q(t) = q + It is evident that Lévy-type markets pose theoretical questions that had never been raised in the Brownian motion framework. We conclude by reenforcing the idea that the impossibility of replicating every potential contingent claim is rather an expected characteristic due to the complexity of the price formation in Lévy-type markets, than a drawback of our model. Lemma 2 . 1 . 21The process H (i) t can be represented as follows: Lemma 3.3. Let Assumptions A0 and A1 hold. Then, there exists a constant K > 0 that does not depend on s, t, ρ, and ρ ′ , so that for any function ϕ ∈ D, sup s∈[t,T ] Theorem 3. 4 . 4Let Assumptions A0 and A1 hold. Then, there exists a T 0 > 0 so that for all T ∈ (0, T 0 ], Problem (3.9) has a unique solution on [0, T ]. t, s, θ) g(s, · , θ(s, · ))](x) ds.(3.17) The proof of the existence and uniqueness of a solution to (3.17) is equivalent to the existence of a unique fixed point of map (3.13) in the space E. For a sufficiently small time interval [0, T ], the latter is implied by Assumption A1 and Lemma 3.3. Now let θ be the solution to (3.17) on [0, T ]. Consider the equation θ(t, x) = [U(t, T, θ)h](x) + t, s, θ) g(s, · ,θ(s, · ))](x) ds(3.18) Lemma 3. 5 . 5Assume A2, A3, and A4 hold. Then the functionf (t,x,ȳ, · ) • θ (1) (t, x), where θ (1) (t, x) is given by (3.3), is Lipschitz in xfor all (t,x,ȳ), and the Lipschitz constant does not depend on (t,x,ȳ).Proof. Note that by Assertions 1 and 2 of Lemma 3.1, the function c 1 2 1whereK is a constant that does not depend on i. Now A4 implies the statement of the lemma. Proposition 3. 6 . 6Assume A2, A3, and A4. Then, the SDE one can rewrite the last term in (3.22) ast<s T θ(s, X s− + δ(s, X s− , Y s− , ∆L s )) − θ(s, X s− ) − δ k (s, X s− , Y s− , ∆L s ) ∂ k θ(s, X s− ) ,where δ k is the kth component of δ. Define the random function h(s, y) = θ(s, X s− + δ(s, X s− , θ(s, X s− ), y)) − θ(s, X s− ) − δ k (s, X s− , θ(s, X s− ), y) ∂ k θ(s, X s− ).(3.24) (0) = p i , p i 0, 1 i d, t ∈ [0, T ]. t 0 f 0(s, Q(s), W (s), Z(s)) ds + t 0 σ(s, Q(s), W (s)) dH(s), (4.4) where q = {log p i } d i=1 . Due to relation (4.3), we exclude the dependence on α(t) in (4.2). BSDE (4.2) takes the form: W (t) = h(Q(T )) + T t g(s, Q(s), W (s), Z(s)) ds − T t Z(s) dH(s). (4.5) Theorem 3.7 and relation (4.3) imply the following result. Theorem 4 . 1 . 41Assume A2, A3 and A4. Then, FBSDEs Lévy Processes and Stochastic Calculus. D Applebaum, Cambridge University PressD. Applebaum, "Lévy Processes and Stochastic Calculus", Cambridge Uni- versity Press, 2009. The pricing of options and corporate liabilities. F Black, M Scholes, J. Political Economy. 81F. Black and M. Scholes, "The pricing of options and corporate liabilities", J. Political Economy, Vol. 81, pp. 637-659, 1973. Hedging options for a large investor and forwardbackward SDEs. J Cvitanic, J Ma, The Annals of Applied Probability. 62J. Cvitanic and J. Ma, "Hedging options for a large investor and forward- backward SDEs", The Annals of Applied Probability, Vol 6, No. 2, pp. 370-398, 1996. Hyperbolic distributions in finance. E Eberlein, U Keller, Bernoulli. 13E. Eberlein and U. Keller, "Hyperbolic distributions in finance", Bernoulli, Vol. 1, No. 3, pp. 281-299, 1995. Non-autonomous Kato Classes and Feynman-Kac Propagators. A Gulisashvili, I A Van Casteren, World ScientificA. Gulisashvili and I.A. van Casteren, "Non-autonomous Kato Classes and Feynman-Kac Propagators", World Scientific, 2006. Backward stochastic differential equations associated with Lévy processes and partial integro-differential equations. M Otmani, Communications on Stochastic Analysis. 2M. el Otmani, "Backward stochastic differential equations associated with Lévy processes and partial integro-differential equations", Communica- tions on Stochastic Analysis, Vol. 2, No. 2, pp. 277-288, 2008. Solving forward-backward stochastic differential equations explicitly -a Four Step Scheme. P Protter, J Ma, J Yong, Probability Theory and Related Fields. 98P. Protter, J. Ma and J. Yong, "Solving forward-backward stochastic dif- ferential equations explicitly -a Four Step Scheme", Probability Theory and Related Fields, Vol. 98, pp. 339-359, 1994. Chaotic and predictable representations for Lévy processes. D Nualart, W Schoutens, Stochastic Processes and their Applications. 90D. Nualart and W. Schoutens, "Chaotic and predictable representations for Lévy processes", Stochastic Processes and their Applications, Vol. 90, pp. 109-122, 2000. BSDEs and Feynman-Kac formula for Lévy processes with applications in finance. D Nualart, W Schoutens, Bernoulli. 7D. Nualart and W. Schoutens, "BSDEs and Feynman-Kac formula for Lévy processes with applications in finance", Bernoulli, Vol. 7, pp. 761- 776, 2001. Stochastic processes and orthogonal polynomials. W Schoutens, SpringerW. Schoutens, "Stochastic processes and orthogonal polynomials", Springer, 2000. Equations of evolution. H Tanabe, Pitman, London-San Francisco-MelbourneH. Tanabe, "Equations of evolution", Pitman, London-San Francisco- Melbourne, 1979.

In this article we present the exact representation of a fully correlated electronic wavefunction as the single-particle basis approaches completeness. It consists of a half-infinite chain of matrices of exponentially increasing size. The complete basis limit is illustrated numerically using the density matrix renormalization group method by computing the core-valence entanglement in the C2 ground state in increasing subsets of cc-pVTZ and pVQZ bases until convergence is reached.

10.1103/physrevb.105.165144

2112.10210

77dea023ade7279152c495f4a99d2a564970e3ad

Exact matrix product state representation and convergence of a fully correlated electronic wavefunction in the infinite basis limit Gero Friesecke Department of Mathematics Technical University of Munich Germany Benedikt R Graswald Department of Mathematics Technical University of Munich Germany Legeza Wigner Research Centre for Physics H-1525BudapestHungary Institute for Advanced Study Technical University of Munich Germany Exact matrix product state representation and convergence of a fully correlated electronic wavefunction in the infinite basis limit (Dated: April 7, 2022) In this article we present the exact representation of a fully correlated electronic wavefunction as the single-particle basis approaches completeness. It consists of a half-infinite chain of matrices of exponentially increasing size. The complete basis limit is illustrated numerically using the density matrix renormalization group method by computing the core-valence entanglement in the C2 ground state in increasing subsets of cc-pVTZ and pVQZ bases until convergence is reached. INTRODUCTION To achieve controllable quantum devices, understanding the intrinsic quantum entanglement hidden within the fundamental molecular building blocks of nature has been a central goal in the past decades [8,10]. Fully resolving this entanglement in theory requires not just no limitations on excitation ranks, but also -owing to the fact that a single electron can assume infinitely many quantum states -reaching the complete basis set (CBS) limit. The difficulty of this latter aim in correlated models was emphasized many decades ago by Pople [13]; due to the curse of dimensionality progress has been limited to a fixed excitation level (and global quantities like energy) [6,7,12,16]. Recent advances in methods based on matrix product state (MPS) factorization have made it possible to calculate electronic states of small molecules accurately in large basis sets compared to conventional methods [1,3,15]. However, the question of what happens to the MPS representation in the CBS limit has remained open both theoretically and numerically. Here we prove that electronic wavefunctions possess an exact MPS representation in a complete basis. It is given by a half-infinite chain of matrices and is thus loosely reminiscent of certain infinite MPS that have been used in solid state physics [9,17], but here the size of the matrices is found to increase exponentially along the chain. We also demonstrate this result numerically by computing the dicarbon ground state in increasingly large size-K subsets of cc-pVTZ and pVQZ bases (108 and 216 spin orbitals, respectively) and exhibiting the convergence of the MPS matrices as K is increased. To the best of our knowledge, our work is the first demonstration of convergence of a fully correlated wavefunction with more than two electrons in the CBS limit. The entanglement for a bipartite decomposition of a pure state is completely characterized by the Schmidt spectrum, i.e., the associated singular values. When one of the parts is finite-dimensional and the other is infinite-dimensional, the Schmidt spectrum still consists of finitely many values only, and can be numerically computed. Here we do so for the finite-dimensional 4-electron core space of C 2 and the infinite-dimensional valence space, in which case there are exactly 256 Schmidt values, all of which are accurately calculated and demonstrated to converge. Our work sheds new light on quantum entanglement in molecules by representing the exact (complete basis) many-electron wavefunction introduced by Dirac [4] as an infinite product, which allows us to overcome the cutoff on the excitation level and finally bring together theory and computational feasibility. By contrast, the fundamental work of Löwdin [11] on Full-CI in the CBS limit, which leads to an in-arXiv:2112.10210v3 [quant-ph] 6 Apr 2022 finite series representing the wave function, remained computationally elusive. SET-UP AND NOTATION Let H be an infinite-dimensional separable Hilbert space spanned by orthonormal orbitals {ϕ i } ∞ i=1 . Let V N be the N -fold antisymmetric product N i=1 H, and let F be the ensuing Fock space, F := ∞ N =0 V N . The prototypical case we have in mind is that H is the full (untruncated) Hilbert space H = L 2 (R 3 ; C 2 ) of square-integrable single-electron wavefunctions, in which case V N is the space of square-integrable antisymmetric N -electron wavefunctions. We denote the Fock space over the basis-truncated Hilbert space spanned by the first k orbitals by F k . Slater determinants indexed by their binary label are denoted Φ µ1...µ K , that is to say Φ µ1...µ K := |ϕ i1 ...ϕ i N if µ i = 1 exactly when i ∈ {i 1 , ..., i N }, i 1 < . . . < i N . (1) MPS FOR INFINITE BASIS For a finite basis set (or finite spin chain), every quantum state in the Fock space has an exact MPS representation provided the size of successive MPS matrices is allowed to increase exponentially until the middle basis function (or site) before decreasing again (see e.g. [14]). We now construct an analogous exact MPS representation for every quantum state in the Fock space over an infinite basis set (or half-infinite spin chain). In this case one obtains an infinite sequence of succesive MPS matrices of exponentially increasing size; to recover the state, which is a scalar, one has to insert a closure vector after the first k matrices and take the limit k → ∞. See Fig. 1. More precisely: Theorem 1 (MPS for infinite basis). Given any normalized quantum state Ψ belonging to the full infinite-basis Fock space F, there exist left-normalized tensors A k ∈ C 2 k−1 ×2×2 k such that lim k→∞ µ1,...,µ k A 1 [µ 1 ]...A k [µ k ]      0 . . . 0 1      Φ µ1...µ k = Ψ. (2) Here the sums run over µ 1 , ..., µ k ∈ {0, 1}, and Φ µ1...µ k is given by (1). Moreover for every k, µ1,...,µ k A 1 [µ 1 ]...A k [µ k ]      0 . . . 0 1      Φ µ1...µ k = P F k Ψ P F k Ψ (3) where P F k denotes the projector onto the Fock space over the first k basis functions, and we use the convention v ||v|| = 0 when v = 0. Bottom: Inserting a closure vector after the k th node tensor filters out its projection onto the Fock space over the first k orbitals, eq. (3), and corresponds to setting the physical variables associated to all higher orbitals to zero. μ 1 μ k A k A 1 A k+1 … μ k+1 μ 1 μ k A k A 1 … 0 basis−truncated Fock space μ k+2 0 A k+2 Proof. The idea is to approximate Ψ by states Ψ K in the basis-truncated Fock space F K , obtain corresponding K-dependent MPS matrices A (K) 1 , A (K) 2 , . .., use the gauge freedom to achieve eq. (3) for Ψ K in place of Ψ when k < K, and let the basis truncation parameter K go to infinity to arrive at universal (K-independent) matrices A 1 , A 2 , ... (see Fig. 2). Now in more detail, start from any sequence Ψ K of approximations to Ψ belonging to the basis-truncated Fock space F K such that lim K→∞ Ψ K = Ψ. Such approximations always exist; for instance one could take Ψ K to be the orthogonal projection of Ψ onto F K . Alternatively, when Ψ is the unique ground state of some Hamiltonian H, Ψ K could be taken to be any ground state of H in F K , see the next section. It is a standard fact (see e.g. [14]) that Ψ K can be represented in MPS form, µ1,...,µ K A (K) 1 [µ 1 ] . . . A (K) K [µ K ] Ψ K Φ µ1...µ K ,(4) where the matrices A (K) k [µ k ] are left normalized, 1 µ k =0 A (K) k [µ k ] † A (K) k [µ k ] = Id,(5) and of size 2 k−1 × 2 k for k ≤ K/2. We use the remaining gauge freedom A (K) k [µ], A (K) k+1 [µ ] → A (K) k [µ]Q, Q † A (K) k+1 [µ ]) with Q unitary to achieve that the vector A (K) k+1 [0] . . . A (K) K [0] =: v (K) k has the normal form v (K) k = v (K) k e k ,(6) where e k = (0, . . . , 0, 1) ∈ R 2 k is the closure vector from the theorem. Replacing the matrix product (4) by its value at µ k+1 = . . . = µ K = 0 and using (6), we have for any k < K µ1,...,µ K A (K) k+1 [µ k+1 ] . . . A (K) K [µ K ] inA (K) 1 [µ 1 ] . . . A (K) k [µ k ] Ψ K v (K) k Φ µ1...µ k = P F k Ψ K , where P F k is the orthogonal projector onto the Fock space over the first k basis functions. In particular, ||Ψ K || · ||v (K) k || = ||P F k Ψ K ||. Combining this with (6) yields P F k Ψ K µ1,...,µ k A (K) 1 [µ 1 ]...A (K) k [µ k ]e k Φ µ1...µ k = P F k Ψ K . (7) The idea now is to fix k and let K → ∞. For K ≥ 2k the matrices A (K) k have fixed size 2 k−1 × 2 k , and are uniformly bounded thanks to (5). Therefore, for any fixed k there is a sequence of truncation parameters K 1 < K 2 < ... such that the tensors A (K1) k , A (K2) k , ... converge to some A k , where A k is again left normalized. By a standard diagonal argument, one can in fact find a k-independent sequence of truncation parameters such that the above convergence occurs for all k. Passing to the limit K → ∞ in (7) and using the convergence of Ψ K to Ψ and of P F k Ψ K to P F k Ψ gives eq. (3). Finally, letting k → ∞ and using that P F k Ψ tends to Ψ yields the representation (2). A k 2 k D K 1 K 2 K 3 K 4 size of A k k full bond dimension basis limit complete logarithmic scale FIG. 2. Construction of the exact MPS matrices A k of a quantum state associated with the basis functions ϕ k in the complete basis (K → ∞) limit, using basis-truncated approximations with K basis functions. The A k have size min(2 k , 2 K−k ) in the finite basis (colored lines) and hence maximal size 2 k (dashed line); in the numerical results we also used increasing finite-D cutoffs (dotted line) until convergence. The above proof shows that for any approximations Ψ K in the basis-truncated Fock space F K which converge to Ψ as K tends to infinity, there exist truncation parameters K 1 < K 2 < ... such that the corresponding tensors A (K 1 ) k , A (K 2 ) k , ... converge to the A k in (2). See Fig. 2. In quantum chemistry it is customary to absorb the spin functions into the physical variables and only use spatial orbitals as MPS nodes. The new MPS matrices subsume two old MPS matrices via B [ν ] = A 2 −1 [µ 2 −1 ]A 2 [µ 2 ] , with ν ranging over 0, 1, 2, 3 corresponding to (µ 2 −1 , µ 2 ) = (0, 0), (1, 0), (0, 1), (1, 1). See Fig. 3. The exact representation of Ψ in Theorem 1 then takes the following form: there exist left-normalized tensors B k ∈ C 4 k−1 ×4×4 k such that, for every k, B 1 [ν 1 ] . . . B k [ν k ]e k Φ ν1...ν k = P F k Ψ P F k Ψ ,(8) and lim k→∞ ν1,...,ν k B 1 [ν 1 ] . . . B k [ν k ]e k Φ ν1...ν k = Ψ. (9) Here Φ ν1,...,ν k denotes the Slater determinant in which the orbital ψ is either empty or singly occupied with up or down spin or doubly occupied, according to whether ν = 0, 1, 2, 3. A 2ℓ−1 μ 2ℓ−1μ ℓ B ℓ μ 2ℓ A 2ℓ FIG. 3. MPS matrix B associated with a spatial orbital ψ . NUMERICAL RESULTS In order to validate the theory we have performed large-scale simulations on the C 2 molecule using the quantum chemistry density matrix renormalization group (QC-DMRG) method [18]. The dicarbon is modeled at 1.25Å bondlength in a frozen core cc-pVTZ [5] basis. As is customary, we then take the ensuing energy-ordered Hartree-Fock orbitals as basis functions corresponding to the MPS nodes. Thus, one is dealing with the correlation of the 8 valence electrons on 58 spatial orbitals, corresponding to a single-particle Hilbert space with dimension d = 116 and a Fock space dimension ≈ 8 × 10 34 . In addition, an almost twice as large frozen-core cc-pVQZ basis with 108 spatial orbitals has been investigated, yielding a singleparticle Hilbert space of dimension d = 216. We have chosen this system because of the presence of strong static and dynamic correlations; see [2] for a recent benchmark study comparing different methods. We tested the convergence of the MPS matrices for the C 2 ground state as the basis size gets large. We focused on the singular value distribution across the bond between the matrices A k and A k+1 for k = 8 (corresponding to the matrices B k and B k+1 for k = 4). This distribution fully represents the entanglement between the Hartree-Fock occupied and virtual orbitals, up to gauge freedom. The infinite basis limit depicted in Figure 2 can be emulated by calculating the ground state in finite bases with K k basis functions. To mimic passage to the large K limit we present results for various K values ranging from 16 to d = 116. Beyond the lowest K value, we also need to limit the bond dimension D (note that exact representation of general states in the Fock space over 116 basis functions would require D ≈ 3 × 10 17 ). Thus, we have also investigated convergence with respect to the bond dimension. The results in Fig. 4 show that the singular value distribution is already converged with respect to the size K of the basis when K ≈ 64, regardless of the bond dimension D. As for the effect of bond dimension truncation, we find that D = 256 is sufficient to capture the singular values above a threshold of about 10 −5 , corresponding to approximately 100 singular values. The tail of the singular value distribution rep- resenting dynamic correlation effects is seen to slightly shift upwards as D is increased to 512 respectively 1024. Further calculations up to D = 4096 lead to a numerically converged spectrum, but are not shown as the difference would be hardly visible at the given axis scale. Results for d = 216 up to D = 1024 also revealed the fast convergence of the Schmidt spectrum already at K ≈ 64. This together with the observed exponential decay provides a theoretical foundation for the long-standing use of small-basis calculations in quantum chemistry. FIG. 1 . 1Top: MPS representation of a quantum state in the full infinite-basis Fock space, eq. (2). Acknowledgements. G.F. and B.R.G. have been supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project number 188264188/GRK1754 within the International Research Training Group IGDK 1754.ÖL. has been supported by the Hungarian National Research, Development and Innovation Office (NK-FIH) through Grants Nos. K120569 and K134983, by the Quantum Information National Laboratory of Hungary, and by the Hans Fischer Senior Fellowship programme funded by the Technical University of Munich -Institute for Advanced Study. The development of DMRG libraries has been supported by the Center for Scalable and Predictive methods for Excitation and Correlated phenomena (SPEC), funded as part of the Computational Chemical Sciences Program by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences at Pacific Northwest National Laboratory. FIG. 4. Singular value distribution of the C2 ground state, ωα(k, K), between core and valence space, i.e. MPS matrices A k and A k+1 for k = 8, and different basis sizes K k. Inside each panel one sees the fast convergence as K gets large, for fixed bond dimension D. Across the panels we increased the bond dimension until convergence.0 100 200 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 (8,16) (8,32) (8,64) (8,116) 0 100 200 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 (8,16) (8,32) (8,64) (8,116) 0 100 200 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 (8,16) (8,32) (8,64) (8,116) The density matrix renormalization group in chemistry and molecular physics: Recent developments and new challenges. A Baiardi, M Reiher, J. Chem. Phys. 152440903A. Baiardi and M. Reiher. The density ma- trix renormalization group in chemistry and molecular physics: Recent developments and new challenges. J. Chem. Phys., 152(4):040903, 2020. Towards large-scale restricted active space calculations inspired by the Schmidt decomposition. G Barcza, M A Werner, G Zaránd, Ö Legeza, T Szilvási, arXiv:2111.06665arXiv preprintG. Barcza, M. A. Werner, G. Zaránd,Ö. Leg- eza, and T. Szilvási. 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Sandoghdar. Nanome- ter resolution and coherent optical dipole cou- pling of two individual molecules. Science, 298(5592):385-389, 2002. Classical simulation of infinitesize quantum lattice systems in two spatial dimensions. J Jordan, R Orús, G Vidal, F Verstraete, J I Cirac, Phys. Rev. Lett. 101250602J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac. Classical simulation of infinite- size quantum lattice systems in two spatial di- mensions. Phys. Rev. Lett., 101:250602, Dec 2008. Quantum entanglement between an atom and a molecule. Y Lin, D R Leibrandt, D Leibfried, C.-W Chou, Nature. 5817808Y. Lin, D. R. Leibrandt, D. Leibfried, and C.- w. Chou. Quantum entanglement between an atom and a molecule. Nature, 581(7808):273- 277, 2020. Quantum theory of manyparticle systems. I. Physical interpretations by means of density matrices, natural spinorbitals, and convergence problems in the method of configurational interaction. P.-O Löwdin, Phys. Rev. 9761474P.-O. Löwdin. Quantum theory of many- particle systems. I. Physical interpretations by means of density matrices, natural spin- orbitals, and convergence problems in the method of configurational interaction. Phys. Rev., 97(6):1474, 1955. Approaching the Basis Set Limit of CCSD(T) Energies for Large Molecules with Local Natural Orbital Coupled-Cluster Methods. P R Nagy, M Kállay, J. Chem. Theory Comput. 1510P. R. Nagy and M. Kállay. Approaching the Basis Set Limit of CCSD(T) Energies for Large Molecules with Local Natural Orbital Coupled- Cluster Methods. J. Chem. Theory Comput., 15(10):5275-5298, 2019. Theoretical models for chemistry. J A Pople, Proceedings of the Summer Research Conference on Theoretical Chemistry: Energy, Structure and Reactivity. D. W. Smiththe Summer Research Conference on Theoretical Chemistry: Energy, Structure and ReactivityNew YorkJohn Wiley & SonsJ. A. Pople. Theoretical models for chem- istry. In D. W. Smith, editor, Proceedings of the Summer Research Conference on Theoreti- cal Chemistry: Energy, Structure and Reactiv- ity. John Wiley & Sons New York, 1973. The density-matrix renormalization group in the age of matrix product states. U Schollwöck, Ann. Phys. 3261U. Schollwöck. The density-matrix renormal- ization group in the age of matrix product states. Ann. Phys., 326(1):96-192, 2011. Tensor product methods and entanglement optimization for ab initio quantum chemistry. S Szalay, M Pfeffer, V Murg, G Barcza, F Verstraete, R Schneider, Legeza, Int. J. Quantum Chem. 115S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, andÖ. Legeza. Tensor product methods and entanglement op- timization for ab initio quantum chemistry. Int. J. Quantum Chem., 115(19):1342-1391, 2015. Canonical versus explicitly correlated coupled cluster: Post-completebasis-set extrapolation and the quest of the complete-basis-set limit. A J C Varandas, Int. J. Quantum Chem. 121926598A. J. C. 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The spin-isospin response of stable and exotic nuclei is investigated in the framework of the proton-neutron relativistic quasiparticle time-blocking approximation (pn-RQTBA). Based on the Covariant Density Functional Theory, this method extends the proton-neutron Relativistic Quasiparticle Random-Phase Approximation (pn-RQRPA) by including the coupling between single quasiparticles and collective nuclear vibrations. In the charge-exchange channel, this coupling generates a time-dependent effective interaction between proton and neutron quasiparticles. The particle-hole component of this interaction adds to the static pion and rho-meson exchange, while the particle-particle component provides a microscopic and consistent proton-neutron pairing interaction. We find that such dynamical effects induce fragmentation and spreading of the Gamow-Teller transition strength which are important for a better agreement with the experimental measurements and for an accurate description of β -decay rates. The new developments include the coupling of single nucleons to isospin-flip vibrations in doubly-magic nuclei. We find that these phonons can have a non-negligible effect on β -decay half-lives and "quenching" of the strength.

10.22323/1.281.0020

1702.00044

9251a4c5d730c3f9111c85a8f063115964a5b64f

Relativistic approach to nuclear spin-isospin excitations including quasiparticle-vibration coupling Relativistic approach to nuclear spin-isospin excitations including quasiparticle-vibration coupling September, 2016 Caroline Robin [email protected] Elena Litvinova [email protected] Adelaide Australia Caroline Robin Department of Physics Department of Physics Western Michigan University 49008KalamazooMIUSA and National Superconducting Cyclotron Laboratory Western Michigan University 49008KalamazooMIUSA Michigan State University 48824East LansingMIUSA Relativistic approach to nuclear spin-isospin excitations including quasiparticle-vibration coupling Relativistic approach to nuclear spin-isospin excitations including quasiparticle-vibration coupling The 26th International Nuclear Physics Conference September, 2016Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). The spin-isospin response of stable and exotic nuclei is investigated in the framework of the proton-neutron relativistic quasiparticle time-blocking approximation (pn-RQTBA). Based on the Covariant Density Functional Theory, this method extends the proton-neutron Relativistic Quasiparticle Random-Phase Approximation (pn-RQRPA) by including the coupling between single quasiparticles and collective nuclear vibrations. In the charge-exchange channel, this coupling generates a time-dependent effective interaction between proton and neutron quasiparticles. The particle-hole component of this interaction adds to the static pion and rho-meson exchange, while the particle-particle component provides a microscopic and consistent proton-neutron pairing interaction. We find that such dynamical effects induce fragmentation and spreading of the Gamow-Teller transition strength which are important for a better agreement with the experimental measurements and for an accurate description of β -decay rates. The new developments include the coupling of single nucleons to isospin-flip vibrations in doubly-magic nuclei. We find that these phonons can have a non-negligible effect on β -decay half-lives and "quenching" of the strength. Introduction Nuclear transitions involving transfer of isospin have many applications in nuclear and particle physics, as well as in astrophysics as they determine weak-interaction rates governing r-process nucleosynthesis and stellar evolution. Such types of modes as Fermi, Gamow-Teller (GT) or spindipole (SD) transitions are very sensitive to the spin-isospin dependence of the nucleon-nucleon interaction and have been extensively studied in many theoretical methods (see e.g. Refs [1] to [20]). In the Relativistic Random-Phase Approximation (RRPA) based on the Relativistic Mean-Field (RMF), where the exchange interaction is not explicitly treated, isospin-transfer modes are fully determined by isovector rho-and pion-exchange. While the proton-neutron (R)RPA (pn-(R)RPA) usually describes the position of GT and SD giant resonances quite well, extensions are necessary to include spreading mechanism essential for the description of width and details of the transition strength. In open-shell nuclei with superfluid pairing correlations, it has been shown that an isoscalar residual pairing interaction is important for an accurate description of the GT strength and weak-interaction rates [5,6,7,11]. In proton-neutron (Relativistic) Quasiparticle-RPA (pn-(R)QRPA) studies, the strength of this interaction is however usually fitted to β -decay half-lives which lowers the predictive power of the method. In this work we study the GT response of a few mid-mass nuclei within the Relativistic Quasiparticle-Vibration Coupling (RQVC) framework presented in Ref. [20]. This approach, applied in the proton-neutron Quasiparticle Time-Blocking Approximation (pn-RQTBA), extends the pn-RQRPA by accounting for the interplay between quasiparticle and collective degrees of freedom, and naturally generates a dynamical pn pairing interaction, without extra empirical factor. We compare our results to the pn-RQRPA and to the available experimental data. For the first time we include the coupling of nucleons to isospin-flip vibrations in doubly-magic nuclei. We investigate their effect on the quenching of the GT strength and β -decay half-lives. Spin-isospin response of nuclei The study of the response of nuclei to a weak external fieldF is a great tool to access their excitation spectra. At first order in the perturbation, such field excites the nucleus from its ground state by inducing particle-hole excitations. The corresponding transition strength distribution S(E) can be directly obtained from the knowledge of the propagator of a correlated particle-hole pair in the nuclear medium, or response function R, since S(E) = ∑ f | Ψ f |F|Ψ i | 2 δ (E − E f + E i ) = − 1 π lim ∆→0 + Im Ψ i |F † R(E + i∆)F|Ψ i ,(2.1) where |Ψ i and |Ψ f denote the nuclear ground and excited states respectively. In the framework of the linear response theory, the response function is the solution of the Bethe-Salpeter equation: R(14, 23) = G(1, 3)G(4, 2) − i d5...d8G(1, 5)G(6, 2)V (58, 67)R(74, 83) ,(2.2) where the indices i = 1, 2, 3... ≡ (t i , k i ) denote the time variable t i along with a complete set of nucleonic quantum numbers k i . In open-shell nuclei with superfluid correlations, the single-nucleon space acquires an extra dimension and these indices are supplemented by an additional quantum number η i = ± denoting the upper and lower component in the Nambu-Gorkov space. In Eq. (2.2), G denotes in principle the exact one-nucleon propagator, while V is an effective two-body interaction induced by the medium. This interaction is consistently determined from the singlenucleon self-energy Σ as V (58, 67) = i δ Σ(5, 6) δ G(7, 8) . (2.3) In the present relativistic quasiparticle-vibration coupling (RQVC) model the static part of the selfenergy is determined within the relativistic mean-field approximation with pairing correlations, or relativistic Hartree-Bogoliubov (RHB) approximation, assuming a static meson-exchange interaction. This leads to a description of the ground state where the (quasi)nucleons evolve independently in classical meson fields. In reality, however, many virtual interaction processes can occur during the propagation of a nucleon in the nucleus. In particular, due to correlations, the excitation of a (quasi)particle-(quasi)hole (qp-qh) pair can induce the coherent excitation of all nucleons, leading to a vibration of the nucleus (phonon), which in turn, produces polarization of the medium and modification of the single-particle motion. Such effects are taken into account by introducing an energy-dependent term to the self-energy, describing the emission and re-absorption of a nuclear vibration by a (quasi)particle. Subsequently, the effective interaction in the BSE (2.2) is supplemented by the dynamical exchange of a phonon between two (quasi)particles. In the chargeexchange channel, and in the time-blocking approximation (TBA) described in [21,22], the BSE for the qp-qh propagator takes the following form in the energy representation: R η 1 η 4 ,η 2 η 3 k 1p k 4n ,k 2n k 3p (ω) = R (0)η 1 η 4 ,η 2 η 3 k 1p k 4n ,k 2n k 3p (ω) + ∑ k 5p k 6n k 7p k 8n η 5 η 6 η 7 η 8 R (0)η 1 η 6 ,η 2 η 5 k 1p k 6n ,k 2n k 5p (ω)W η 5 η 8 ,η 6 η 7 k 5p k 8n ,k 6n k 7p (ω)R η 7 η 4 ,η 8 η 3 k 7p k 4n ,k 8n k 3p (ω) ,(2.4) where (p, n) refer to proton and neutron indices respectively, ω = E + i∆ is the energy variable, R (0) denotes the free qp-qh propagator in the proton-neutron channel, and W is the effective interaction given by the sum of the static meson exchange and the energy-dependent amplitude Φ(ω) containing the effect of QVC: W (ω) = V ρ + V π + V δ π + Φ(ω) . (2.5) The sum V ρ + V π is the finite range isovector meson-exchange interaction, while V δ π denotes the zero-range Landau-Migdal term taken with parameter g = 0.6, as the exchange interaction is not treated in the present work [12]. When considering this static interaction alone, one gets back to the pn-RQRPA. The QVC amplitude Φ(ω) is represented in Fig. 1 in terms of Feynman diagrams. This dynamical interaction introduces 1(q)p-1(q)h⊗phonon configurations and is responsible for damping of the transition strength. A detailed expression of W (ω) can be found in Ref. [20]. In the following the extension of the pn-RQRPA including QVC effects in the TBA is referred to as pn-RQTBA. Gamow-Teller response of stable and exotic nuclei Gamow-Teller (GT) modes describe the nuclear response to the following spin-isospin flip operatorF GT ± = A ∑ i=1 Σ (i) τ (i) ± , with Σ (i) = σ (i) 0 0 σ (i) . (3.1) For such unnatural parity modes the pion gives almost the full contribution of the static meson exchange interaction (2.5). As it is absent in the RHB ground-state, the pion is considered with the free-space coupling constant f 2 π 4π = 0.08, which makes GT a great study-case where no doublecounting effect is expected when going beyond the pn-RQRPA description. In this work we calculate the response of stable and neutron-rich nuclei to theF GT − operator, using the numerical scheme described in Ref. [20]. Starting from the RHB approximation with the NL3 meson parametrization [23] and a monopole-monopole pairing force [22], the set of phonons that are coupled to the quasiparticles is calculated in the RQRPA. As a first step, we consider the coupling to non-isospin-flip phonons with J π = 2 + , 3 − , 4 + , 5 − , 6 + in an energy window of 30 MeV. The GT response is then calculated by solving Eq. (2.4), and the strength distribution is obtained from Eq. (2.1). In Ref. [20] we calculated the GT − response of neutron-rich Nickel isotopes within this approach. In Figs. 2a and 2b we show the strength distributions in 68 Ni and 78 Ni calculated with a smearing parameter ∆ = 200 keV. For comparison we show the results obtained at the pn-RQRPA level in blue, while the full calculations including QVC (pn-RQTBA) are displayed with red lines. In order to obtain the excitation spectrum with respect to the daughter ground-state, all distributions have been shifted by the binding energy difference ∆B = B(parent) − B(daughter) calculated in the RHB approximation. Clearly, the QVC induces fragmentation and spreading of the pn-RQRPA transition strength over a larger energy range including the low-energy region. We saw that such effects resulted in a considerable improvement of the β -decay half-lives as compared to the pn-RQRPA description [20]. We also show in Fig. 2a the low-lying strength distribution in 68 Ni calculated with ∆ = 20 keV in order to disentangle the low-lying states. Experimentally it is known that the ground-state of 68 Cu has angular momentum and parity 1 + [24]. The shift induced by QVC clearly improves the position of the 1 + 1 state, as it is found to lie at 330 keV versus 3.88 MeV in pn-RQRPA. We remind that we do not introduce any static proton-neutron pairing. In pn-(R)QRPA studies such pairing is usually introduced through an attractive residual isoscalar pn interaction in the particle-particle channel which is fitted to e.g. β -decay half-lives [5,6,7,11]. When the QVC interaction of Fig. 1 is taken into account, pn pairing appears naturally due to the presence of particle-like pairing which generates the particle-particle component of the dynamical pn interaction Φ(ω). This can explain why the QVC induces a similar effect on the low-energy strength as the phenomenological static pn pairing. We show in Figs. 2c and 2d the GT − strength distributions in 90 Zr and 48 Ca which have been measured experimentally. The strength in 90 Zr has been calculated using a smearing ∆ = 1 MeV, to simulate the experimental resolution of Ref. [25]. When QVC is included, a nice agreement with the data is obtained up to an excitation energy of ∼ 25 MeV. At higher energy the experimental strength also contains the isovector spin-monopole response [25]. The GT − strength in 48 Ca has been measured with a higher energy resolution corresponding to ∆ = 210 keV in Ref. [26]. At such high resolution, the pn-RQRPA is not able to reproduce the details of the experimental spectrum. The position of the first low-lying state and giant resonance region are also overestimated by ∼ 3 − 4 MeV. The fragmentation induced by QVC leads to the appearance of finer details in the transition strength, which is also shifted down in accordance with the data. The peak around ∼ 13 MeV however still lacks fragmentation. Up to now the phonon spectrum coupled to the quasiparticles was limited to non-isospin-flip phonons with natural parity. Ref. [27] however showed that the presence of low-lying isospin-flip modes can have an impact on the single-nucleon shell structure. Recent developments now make it possible to include the coupling to charge-exchange phonons in the description of the response in doubly magic nuclei. Such vibrations introduce new diagrams in the QVC interaction that are displayed in Fig. 3. We note that due to charge conservation they only generate additional self-energy insertions, and do not appear in the form of a phonon-exchange. Moreover, we see that these new self-energy terms involve proton-neutron particle-particle elements of the particle-vibration coupling vertex, and thus can also be interpreted as a (virtual) energy-dependent pn pairing interaction in doubly-magic nuclei. 78 Ni calculated when adding the coupling to charge-exchange vibrations with J π = 0 ± , 1 ± , 2 ± , 3 ± , 4 ± , 5 ± , 6 ± . Although small, the effect of such phonons is noticeable as they slightly modify the giant resonance region, introduce a new state around 26 MeV and shift further down the low-energy part of the strength. The resulting half-life calculated with the effective weak axial vector coupling constant g A = 1 is shown in Fig. 5 and is now in great accordance with the experimental value [28]. In Fig. 4c, we show the corresponding cumulative integrated strength, or cumulative sum of B(GT − ) values. In the particle sector, above the giant resonance region around 30 MeV we note that the pn-RQRPA strength has almost reached its saturated value (97%). Introducing the coupling to isoscalar and isovector phonons (plain red curve) leads to a "quenching" of ∼ 16% of the RQRPA strength at this energy due to fragmentation and redistribution of the strength. We also note from Fig. 4a the presence of states at very large negative energy, corresponding to transitions to the Dirac sea of negative-energy states. As already mentioned in [29], such transitions are the result of the "empty Dirac-sea approximation" which is made at the relativistic mean-field level to avoid divergences. In the present case of 78 Ni these transitions to the antiparticle sector take away about 10% of the total GT − strength. In order to reproduce details of the excitation spectra, and to tackle the quenching problem of the GT strength, it is important to include complex nucleonic configurations in a model space which is as large as possible. In the doubly magic nucleus 78 Ni, it is possible to increase the QVC energy window, in which 1p-1h⊗phonon excitations are included, up to E win = 100 MeV that is the same energy cut-off which is used to select the 1p-1h pairs in pn-RQRPA. The final results are shown with black lines. The main effect of increasing the value of E win is an enhanced "quenching" at high excitation energy above 30 MeV, (Fig. 4c), and a slight redistribution of the low-lying strength which leads to an additional decrease of the half-life (Fig. 5), that remains however within the experimental error bars. Finally we show in Fig. 6a and 6b phonons brings further fragmentation of the peak at ∼ 13 MeV which is counterbalanced by an increase of the strength around ∼ 7 MeV. We now see a clear state at ∼ 18.3 MeV which is ∼ 1.3 MeV too high than the one seen experimentally. The first state around 2 MeV is shifted down when introducing charge-exchange phonons and increasing the QVC window E win to a value of 100 MeV. Around 30 MeV, which is the excitation energy reached experimentally, we note again from Fig. 6b that the pn-RQRPA distribution has almost saturated as it reaches ∼ 98% of the total GT − strength in the particle sector. The strength with QVC (in black) introduces some quenching since it exhausts only ∼ 91% of that value. Such redistribution of the transition strength is however not sufficient to reach the experimental value of only 71%. Summary and outlook In this work we applied the pn-RQTBA approach to the description of Gamow-Teller (GT) transitions in a few mid-mass nuclei. We found that the coupling between quasiparticles and collectives vibrations, which introduces complex 1qp-1qh⊗phonons configurations, systematically induces fragmentation and spreading of the pn-RQRPA transition strength, and leads to a more detailed description of the GT states, in better accordance with the data. We included for the first time the coupling to charge-exchange vibrations in doubly magic nuclei and found that such phonons can have a non-negligible impact on the β -decay half-lives and on the quenching of the GT strength. When the experimental resolution is high, it is however clear that the RQVC framework implemented in the present Time-Blocking Approximation is often not sufficient to reproduce the very fine details of the strength. In order to further improve the description of spin-isospin modes we plan for the future to include higher-order configurations in the response [30] and in the ground state. Figure 1 : 1Quasiparticle-Vibration Coupling interaction in the proton-neutron channel. GT − strength in 68 Ni calculated with a smearing ∆ = 200 keV. We also show the low-lying strength calculated with ∆ = 20 keV. The arrow denotes the position of the 1 + 1 state in 68 Cu[24]. GT − strength in 78 Ni calculated with a smearing ∆ = 200 keV. GT − strength in 90 Zr calculated with a smearing ∆ = 1 MeV. The experimental distribution has been extracted from Ref.[25]. GT − strength in 48 Ca calculated with a smearing ∆ = 210 keV. The experimental distribution has been extracted from Ref.[26]. Figure 2 : 2GT − strength distribution in a few stable and neutron-rich nuclei. The pn-RQRPA results are shown with blue lines while the pn-RQTBA results (with QVC) are shown with red lines. Figure 3 : 3QVC interaction in the proton-neutron channel. The wavy lines represent isoscalar (nonisospin-flip) phonons while the springs represent isovector (isospin-flip) phonons. Fig . 4b displays in plain red lines the GT − strength in the GT − strength distribution in 48 Ca in the particle sector and the corresponding cumulative sum of the B(GT − ) values. The coupling to charge-exchange -RQTBA IS phon E win =30MeV pn-RQTBA all phon E win =30MeV pn-RQTBA all phon E win =100MeV (c) 78 Ni cumulative sum of the B(GT − ) values. Figure 4 : 4GT − strength distribution in78 Ni and corresponding cumulative sum of B(GT − ). The results are shown at different theoretical levels: at the pn-RQRPA level (in blue), at the pn-RQTBA level with coupling to isoscalar (IS) phonons in a QVC energy window E win = 30 MeV in dashed red, at the pn-RQTBA level with coupling to IS and charge-exchange phonons in a QVC energy window E win = 30 MeV in plain red, and at the pn-RQTBA level with coupling to IS and chargeexchange phonons in a QVC energy window E win = 100 MeV in black. Figure 5 : 5Half-life of 78 Ni in seconds. The experimental value is taken form Ref.[28]. The error bars are experimental ones. GT − strength in 48 Ca in the particle sector. pn-RQTBA IS phon E win =30MeV pn-RQTBA all phon E win =30MeV pn-RQTBA all phon E win =100MeV (b) 48 Ca cumulative sum of the B(GT − ). Figure 6 : 6GT − strength in 48 Ca in the particle sector and cumulative sum of the B(GT − ) values. AcknowledgmentsWe thank T. Marketin for providing part of the code for pn-RQRPA matrix elements. This work was supported by US-NSF Grants PHY-1404343 and PHY-1204486. . B A Brown, Prog. Part. Nucl. Phys. 47517B.A. Brown, Prog. Part. Nucl. Phys. 47, 517 (2001). . E Caurier, Rev. Mod. Phys. 77427E. Caurier et al., Rev. Mod. Phys. 77, 427 (2005). . S E Koonin, D J Dean, K Langanke, Phys. Rep. 2781S.E. Koonin, D.J. Dean and K. Langanke, Phys. Rep. 278, 1 (1997). . D L Fang, B A Brown, T Suzuki, Phys. Rev. C. 8824314D. L. Fang, B. A. Brown and T. Suzuki, Phys. Rev. C 88, 024314 (2013). . J Engel, Phys. Rev. C. 6014302J. Engel et al., Phys. Rev. C 60 014302 (1999). . 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C. 8534314Y. F. Niu, G. Colò, M. Brenna, P. F. Bortignon and J. Meng, Phys. Rev. C 85, 034314 (2012). . Y F Niu, G Colò, E Vigezzi, Phys. Rev. C. 9054328Y. F. Niu, G. Colò and E. Vigezzi, Phys. Rev. C 90, 054328 (2014). . Y F Niu, Z M Niu, G Colò, E Vigezzi, Phys. Rev. Lett. 114142501Y. F. Niu, Z. M. Niu, G. Colò and E. Vigezzi, Phys. Rev. Lett. 114, 142501 (2015). . T Marketin, E Litvinova, D Vretenar, P Ring, Phys. Lett. B. 706477T. Marketin, E. Litvinova, D. Vretenar and P. Ring, Phys. Lett. B 706, 477 (2012). . E Litvinova, Phys. Lett. B. 730307E. Litvinova et al. Phys. Lett. B 730, 307 (2014). . C Robin, E Litvinova, Eur. Phys. J. A. 52205C. Robin and E. Litvinova, Eur. Phys. J. A 52, 205 (2016). . V I Tselyaev, Sov. J. Nucl. Phys. 501252Yad. Fiz.V. I. Tselyaev, Yad. Fiz. 50, 1252 (1989) [Sov. J. Nucl. Phys. 50, 780 (1989)]. . E Litvinova, P Ring, V Tselyaev, Phys. Rev. C. 7814312E. Litvinova, P. Ring and V. Tselyaev, Phys. Rev. C 78, 014312 (2008). . J Boguta, A R Bodmer, Nucl. Phys. 292413J. Boguta and A. R. Bodmer, Nucl. Phys. A292, 413 (1977). . National Nuclear Data Center (NNDC) database. National Nuclear Data Center (NNDC) database, www.nndc.bnl.gov . T Wakasa, Phys. Rev. C. 552909T. Wakasa et al., Phys. Rev. C 55, 2909 (1997). . K Yako, Phys. Rev. Lett. 10312503K. Yako et al. ,Phys. Rev. Lett. 103, 012503 (2009). . E Litvinova, Phys. Lett. B. 755138E. Litvinova, Phys. Lett. B 755, 138 (2016). . P T Hosmer, Phys. Rev. Lett. 94112501P. T. Hosmer et al., Phys. Rev. Lett. 94, 112501 (2005). . N Paar, T Nikšić, D Vretenar, P Ring, Phys. Rev. C. 6954303N. Paar, T. Nikšić, D. Vretenar and P. Ring, Phys. Rev. C 69, 054303 (2004). . E Litvinova, Phys. Rev. C. 9134332E. Litvinova, Phys. Rev. C 91, 034332 (2015).

In the purpose of making the consensus algorithm robust to outliers, consensus on the median value has recently attracted some attention. It has its applicability in for instance constructing a resilient distributed state estimator. Meanwhile, most of the existing works consider continuous-time algorithms and uses high-gain and discontinuous vector fields. This issues a problem of the need for smaller time steps and yielding chattering when discretizing by explicit method for its practical use. Thus, in this paper, we highlight that these issues vanish when we utilize instead Implicit-Explicit Method, for a broader class of networks designed by the blended dynamics approach. In particular, for undirected and connected graphs, we propose a discrete-time distributed median solver that does not suffer from chattering. We also verify by simulation that it has a smaller iteration number required to arrive at a steady-state.

10.2139/ssrn.4116081

2106.09899

487d9ad0ba7182773d111f1d53b0224e9979ed74

Networks obtained by Implicit-Explicit Method: Discrete-time distributed median solver Member, IEEEJin Gyu Lee Networks obtained by Implicit-Explicit Method: Discrete-time distributed median solver 1Index Terms-consensus protocolsresilient consensusmulti- agent systemsblended dynamics In the purpose of making the consensus algorithm robust to outliers, consensus on the median value has recently attracted some attention. It has its applicability in for instance constructing a resilient distributed state estimator. Meanwhile, most of the existing works consider continuous-time algorithms and uses high-gain and discontinuous vector fields. This issues a problem of the need for smaller time steps and yielding chattering when discretizing by explicit method for its practical use. Thus, in this paper, we highlight that these issues vanish when we utilize instead Implicit-Explicit Method, for a broader class of networks designed by the blended dynamics approach. In particular, for undirected and connected graphs, we propose a discrete-time distributed median solver that does not suffer from chattering. We also verify by simulation that it has a smaller iteration number required to arrive at a steady-state. I. INTRODUCTION Consensus problem; a design problem for a network that yields agreement on their states has attracted much attention during the decades [1]. Such popularity comes from its various uses in, for instance, mobile multi-robot systems and sensor networks for the purpose of coordination and estimation respectively. In the meantime, the consensus value obtained by such couplings were usually designed or obtained as (weighted) averages of initial values or external inputs. Meanwhile, considering its application, a large network of cheap robots or sensors, it is hard to assume the reliability of individuals, and such large-scale distributed algorithms should be resilient to faults, outliers, and malicious attacks. In this respect, consensus on the average is inappropriate, as the mean statistic is weak to these abnormalities. On the contrary, what is robust to outliers, is the median statistic. By this observation, such a problem to design a network to achieve consensus to the median of initial values or external inputs has been recently tackled [2]- [6]. Consensus on the median is still useful in the same manner illustrated earlier. In particular, [5] introduces its application to distributed estimation under malicious attacks. However, most of the existing works deal with continuoustime algorithms and uses high-gain and discontinuous vector fields. Therefore, to adjust it to implement in a discrete-time framework, the usual explicit method yields some trouble. In particular, for stiff dynamics (high-gain), the explicit method requires a smaller time step (that depends on the gain). But, most importantly, the used discontinuous dynamics yields chattering, which arises from the theoretical use of Filippov solution [7] in continuous-time. This means a vector field can take any value in the interval, but in the end, it takes a particular value that ensures the existence of a solution. Since such a particular value is usually hard to find, implementing this in a discrete-time framework requires an alternative. Otherwise, it requires a sufficiently small time step. One way of resolving these issues is to use the implicit method. Thus, in this paper, we will illustrate that the networks constructed by the blended dynamics approach (approach using strong diffusive coupling), such as the distributed median solver given in [5], can be successfully discretized by the Implicit-Explicit Method as in [8]. In particular, the obtained network does not suffer from an excess of parameters to tune; it does not have to choose an appropriate time step at each time we choose the coupling gain, unlike the explicit method. We will concentrate on the network introduced in [5], as this contains discontinuity only in the individual vector field, hence makes the application of the Implicit-Explicit Method easier. But, this is also to illustrate that this method applies well to the class of networks designed by the well-developed blended dynamics approach [9]. One exception in the previous works that has introduced discrete-time algorithm is [6]. However, they have another layer of network to perform the task. The network proposed in this paper has a smaller dimension compared to this network but instead achieves only approximate consensus. This paper is organized as follows. In Section II, we briefly introduce the blended dynamics approach and illustrate how such a class of networks can be discretized by the Implicit-Explicit Method. Then, in Section III, we propose our discretetime distributed median solver following the given outline and then prove its convergence. Section IV verifies by simulation, its ability to remove chattering, and then we conclude in Section V. II. DISCRETIZATION OF NETWORKS UNDER STRONG DIFFUSIVE COUPLING Recently developed blended dynamics approach [9] is based on the observation that a strong diffusive coupling makes heterogeneous multi-agent systems behave like a single dynamical system which has its vector field as the average of all the individual vector fields in the network [10], [11]. In particular, consider a network given aṡ where N := {1, . . . , N } is the set of agent indices with the number of agents N , and N i is a subset of N whose elements are the indices of the agents that send the information to agent i. Here, the coefficient α ij is the ijth element of the adjacency matrix that represents the interconnection graph, and we also assume hereafter that the graph is undirected and connected. Then, as the coupling gain k approaches infinity, the network achieves arbitrary precision approximate synchronization and its synchronized behavior can be characterized by the single dynamicsẋ x i = f i (x i ) + k j∈Ni α ij (x j − x i ), i ∈ N ,= 1 N N i=1 f i (x) which is called blended dynamics, under the only assumption that the blended dynamics is stable, e.g., it has contraction property or has an asymptotically stable limit cycle. The entire theory is based on the singular perturbation argument, and it has wide applicability in network design such as distributed optimization, distributed estimation, and formation control. We refer to [12] for an exhaustive review on the topic. Now, to avoid the problem of using the explicit method in discretizing stiff dynamics, which in this case arises as to the use of smaller time steps for increasing coupling gain k (hence making it harder to perform decentralized design), 1 we propose in this section the following, which makes use of the Implicit-Explicit Method. In particular, we discretize in the manner given as x i [n + 1] − x i [n] = f i (x i [n + 1]) + k j∈Ni α ij (x j [n] − x i [n + 1]) or equivalently as x i [n + 1] + k j∈Ni α ij x i [n + 1] − f i (x i [n + 1]) = x i [n] + k j∈Ni α ij x j [n]. Now, this yields the network of form x i [n + 1] = F k i   x i [n] + k j∈Ni α ij x j [n]   , i ∈ N where F k i (·) is the inverse of (1 + kd i )x − f i (x), where d i = j∈Ni α ij . By the implicit function theorem, such function F k i is well-defined semi-globally for sufficiently large k. By its definition, it satisfies the following identity. F k i ((1 + kd i )x − f i (x)) ≡ x The utility of this approach will be found in the particular example of distributed median solver, in Section III. III. DISCRETE-TIME DISTRIBUTED MEDIAN SOLVER The continuous-time distributed median solver that we want to discretize in the manner illustrated in Section II is motivated by the blended dynamics approach and is given in [5] aṡ x i = sgn(o i − x i ) + k j∈Ni α ij (x j − x i )(1) where by increasing k sufficiently large, the network gets approximately synchronized to the median value of a collection O of real numbers o i , i = 1, . . . , N . The function sgn : R → R denotes the signum function defined as sgn(s) = s/|s| for non-zero s, and sgn(s) = 0 for s = 0. In this paper, the median is defined as a real number that belongs to the set M O := {o s (N +1)/2 }, if N is odd [o s N/2 , o s N/2+1 ], if N is even where o s i 's are the elements of O with its index being sorted (rearranged) such that o s 1 ≤ o s 2 ≤ · · · ≤ o s N . With the help of this relaxed definition of the median, finding the median of O becomes solving a simple optimization problem minimize x N i=1 |o i − x|. Then, the gradient descent algorithm given bẏ x = N i=1 sgn(o i −x) solves the minimization problem; lim t→∞ x(t) M O = 0. 2 This motivates the network (1) according to the blended dynamics approach introduced in Section II. In this manner, we discretize the network (1) accordingly as x i [n + 1] = S k i   x i [n] + k j∈Ni α ij x j [n]   , where S k i (·) is a left inverse of (1+kd i )x−sgn(o i −x). Since (1 + kd i )x − sgn(o i − x) =      (1 + kd i )x − 1, if x < o i , (1 + kd i )x, if x = o i , (1 + kd i )x + 1, if x > o i , we obtain S k i (x) =      x+1 1+kdi , if x < (1 + kd i )o i − 1, x−1 1+kdi , if x > (1 + kd i )o i + 1, x 1+kdi , otherwise. In particular, the network is simply x i [n + 1] = 1 1 + kd i   1 + x i [n] + k j∈Ni α ij x j [n]   when x i [n] + k j∈Ni α ij x j [n] < (1 + kd i )o i − 1 = 1 1 + kd i   −1 + x i [n] + k j∈Ni α ij x j [n]   when x i [n] + k j∈Ni α ij x j [n] > (1 + kd i )o i + 1 = 1 1 + kd i   x i [n] + k j∈Ni α ij x j [n]   otherwise(2) for i ∈ N . We have the following convergence result. Theorem 1: Under the assumption that the communication graph induced by the adjacency element α ij is undirected and connected, for any > 0, there exists k * > 0 such that, for each k > k * and initial condition x i [0] ∈ R, i ∈ R, the solution to (2) exists for all n ∈ N, and satisfies lim sup n→∞ x i [n] M O ≤ for all i ∈ N . Proof: Note first that the network (2) can be written as w k :=    (1 + kd 1 )/ N i=1 (1 + kd i ) . . . (1 + kd N )/ N i=1 (1 + kd i )    is the left eigenvector of B k associated with the unique eigenvalue 1. The uniqueness comes from the connectivity of the network. Therefore, we obtain w T k X[n + 1] = w T k X[n] + N i=1 (1 + kd i )s i [n] N i=1 (1 + kd i ) and B n k − 1 N w T k < C k q n k with some C k > 0 and q k ∈ (0, 1) [13], [14]. In particular, lim k→∞ C k = max i d i / min i d i =: C ∞ < ∞ and lim k→∞ q k =: q ∞ ∈ (0, 1). See Appendix A for its illustration. Now, since X[n] = B n k X[0] + n i=1 B n−i k S[i − 1] we can conclude that (I N − 1 N w T k )X[n] = (B n k − 1 N w T k )X[0] + n i=1 (B n−i k − 1 N w T k )S[i − 1]. Therefore, (I n − 1 N w T k )X[n] ≤ B n k − 1 N w T k X[0] + N i=1 B n−i k −1 N w T k S[i − 1] ≤ C k q n k X[0] + n i=1 C k q n−i k √ N 1 + k min i d i ≤ C k q n k X[0] + C k 1 − q k √ N 1 + k min i d i . This implies that for any > 0, there exists k * such that for each k ≥ k * and initial condition X[0], there exists n * ∈ N such that (I n − 1 N w T k )X[n] ≤ 3(3) for all n ≥ n * . Note that C k /(1 − q k ) > 1, and thus, we have 3 > 1 1 + kd i , ∀i ∈ N for such k. Now, since we have proved arbitrary precision approximate synchronization, let us recall how the averaged variable w T k X[n] behaves; w T k X[n + 1] = w T k X[n] + N i=1ŝ i [n] N i=1 (1 + kd i ) whereŝ i [x i [n] + k j∈Ni α ij x j [n] > (1 + kd i )o i + 1, and thus, for n ≥ n * , if w T k X[n] ≥ o i + 2 3 , then by (3), we have which concludes the proof with the help of (3). Now, in the next section, we observe in the simulation result, a dramatic removal of the chattering phenomenon, compared to the network obtained by the explicit method. x i [n] + k j∈Ni α ij x j [n] ≥ (1 + kd i ) w T k X[n] − 3 ≥ (1 + kd i ) o i + 3 > (1 + kd i )o i + 1 henceŝ i [n] = −1. Similarly, if w T k X[n] ≤ o i − 2 3 ,then IV. SIMULATION To compare the chattering phenomenon in the network, we consider a simple network consisting of three agents, where o 1 = 0, o 2 = 1, and o 3 = 100. The communication graph is complete and unitary; α ij = 1 for all i = j. The simulation result of the network (2) with k = 10 is given in Figure 1. On the other hand, if we simulate a network obtained by discretizing (1) with the explicit method, given as x i [n + 1] = x i [n] + T s sgn(o i − x i [n]) + kT s j∈Ni α ij (x j [n] − x i [n])(4) for sufficiently small time step T s > 0, then with T s = 0.05, we get the trajectories illustrated in Figure 2. Here, by slightly increasing the time step to T s = 0.07, we get unstable trajectories. Note that we observe not only a bigger steady-state error, but also the chattering phenomenon. To recover the accuracy in the steady-state limit, we should employ T s = 0.005, which results in the simulation result given in Figure 3, but it requires a larger number of iteration. V. CONCLUSION By studying for a particular example of distributed median solver, we have seen the utility of the Implicit-Explicit Method, in discretizing a network constructed by the blended dynamics approach, which for the explicit method, by its stiffness in the dynamics, suffers from a problem like the need for a smaller time step, which depends on the coupling gain. Moreover, the method has proven its ability to remove the chattering phenomenon when discretizing a system having discontinuity in its vector field. The future consideration will be on the general conclusion of the use of the Implicit-Explicit Method on such class of networks and also on the analytical verification of its advantages compared with other methods. APPENDIX First define the Laplacian matrix L = [l ij ] ∈ R N ×N of a graph as L := D−A, where A = [α ij ] is the adjacency matrix of the graph and D is the diagonal matrix whose diagonal entries are d i , i ∈ N . By its construction, it contains at least one eigenvalue of zero, whose corresponding eigenvector is 1 N := [1, . . . , 1] T ∈ R N , and all the other eigenvalues have nonnegative real parts. For undirected graphs, the zero eigenvalue is simple if and only if the corresponding graph is connected. Moreover, I N −D −1 L has its eigenvalues contained inside the unit circle and the eigenvalue with magnitude one becomes unique if and only if the graph is connected. Then, we have the representation I N − B k = diag k 1 + kd 1 , . . . , k 1 + kd N L =: D k L, and thus, I N − D k −1 B k D k = D k L D k =: L k . Since L k is a symmetric positive semi-definite matrix, there exists normalized eigenvectors v 1,k , . . . , v N,k associated with eigenvalues 0 < λ 2,k ≤ · · · ≤ λ N,k such that L k v i,k = λ i,k v i,k and v T i,k L k = λ i,k v T i,k for i = 2, . . . , N and L k v 1,k = 0, v T 1,k L k = 0. This implies that D k −1 B k D k = V k diag(1, 1 − λ 2,k , . . . , 1 − λ N,k )V T k where V k = [v 1,k · · · v N,k ]. Therefore, by noting that v 1,k is the normalized vector of √ D k −1 1 N , hence D k V k diag(1, 0, . . . , 0)V T k D k −1 = 1 N w T k we can conclude that B n k − 1 N w T k = D k V k diag (0, (1 − λ 2,k ) n , . . . , (1 − λ N,k ) n )V T k D k −1 . This finally implies B n k − 1 N w T k ≤ C k q n k where C k := D k D k −1 = 1 + k max i d i 1 + k min i d i , q k := max {|1 − λ 2,k | , |1 − λ N,k |} . Now, noting that L k has the same set of eigenvalues with the matrix D k L, by the Gershgorin circle theorem, we can certify that λ i,k ∈ [0, 2), hence q k < 1. In particular, we have lim k→∞ C k = max i d i min i d i lim k→∞ q k = max {|1 − λ 2,∞ | , |1 − λ N,∞ |} where 0 < λ 2,∞ ≤ · · · ≤ λ N,∞ < 2 are the eigenvalues of diag(1/d 1 , . . . , 1/d N )L = D ∞ L = D −1 L. arXiv:2106.09899v1 [eess.SY] 18 Jun 2021 X[n + 1 ] 1= B k X[n] + S[n] where X[n] = [x 1 [n] · · · x N [n]] T , S[n] = [s 1 [n] · · · s N [n]] T , (1 + kd i )s i [n] ∈ {−1, 0, 1}, and B k is a stochastic matrix. Note also that n] := (1 + kd i )s i [n] ∈ {−1, 0, 1}. This implies that if the overall balance N i=1ŝ i [n] is positive, then the averaged value increases, while if the balance is negative, then the value decreases. On the other hand, by its constructionŝ i [n] = −1 if and only if (see (2)) Fig. 1 . 1Simulation result of the network (2) with initial conditions x 1 [0] = 0, x 2 [0] = 1, and x 3 [0] = 1.5. Fig. 2 . 2Simulation result of the network (4) with Ts = 0.05 and initial conditions x 1 [0] = 0, x 2 [0] = 1, and x 3 [0] = 1.5. This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(Ministry of Science and ICT) (No. 2019R1A6A3A12032482). J. G. Lee is with Control Group, Department of Engineering, University of Cambridge, United Kingdom. [email protected] In particular, even to ensure stability in the obtained network by the explicit method, the time step has to be well-selected to be sufficiently small. This problem does not arise in the following approach, hence leaves the coupling gain k to be the only global parameter that we have to tune. For a set Ξ, x Ξ denotes the distance between the vector x and Ξ, i.e., x Ξ := inf y∈Ξ x − y . Distributed coordination of multi-agent networks: emergent problems, models, and issues. W Ren, Y Cao, SpringerW. Ren and Y. Cao, Distributed coordination of multi-agent networks: emergent problems, models, and issues. Springer, 2010. Finite-time consensus on the median value by discontinuous control. 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We prove a 'resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever d is sufficiently large compared to ε > 0, a.a.s. the following holds: let G ′ be any subgraph of the random n-vertex d-regular graph G n,d with minimum degree at least (1/2 + ε)d. Then G ′ is Hamiltonian.This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that d is large cannot be omitted, and secondly, the minimum degree bound cannot be improved.

10.1017/s0963548320000346

1903.05052

04256e2b64475ff7896a8f45e8cc3be90435378b

DIRAC'S THEOREM FOR RANDOM REGULAR GRAPHS 12 Mar 2019 Padraig Condon AND DERYK OSTHUS Alberto Espuny Díaz AND DERYK OSTHUS António Girão AND DERYK OSTHUS Daniela Kühn AND DERYK OSTHUS DIRAC'S THEOREM FOR RANDOM REGULAR GRAPHS 12 Mar 2019arXiv:1903.05052v1 [math.CO] We prove a 'resilience' version of Dirac's theorem in the setting of random regular graphs. More precisely, we show that, whenever d is sufficiently large compared to ε > 0, a.a.s. the following holds: let G ′ be any subgraph of the random n-vertex d-regular graph G n,d with minimum degree at least (1/2 + ε)d. Then G ′ is Hamiltonian.This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly, the condition that d is large cannot be omitted, and secondly, the minimum degree bound cannot be improved. Introduction The study of Hamiltonicity has been at the core of graph theory for the past few decades. A graph G is said to be Hamiltonian if it contains a cycle which covers all of the vertices of G, and this is called a Hamilton cycle. It is well-known that the problem of determining whether a graph is Hamiltonian is NP-complete, and thus most results about Hamiltonicity deal with sufficient conditions which guarantee this property. One of the most well-known examples is due to Dirac, who proved that any graph G on n ≥ 3 vertices with minimum degree at least n/2 is Hamiltonian. 1.1. Hamilton cycles in random graphs. The search for Hamilton cycles in various models of random graphs has also been a driving force in the development of this theory. The classical binomial model G n,p , in which each possible edge is added to an n-vertex graph with probability p independently of the other edges, has seen many results in this direction. In particular, Komlós and Szemerédi [23] showed that p = log n/n is the 'sharp' threshold for the existence of a Hamilton cycle. This can be strengthened to obtain the following hitting time result. Consider a random graph process as follows: given a set of n vertices, add each of the n 2 possible edges, one by one, by choosing the next edge uniformly at random among those that have not been added yet. In this setting, Ajtai, Komlós and Szemerédi [1] and Bollobás [10] independently proved that a.a.s. the resulting graph becomes Hamiltonian as soon as its minimum degree is at least 2. The search for Hamilton cycles in other random graph models has proven more difficult. In this paper we will deal with random regular graphs: given n, d ∈ N such that d < n and nd is even, G n,d is chosen uniformly at random from the set of all d-regular graphs on n vertices. The study of this model is often more challenging than that of G n,p due to the fact that the presence and absence of edges in G n,d are correlated. Several different techniques have been developed to deal with this model, such as the configuration model (see Section 3.3) or edgeswitching techniques. Robinson and Wormald [33] proved that G n,3 is a.a.s. Hamiltonian, and later extended this result to G n,d for any fixed d ≥ 3 [34]. This is in contrast to G n,p , where the average degree must be logarithmic in n to ensure Hamiltonicity. These results were later generalised by Cooper, Frieze and Reed [14] and Krivelevich, Sudakov, Vu and Wormald [26] for the case when d is allowed to grow with n, up to d ≤ n − 1. Many further results can be found in the recent survey of Frieze [16]. 1.2. Local resilience. More recently, several extremal results have been translated to random graphs via the concept of local resilience. The local resilience of a graph G with respect to some property P is the maximum number r ∈ N such that, for all H ⊆ G with ∆(H) < r, the graph G \ H satisfies P. We say that G is r-resilient with respect to a property P if the local resilience of G is greater than r. The systematic study of local resilience was initiated by Sudakov and Vu [36], and the subject has seen a lot of research since. Note that Dirac's theorem can be restated in this terminology to say that the local resilience of the complete graph K n with respect to Hamiltonicity is ⌊n/2⌋. This concept of local resilience then naturally suggests a generalisation of Dirac's theorem to random graphs. In the binomial model, Lee and Sudakov [27] showed that, for any constant ε > 0, if p ≥ C log n/n and C is sufficiently large, then a.a.s. G n,p is (1/2 − ε)np-resilient with respect to Hamiltonicity. This improved on earlier bounds [7,8,17,36]. Very recently, Montgomery [30] and independently Nenadov, Steger and Trujić [32], proved a hitting time result for the local resilience of G n,p with respect to Hamiltonicity. In a different direction, Condon, Espuny Díaz, Kim, Kühn and Osthus [12] considered 'resilient' versions of Pósa's theorem and Chvátal's theorem for G n,p . The resilience of random regular graphs with respect to Hamiltonicity is less understood. Ben-Shimon, Krivelevich and Sudakov [7] proved that, for large (but constant) d, a.a.s. G n,d is (1 − ε)d/6-resilient with respect to Hamiltonicity. They conjectured that the true value should be closer to d/2. Conjecture 1.1 (Ben-Shimon, Krivelevich and Sudakov [7]). For every ε > 0 there exists an integer D = D(ε) > 0 such that, for every fixed integer d > D, the local resilience of G n,d with respect to Hamiltonicity a.a.s. lies in the interval ((1/2 − ε)d, (1/2 + ε)d). They also suggested to study the same problem when d is allowed to grow with n. In this direction, Sudakov and Vu [36] showed that, for any fixed ε > 0, and for any (n, d, λ)-graph G (that is, a d-regular graph on n vertices whose second largest eigenvalue in absolute value is at most λ) with d/λ > log 2 n, we have that G is (1/2 − ε)d-resilient with respect to Hamiltonicity. This, together with a result of Krivelevich, Sudakov, Vu and Wormald [26] and recent results of Cook, Goldstein and Johnson [13] and Tikhomirov and Youssef [37] about the spectral gap of random regular graphs, implies that, for log 4 n ≪ d ≤ n − 1, a.a.s. G n,d is (1/2 − ε)d-resilient with respect to Hamiltonicity. One can extend this to d ≫ log n by combining a result of Kim and Vu [22] on joint distributions of binomial random graphs and random regular graphs with the result of Lee and Sudakov [27] about the resilience of G n,p with respect to Hamiltonicity. The study of local resilience has not been restricted to Hamiltonicity. Other properties that have been considered include the containment of perfect matchings [12,32], directed Hamilton cycles [15,18,31], cycles of all possible lengths [25], k-th powers of cycles [35], bounded degree trees [5], triangle factors [6], and bounded degree graphs [2,20]. 1.3. New results. In this paper, we completely resolve Conjecture 1.1, as well as its extension to d growing slowly with n (recall that the case when d ≫ log n is covered by earlier results). This can be seen as a version of Dirac's theorem for random regular graphs. Our main result gives the lower bound in Conjecture 1.1. Theorem 1.2. For every ε > 0 there exists D such that, for every D < d ≤ log 2 n, the random graph G n,d is a.a.s. (1/2 − ε)d-resilient with respect to Hamiltonicity. While we do not try to optimise the dependency of D on ε, we remark that D in Theorem 1.2 can be taken to be polynomial in ε −1 . This is essentially best possible in the sense that Theorem 1.2 fails if d ≤ (2ε) −1 . Our proof also shows that G n,d is not a.a.s. (d−1)/2-resilient with respect to the containment of a perfect matching. Moreover, one can adapt the proof of Theorem 1.3 to show that, for every even d, the random graph G n,d is not a.a.s. d/2-resilient with respect to Hamiltonicity (or the containment of a perfect matching). It would also be interesting to obtain bounds on the resilience for small d. In particular, here are some questions: i) Given any fixed even d, determine whether the graph G n,d is a.a.s. (d/2 − 1)-resilient with respect to Hamiltonicity. ii) What is the likely resilience of G n,4 with respect to Hamiltonicity or the containment of perfect matchings? Is a graph obtained from G n,4 by removing any matching a.a.s. Hamiltonian? Finally, we observe (as is well known) that the upper bound of (1/2 + ε)d in Conjecture 1.1 follows easily from edge distribution properties of random regular graphs. Indeed, we note that for every ε > 0 there exists a constant D such that for every D ≤ d ≤ log 2 n, a.a.s. the graph G = G n,d has the property that between any two disjoint sets A, B of size ⌊n/2⌋ and ⌈n/2⌉, respectively, the number of edges in G[A, B] is a.a.s. bounded from above by (1/2 + ε/2)nd/2 (see Proposition 4.4). Now let A, B be a maximum cut in G. Thus e G (a, B) ≥ d/2 for all a ∈ A, and similarly for all b ∈ B. If |A| = |B|, then by deleting the edges in G[A] ∪ G[B], the remaining graph is not Hamiltonian since it forms an unbalanced bipartite graph. If |A| = |B|, then by the above property, there must exist a vertex x ∈ A such that e G (x, B) ≤ (1/2 + ε/2)d. 1.4. Organisation of the paper. The remainder of the paper is organised as follows. In Section 2 we give a sketch of the proof of Theorem 1.2. In Section 3 we collect notation, some probabilistic tools, and observations about the configuration model. Section 4 is devoted to proving different edge-distribution and expansion properties of random regular graphs and their subgraphs, and the proof of Theorem 1.2 is given in Section 5, using all the techniques that have been introduced before. Finally, we prove Theorem 1.3 in Section 6. 2. Outline of the proof of Theorem 1.2 Consider G = G n,d . Let H ⊆ G be such that ∆(H) ≤ (1/2 − ε)d and let G ′ := G \ H. We will prove that G ′ contains a 'sparse' spanning subgraph R which has strong edge expansion properties. These properties will then be used to provide a lower bound on the number of edges in G whose addition would make R Hamiltonian, or increase the length of a longest path in R (such edges are commonly called 'boosters', see e.g. [24]). We then argue that some of these edges must in fact be retained when passing to G ′ . We then add such edges to R and iterate the above process (at most n times) until R becomes Hamiltonian. More specifically, as a preliminary step we 'thin' the graph G ′ , that is, we take a subgraph R ⊆ G ′ with ∆(R) ≤ δd, for some δ ≪ ε. As described above, we consider a longest path in R and then argue that it can be extended via edges in G ′ \ R. The fact that R is relatively 'sparse' with respect to G ′ will be important when calculating union bounds over all graphs R of this type, at a later stage in the proof. Given many paths of maximum length and with different endpoints in R, it follows that there will be many edges whose addition will increase the length of a longest path (or make R Hamiltonian). A theorem of Pósa implies that graphs with strong expansion properties will indeed contain many of such paths. These expansion properties are captured by the notion of a 3-expander (see Definition 4.1). Therefore, we wish to show that our thinned graph R can be chosen to be a 3-expander. This is one point where working with the random graph G n,d proves more difficult than working with G n,p , due to the fact that the appearance of edges in G n,d is correlated. The next step is to provide a lower bound on the number of edges whose addition to R would increase the length of a longest path (or make R Hamiltonian). Here we further develop an approach of Montgomery [30] who, instead of considering single edges that would bring R closer to being Hamiltonian, considered 'booster' edge pairs whose addition would yield the same result. For example, if R is connected and P is a longest path in R with endpoints x and y, and ab is an edge of P (with b closer to y on P ), then {ya, xb} is a booster pair. The main advantage of considering such pairs of edges is that it results in a much larger set of boosters for R. More precisely, we show the existence of another thinned graph F ⊆ G ′ \ R for which each booster we consider is of the form {e, e ′ }, where e ∈ E(F ) and e ′ ∈ E(G ′ ) (see Corollary 5.6). Finally, we can complete the proof of the main theorem by iteratively adding booster pairs to the thinned graph R, increasing the length of a maximum path in each step until R becomes Hamiltonian. Two points are important here as to why we can iterate this process. First, proving the existence of boosters (see Lemma 5.7) involves a union bound over all pairs of thinned graphs R and F. To bound this efficiently, we need that both R and F are relatively 'sparse' with respect to G ′ . But in each step we only add two booster edges to R, so it remains sparse. Secondly, we take special care to ensure that no vertex is contained in too many of the boosters we add to R, ensuring that its degree in successive iterations remains small. This process terminates after at most n iterations, resulting in a graph R ′ ⊆ G ′ which is Hamiltonian. That is, whenever we claim that a result holds for 0 < a ≪ b ≤ 1, we mean that there exists a non-decreasing function f : [0, 1) → [0, 1) such that the result holds for all a > 0 and all b ≤ 1 with a ≤ f (b). We will not compute these functions explicitly. Throughout this paper, the word graph will refer to a simple, undirected graph. Whenever the graphs are allowed to have parallel edges or loops, we will refer to these as multigraphs. Given any (multi)graph G = (V, E) and sets A, B ⊆ V , we will denote the (multi)set of edges of G spanned by A as E G (A), and the (multi)set of edges of G having one endpoint in A and one endpoint in B as E G (A, B). The number of such edges will be denoted by e G (A) and e G (A, B), respectively. We will also write e(G) for e G (V ). Given two (multi)graphs G 1 and G 2 on the same vertex set V , we write G 1 + G 2 := (V, E(G 1 ) ∪ E(G 2 )), where the union represents set union for graphs and multiset union for multigraphs. When G 1 and G 2 are graphs, we write G 1 \ G 2 := (V, E(G 1 ) \ E(G 2 ) ). Given any vertex v ∈ V , we will denote the set of vertices which are adjacent to v in G by N G (v). We define N G (A) := v∈A N G (v). The degree of vertex v in a multigraph G is d G (v) := |{e ∈ E(G) : v ∈ e}| + |{e ∈ E(G) : e = vv}| (i.e. each loop at v contributes two to d G (v)). We denote ∆(G) := max v∈V d G (v) and δ(G) := min v∈V d G (v). The (multi)graph G is said to be d-regular for some d ∈ N if all vertices have degree d. Given a multigraph G on [n], we refer to the vector d = (d G (1), . . . , d G (n)) as its degree sequence. In general, a vector d = (d 1 , . . . , d n ) with d i ∈ Z ≥0 for all i ∈ [n] is called graphic if there exists a graph on n vertices with degree sequence d (note that, as long as n i=1 d i is even, there is always a multigraph with degree sequence d). Given a graph G and a real number α > 0, let H α (G) be the collection of all spanning subgraphs H ⊆ G for which d H (v) ≤ αd G (v), for all v ∈ V (G). We will use G n,d to denote the set of all d-regular graphs on vertex set [n], and G n,d will denote a graph chosen from G n,d uniformly at random. Whenever we use this notation, we implicitly assume that nd is even. In more generality, given a graphic degree sequence d = (d 1 , . . . , d n ), we will denote the collection of all graphs on vertex set [n] with degree sequence d by G n,d , and G n,d will denote a graph chosen from G n,d uniformly at random. We use a.a.s. as an abbreviation for asymptotically almost surely. Given a sequence of events {E n } n∈N , whenever we claim that E n holds a.a.s., we mean that the probability that E n holds tends to 1 as n tends to infinity. For the purpose of clarity, we will ignore rounding issues when dealing with asymptotic statements. By abusing notation, given p ≥ 0 and n ∈ N, we write Bin(n, p) for the binomial distribution with parameters n and min{p, 1}. 3.2. Probabilistic tools. We will need the following Chernoff bound (see e.g. [ ≤ δ ≤ 1 we have that P[|X − µ| ≥ δµ] ≤ 2e −δ 2 µ/3 . The following bound will also be used repeatedly (see e.g. [3, Theorem A.1.12]). Lemma 3.2. Let X ∼ Bin(n, p), and let β > 1. Then, P[X ≥ βnp] ≤ (e/β) βnp . Given any sequence of random variables X = (X 1 , . . . , X n ) taking values in a set A and a function f : A n → R, for each i ∈ [n] ∪ {0} define Y i := E[f (X) | X 1 , . . . , X i ]. The sequence Y 0 , . . . , Y n is called the Doob martingale for f . All the martingales that appear in this paper will be of this form. To deal with them, we will need the following version of the well-known Azuma-Hoeffding inequality. [4,19]). Let X 0 , X 1 , . . . be a martingale and suppose that Lemma 3.3 (Azuma's inequality|X i − X i−1 | ≤ c i for all i ∈ N. Then, for any n, t ∈ N, P[|X n − X 0 | ≥ t] ≤ 2 exp −t 2 2 n i=1 c 2 i . Finally, the Lóvasz local lemma will come in useful. Let E := {E 1 , E 2 , . . . , E m } be a collection of events. A dependency graph for E is a graph H on vertex set [m] such that, for all i ∈ [m], E i is mutually independent of {E j : j = i, j / ∈ N H (i)}, that is, if P[E i ] = P[E i | j∈J E j ] for all J ⊆ [m] \ (N H (i) ∪ {i}) . We will use the following version of the local lemma (it follows e.g. from [3, Lemma 5.1.1]). Lemma 3.4 (Lóvasz local lemma). Let E := {E 1 , E 2 , . . . , E m } be a collection of events and let H be a dependency graph for E. Suppose that ∆(H) ≤ d and P[E i ] ≤ p for all i ∈ [m]. If ep(d + 1) ≤ 1, then P m i=1 E i ≥ (1 − ep) m . 3.3. The configuration model. We will work with the configuration model introduced by Bollobás [9], which can be used to sample d-regular graphs uniformly at random. In more generality, it can be used to produce graphs with any given graphic degree sequence d. The process to generate such graphs is as follows. Given n ∈ N and a degree sequence d = (d 1 , . . . , d n ) with m := n i=1 d i even,e = x ij x kℓ , for some i, k ∈ [n], j ∈ [d i ] and ℓ ∈ [d k ], and add ik to E (if i = k, this adds a loop to E). When we consider a multigraph G obtained via this configuration model, this will be denoted by G ∼ C n,d . In particular, when we obtain a d-regular multigraph via the configuration model, we will denote this by G ∼ C n,d . We refer to the possible perfect matchings on the expanded set of [n] as configurations, and we will denote a configuration obtained uniformly at random by M ∼ C * n,d . By abusing notation, we will sometimes also use C * n,d to denote the set of all configurations with parameters n and d. In order to easily distinguish the setting of graphs from that of configurations, we will call the elements of the expanded sets points, and each element in a configuration will be called a pairing. The above process may produce a multigraph with loops and/or multiple edges. However, if d is a graphic degree sequence, then, when conditioning on the resulting multigraph being simple, the configuration model yields a graph G ∈ G n,d chosen uniformly at random. The following proposition bounds the probability that this happens, and can be proved similarly to (part of) a result of Cooper, Frieze and Reed [14,Lemma 7] (see [11] for details). It will be useful when analysing the distribution of edges in G n,d via the configuration model. d ′ = (d 1 , . . . , d n ) such that d i < δd for all i ∈ [n]. Let d := (d − d 1 , . . . , d − d n ) and let F ∼ C n,d . Then, if n is sufficiently large, P[R + F is simple] ≥ e −3d 2 . Note that, by choosing R to be the empty graph on n vertices, we obtain a lower bound on the probability that the multigraph obtained by a random configuration is simple. When studying the configuration model, it will be useful to consider the following process to generate M ∼ C * n,d . Let d = (d 1 , . . . , d n ) and suppose that m := n i=1 d i is even. Label the points of the expanded set of [n] in any arbitrary order, x 1 , . . . , x m , and identify them naturally with the set [m]. Start with an empty set of pairings M 0 . Inductively, for each i ∈ [m], if i is covered by M i−1 , let M i := M i−1 ; otherwise, choose a point j ∈ [m] \ (V (M i−1 ) ∪ {i}) uniformly at random and define M i := M i−1 ∪ {ij}. We sometimes refer to M i as the i-th partial configuration. Finally, let M := M m . It is clear that the resulting configuration M is generated uniformly at random, independently of the labelling of the expanded set of [n]. We will often be interested in bounding the number of edges in G n,d between two sets of vertices. For this, it will be useful to consider binomial random variables that stochastically dominate the number of edges. We formalise this via the following lemma. In order to give bounds on the distribution of edges in G n,d we will use an edge-switching technique, first introduced by McKay and Wormald [29]. We will consider the following switching. Lemma 3.6. Let n, d ∈ N with d < n, and let δ ∈ [0, 1). Let d = (d 1 , . . . , d n ) with n i=1 d i even be such that (1 − δ)d ≤ d i ≤ d for all i ∈ [n]. Let G ∼ C n,= |E ′ |. We write G ∼ G ′ if there exist u 1 u 2 , v 1 v 2 ∈ E such that E ′ = (E \ {u 1 u 2 , v 1 v 2 }) ∪ {u 1 v 1 , u 2 v 2 }. The following lemma bounds the probability that certain variables on configurations deviate from their expectation. k ∈ [m] \ (V (M i−1 ) ∪ {i, j}), label M k by M k,ℓ := g j,k (M j,ℓ ) for each ℓ ∈ [N ]. By assumption, we have |X(M j,ℓ ) − X(M k,ℓ )| ≤ c for all distinct j, k ∈ [m] \ (V (M i−1 ) ∪ {i}) and ℓ ∈ [N ]. Using this, it is easy to conclude that |Y i (M ) − Y i−1 (M )| ≤ c. The statement now follows by Lemma 3.3. The following proposition implies that the distribution of edges in G n,d behaves roughly as in a binomial random graph G n,d/n , even after conditioning on the containment of some 'sparse' subgraph. Proposition 4.4. For every 0 < ε ≤ 1/2 there exists δ > 0 such that the following holds. Let d ≤ log 2 n be a positive integer and let G = G n,d . Let R be a graph on vertex set [n] with ∆(R) < δd. Moreover, let A ⊆ [n] and, for each a ∈ A, let Z a ⊆ [n] (2) \ E(R) be a collection of edges incident to a such that z := a∈A |Z a | satisfies z > εn 2 . Then, P a∈A |Z a ∩ E(G)| − zd n ≥ ε zd n R ⊆ G ≤ e −(ε/10) 4 nd . Proof. Let 0 < δ ≪ ε. For each i ∈ [n], let d i := d − d R (i) > (1 − δ)d, and let d := (d 1 , . . . , d n ). Let M ∼ C * n,d and let F = ϕ(M ), so that F ∼ C n,d and R + F is a d-regular multigraph. By Lemma 3.6, for each a ∈ A, the random variable Y a ∼ Bin(d a , (n − |Z a |)/((1 − δ)n − 2)) stochastically dominates e F (a, [n] \ (V (Z a ) \ {a})). Let Z(F ) := a∈A |Z a ∩ E(F )|. Note that E[Y a ] < (1 + ε 3 )d a (n − |Z a |)/n for all a ∈ A. It then follows that E[|Z a ∩ E(F )|] ≥ d a − E[Y a ] ≥ (1 + ε 3 )|Z a |d a /n − ε 3 d a . Therefore, we have E[Z(F )] ≥ (1 − ε 2 )zd/n. Now, note that |Z(F ) − Z(F ′ )| ≤ 8 when F ∼ F ′ . Let Z ′ : C * n,d → Z be such that Z ′ (M ) = Z(F ) whenever ϕ(M ) = F . It follows that |Z ′ (M ) − Z ′ (M ′ )| ≤ 8 when M ∼ M ′ . Moreover, E[Z ′ (M )] = E[Z(F )] . Therefore, we can apply Lemma 4.3 to obtain P Z ′ (M ) ≤ (1 − ε) zd n ≤ 2e −ε 4 nd/512 . By definition, the same bound holds for Z(F ). It now follows from Proposition 3.5 that P Z(F ) ≤ (1 − ε) zd n R + F is simple ≤ 2e 3d 2 e −ε 4 nd/512 . (4.1) By a similar argument we can show that P Z(F ) ≥ (1 + ε) zd n R + F is simple ≤ 2e 3d 2 e −ε 4 nd/512 . (4.2) The result follows by combining (4.1) and (4.2). Lemma 4.5. For every 0 < δ < 10 −5 there exists D ∈ N such that for any D < d ≤ log 2 n we have that a.a.s. the random graph G n,d satisfies the following properties. (i) For every S ⊆ [n] with δ 2 d ≤ |S| ≤ 5δ 2 n, we have e G n,d (S) ≤ δd|S|/25. (ii) For every S ⊆ [n] with 5δ 2 n ≤ |S| ≤ n/100, we have e G n,d (S) ≤ d|S|/25. Proof. Let 1/D ≪ δ. For any D ≤ d ≤ log 2 n, let G ∼ C n,d . For each S ⊆ [n] such that δ 2 d ≤ |S| ≤ 5δ 2 n and any multigraph F on [n], let g(S, F ) be the event that e F (S) ≤ δd|S|/25. It follows by Lemma 3.6 that the variable e G (S) is stochastically dominated by Y ∼ Bin(d|S|, 5|S|/(4n)). We denote byP the probability measure associated with the configuration model and let P be the measure associated with the space of (simple) d-regular graphs. Therefore, by Lemma 3.2 we havê P[g(S, G)] ≤P [e G (S) ≥ δd|S|/25] =P e G (S) ≥ (4δn/(125|S|))5d|S| 2 /(4n) < (|S|/en) 2|S| . It follows by Proposition 3.5 that Proposition 4.6. For every 0 < δ < 10 −5 there exists D ∈ N such that for any D < d ≤ log 2 n we have that a.a.s. the random graph G = G n,d satisfies the following properties. P S⊆[n] δ 2 d≤|S|≤5δ 2 n g(S, G n,d ) =P S⊆[n] δ 2 d≤|S|≤5δ 2 n g(S, G) | G is simple ≤ e 3d 2 S⊆[n] δ 2 d≤|S|≤5δ 2 nP [g(S, G)] ≤ e 3d 2 (i) Let R ⊆ G be a spanning subgraph with δ(R) > δd. Then, for every S ⊆ [n] with |S| ≤ δ 2 n, we have |N R (S)| ≥ 3|S|. (ii) For every S, S ′ ⊆ [n] with δ 2 n ≤ |S| ≤ |S ′ | ≤ 3|S| ≤ 3n/400, we have e G (S, S ′ ) ≤ d|S|/5. Proof. Let 1/D ≪ δ and condition on the statement of Lemma 4.5 holding, which occurs a.a.s. We first prove (i). For each S ⊆ [n] such that |S| < δ 2 d, the fact that every vertex has degree at least δd ensures that Proposition 4.7. For every 0 < δ < 10 −5 there exists D ∈ N such that for any D < d ≤ log 2 n we have that a.a.s. the random graph G = G n,d has the following property. Let H ∈ H 1/2 (G) and let G ′ := G \ H. Then, there exists a spanning graph R ⊆ G ′ such that ∆(R) < δd and, for every S ⊆ [n] with |S| ≤ n/400, we have that |N R (S)| ≥ 3|S|. |N R (S)| ≥ δd > 3δ 2 d. Now let S ⊆ [n] with δ 2 d ≤ |S| ≤ δ 2 n. Suppose |N R (S)| < 3|S|. Let Y ⊆ [n] be such that |Y | = 3|S| and N R (S) ⊆ Y . We have by Lemma 4.5(i) that 4|S|δd/25 ≥ e G (S ∪ Y ) ≥ e R (S ∪ Y ) ≥ e R (S, Y ) ≥ |S|δd − e G (S) > |S|δd/2, Proof. Let 1/D ≪ δ and letδ := δ/8. Condition on the event that the statements of Lemma 4.5 and Proposition 4.6 hold withδ playing the role of δ, which happens a.a.s. Suppose G satisfies these events and H ∈ H 1/2 (G), and let G ′ := G \ H. We now construct a suitable R for this G ′ . Consider a random subgraph R of G ′ where each edge is chosen independently and uniformly at random with probability 4δ. Consider the following events. (G1) For all v ∈ [n] we haveδd < d R (v) < 8δd. (G2) For every S ⊆ [n] with |S| ≤ n/400, we have |N R (S)| ≥ 3|S|. Note that, if both (G1) and (G2) hold, then R is a subgraph which satisfies the properties in the statement of the lemma. For each v ∈ [n], let A v be the event that d R (v) / ∈ (δd, 8δd). By Lemma 3.1, we have P[A v ] < 4e −A v ≥ (1 − 12e −δd/6 ) n ≥ 2 −n . Next, for S, S ′ ⊆ [n], let g(S, S ′ ) be the event that N R (S) ⊆ S ′ . Let (G3) be the event that for no pair of subsets S, S ′ ⊆ [n] with S ′ ⊆ N G ′ (S) andδ 2 n ≤ |S| ≤ |S ′ | ≤ 3|S| ≤ 3n/400 the event g(S, S ′ ) occurs. We have by Proposition 4.6(ii) and Lemma 4.5 that e G ′ (S, [n] \ S ′ ) ≥ d|S|/2 − e G ′ (S, S ′ ) − e G ′ (S) ≥ d|S|/2 − d|S|/5 − d|S|/25 ≥ d|S|/5. Therefore, we have P[g(S, S ′ )] ≤ (1 − 4δ) d|S|/5 ≤ e −4δd|S|/5 ≤ 2 −4n . A union bound implies that P[R fails to satisfy (G3)] ≤ 2 2n 2 −4n < 2 −n . Therefore, there exists an instance of R which satisfies both (G1) and (G3) simultaneously. Furthermore, since R satisfies (G1), it follows by Proposition 4.6(i) that for every S ⊆ [n] with |S| ≤δ 2 n we have that |N R (S)| ≥ 3|S|. Combining this with (G3) we see that R also satisfies (G2). Thus, R is a subgraph of the desired form. Proposition 4.8. For every ε > 0 there exists D > 0 such that for any D < d ≤ log n 2 we have that a.a.s. the random graph G = G n,d has the following property. Let H ∈ H 1/2−ε (G) and let G ′ := G \ H. Let R ⊆ G ′ be a spanning graph such that for every S ⊆ [n] with |S| ≤ n/400, we have |N R (S)| ≥ 3|S|. Then, there exists a spanning 3-expander R ′ ⊆ G ′ with e(R ′ ) ≤ e(R)+400. Proof. Let 1/D ≪ ε. We first note that a.a.s., for any A, B ⊆ [n] with |A| = n/400 and |B| = (1/2 − ε/10)n, we have a∈A e G (a, B) > (1/2 − ε/5)|A|d. Indeed, this follows by an application of Proposition 4.4 with R := ∅ and Z a being the star with centre a whose leaves are all the vertices in B \ {a}. We now claim that for any A ⊆ [n] with |A| ≥ n/400 we have that |N G ′ (A)| ≥ (1/2 + ε/10)n. (4.3) To see this, note that if there exists A ⊆ [n] with |A| ≥ n/400 and |N G ′ (A)| < (1/2 + ε/10)n then we may take subsets A ′ ⊆ A with |A ′ | = n/400 and B ⊆ [n] with |B| = (1/2 − ε/10)n such that e G ′ (A ′ , B) = 0. However, we have already noted that for such A ′ and B we have that a∈A ′ e G (a, B) ≥ (1/2 − ε/5)|A ′ |d. It follows that there exists a ∈ A ′ with e G (a, B) > (1/2 − ε/5)d and therefore e G ′ (a, B) > 0. Thus, no such A and B exist. In particular, this implies that G ′ is connected. Indeed, assume that G ′ is not connected and let A [n] be a (connected) component of size |A| ≤ n/2. We must have that |N G ′ (A)| ≤ |A|, but (4.3) and the statement hypotheses imply that |N G ′ (A)| > |A|, a contradiction. Finally, note that R consists of at most 400 components, since each connected component has order at least n/400. Since G ′ is connected, we may choose a set E ⊆ E(G ′ ) with |E| ≤ 400 such that the graph R ′ := ([n], E(R) ∪ E) is connected, and thus is a spanning 3-expander. Lemma 4.9. For every ε > 0 and 0 ≤ δ ≤ 10 −5 there exists D > 0 such that for any D < d ≤ log 2 n we have that a.a.s. the random graph G = G n,d has the following property. Let H ∈ H 1/2−ε (G) and let G ′ := G \ H. Then, there exists a spanning 3-expander R ⊆ G ′ with ∆(R) < δd. Proof. Let 1/D ≪ δ, ε and condition on the statements of Propositions 4.7 and 4.8 both holding with δ/2 playing the role of δ, which happens a.a.s. By Proposition 4.7 we may find a spanning subgraph R ′ ⊆ G ′ with ∆(R ′ ) < δd/2 and such that, for all S ⊆ [n] with |S| ≤ n/400, we have |N R ′ (S)| ≥ 3|S|. Then, by Proposition 4.8 we may find a spanning 3-expander R ⊆ G ′ with ∆(R) < ∆(R ′ ) + 400 < δd. Finding many boosters The following proposition provides an upper bound on the expected number of 'thin' subgraphs that G n,d contains. Proof. For each R ∈ R, let X R be an indicator random variable where X R (G) = 1 if and only if R ⊆ G. Let X R := R∈R X R . Then E[X R ] = R∈R P[R ⊆ G]. Moreover, note that we always have X R ≤ δdn i=1 dn/2 i ≤ e 2δdn log(1/δ) and, therefore, R∈R P[R ⊆ G] = E[X R ] ≤ e 2δdn log(1/δ) , as desired. The following result can easily be proved using "Pósa rotations" (see e.g. [24]). Lemma 5.2. Let R be a 3-expander and let P be a longest path in R, with endpoint v. Then, there exists a set A ⊆ V (P ) with |A| > n/10 4 such that for each a ∈ A there exists a path P a in R with endpoints v and a, and such that V (P a ) = V (P ). Definition 5.3 (Booster). Let H be a graph and let E ⊆ V (H) (2) . Let F := (V (H), E). We call E a booster for H if the graph H + F contains a longer path that H does, or if H + F is Hamiltonian. We will often be interested in the case where E consists of a single edge e / ∈ E(H). In this case we refer to e as a booster for H. Given any path P with endpoints u and v, assume an orientation on its edges (say, from u to v). Given any vertex x ∈ V (P ) \ {v}, we call the vertex that follows x in this orientation its successor, and we denote this by suc P (x). Lemma 5.4. For all 0 < ε < 1/10 5 there exist δ, D > 0 such that for D ≤ d ≤ log 2 n the random graph G = G n,d a.a.s. satisfies the following. Let H ∈ H 1/2−ε (G) and let G ′ := G \ H. Let R ⊆ G ′ be a spanning 3-expander with ∆(R) ≤ 2δd, and let S ⊆ [n] with |S| ≤ δn. Then, there exists a set V R ⊆ [n] with |V R | ≥ n/10 4 with the following property: for each v ∈ V R , there exists a set U v ⊆ [n] with |U v | ≥ (1/2 + ε/8)n such that, for each u ∈ U v , there exists a set E v,u as follows: (a) E v,u ⊆ E((G ′ \ R)[[n] \ S]) with |E v,u | ≥ 50/(εδ), (b) {uv, e} is a booster for R for every e ∈ E v,u , (c) E v,u 1 ∩ E v,u 2 = ∅ for all u 1 = u 2 . Proof. Let 1/D ≪ δ ≪ ε < 1/10 5 . Let R be the set of all n-vertex 3-expander graphs R on [n] with ∆(R) ≤ 2δd. It follows by Lemma 5.2 that for each R ∈ R there exists a set V R ⊆ [n] of size |V R | ≥ n/10 4 such that for every v ∈ V R there exists a longest path in R terminating at v. For each R ∈ R, v ∈ V R and S ⊆ [n] with |S| ≤ δn, let f (R, S, v) be the event that, for every H ∈ H 1/2−ε (G) such that R ⊆ G ′ , there exists a set of vertices U v ⊆ [n] with |U v | ≥ (1/2+ε/8)n and such that for each u ∈ U v there exists a set E v,u satisfying (a)-(c). With this definition, the probability p * that the assertion in the lemma fails is bounded by p * ≤ S⊆[n]:|S|≤δn R∈R v∈V R P[f (R, S, v) | R ⊆ G] P[R ⊆ G]. (5.1) For fixed R ∈ R, v ∈ V R and S ⊆ [n] with |S| ≤ δn, we shall now estimate P[f (R, S, v) | R ⊆ G]. Let P be a longest path in R with endpoint v. As R is a 3-expander, by Lemma 5.2 there must exist a set A ⊆ V (P ) \ S of size |A| = εn/20 such that for each a ∈ A, there is a longest path P a in R starting at v and ending at a with V (P a ) = V (P ) (if there is more than one such path, fix one arbitrarily). Assume that each P a is oriented from v to a. Let B := [n] \ (A ∪ S ∪ {v}). For each u ∈ B ∩ V (P ), let X u := {ab : a ∈ A, b ∈ B, u = suc Pa (b)}. Observe that {uv, ab} is a booster for R for any ab ∈ X u . Clearly, |X u | ≤ |A| and X u ∩ X u ′ = ∅ for all distinct u, u ′ ∈ B ∩ V (P ). Furthermore, for each u ∈ B \ V (P ), let X u := {au : a ∈ A}. Note that au ∈ X u is a booster since its inclusion would result in a longer path in R. We shall now show that, for most vertices u ∈ B, there is a 'large' set of boosters, that is, X u is 'large'. We will then use this to show that many of these boosters must lie in G ′ \ R. For every a ∈ A, there are at least |V (P )| − 2|A| − 2|S| − 2 vertices b ∈ V (P ) such that neither b nor its successor on P a belong to A ∪ S ∪ {v}. It follows that | u∈B∩V (P ) X u | ≥ |A|(|V (P )| − 2|A| − 2|S| − 2). We also have that | u∈B\V (P ) X u | = |A|(n − |V (P ) ∪ S|). Therefore, the following holds: u∈B X u ≥ |A|(|V (P )| − 2|A| − 2|S| − 2) + |A|(n − |V (P ) ∪ S|) ≥ |A|(n − 2|A| − 3|S| − 2) ≥ (1 − ε/9)|A|n. For each u ∈ B, let Y u := X u \ E(R). It follows that u∈B Y u ≥ (1 − ε/9)|A|n − e(R) ≥ (1 − ε/8)|A|n. For each a ∈ A, let Z a be the set of edges in u∈B Y u with a as an endpoint. It is easy to see that a∈A |Z a | = | a∈A Z a | = | u∈B Y u | ≥ (1 − ε/8)|A|n. Consider now the following events: From two applications of Proposition 4.4 we obtain that P[ F 1 ∧ F 2 | R ⊆ G] ≥ 1 − e −(ε/500) 4 dn . To finish the proof we must show that if F 1 ∧ F 2 holds, then f (R, S, v) also holds. Consider any G ∈ G n,d which satisfies both F 1 and F 2 and such that R ⊆ G. Fix any H ∈ H 1/2−ε (G) such that R ⊆ G ′ . For each u ∈ B, let E u := Y u ∩ E(G ′ ). As we have seen above, for each e ∈ E u , the set {uv, e} is a booster for R. Furthermore, none of the endvertices of e lies in S, by construction. Let U ⊆ B be the set of vertices u ∈ B for which |E u | ≥ 50/(εδ). Observe that, by F 2 , if | u∈U Y u ∩ E(G)| ≥ (1/2 + ε/4)|A|d, then |U | ≥ (1/2 + ε/8)n. But u∈U Y u ∩ E(G) ≥ u∈U E u = u∈B Y u ∩ E(G) − u∈B Y u ∩ E(H) − u∈B\U |E u | ≥ a∈A Z a ∩ E(G) − a∈A Z a ∩ E(H) − 50 εδ |B \ U | (F 1) ≥ 1 − ε 4 |A|d − a∈A 1 2 − ε d G (a) − 50 εδ n ≥ 1 − ε 4 |A|d − 1 2 − ε |A|d − 10 3 |A|d ε 2 dδ ≥ 1 2 + ε 4 |A|d. Hence, by F 2 we have that |U | ≥ (1/2 + ε/8)n, as we wanted to show. Since H was arbitrary, it follows that f (R, S, v) holds. Thus, P[f (R, S, v) | R ⊆ G] ≥ P[F 1 ∧ F 2 | R ⊆ G] ≥ 1 − e −(ε/500) 4 dn . We can now use this bound in equation ( Corollary 5.6. For all 0 < ε < 1/10 5 there exist δ, D > 0 such that for D ≤ d ≤ log 2 n the random graph G = G n,d a.a.s. satisfies the following. Let H ∈ H 1/2−ε (G) and let G ′ := G \ H. Let R ⊆ G ′ be a spanning 3-expander with ∆(R) ≤ 2δd, and let S ⊆ [n] with |S| ≤ δn. Then, there exists some subgraph F ⊆ G ′ \ R satisfying ∆(F ) ≤ 2δd, such that R has ε/16-many boosters with help from F , with the property that the set of secondary edges is vertex-disjoint from S. Proof. Let 1/D ≪ δ ≪ ε < 1/10 5 . Condition on the event that G satisfies all the properties in the statement of Lemma 5.4, which happens a.a.s. Let H, G ′ , R, S be as in the statement of Corollary 5.6. By Lemma 5.4, we may find a set V R ⊆ [n] of size |V R | ≥ n/10 4 such that, for each v ∈ V R , there exists a set U v ⊆ [n] with |U v | ≥ (1/2 + ε/8)n such that, for each u ∈ U v , there exists a set E v,u ⊆ E((G ′ \ R)[[n] \ S]) with |E v,u | ≥ 50/(εδ) and such that, for every e ∈ E v,u , {uv, e} is a booster for R, and such that E v,u 1 ∩ E v,u 2 = ∅ for all u 1 = u 2 . Note that each such edge e is vertex-disjoint from S, by construction. Let F be a random subgraph of G ′ \ R where every edge in G ′ \ R is chosen independently at random with probability δ/2. For each v ∈ V R , let U ′ v ⊆ U v be the set of vertices u ∈ U v for which E v,u ∩ E(F ) = ∅. For every u ∈ U v , we have that P[u / ∈ U ′ v ] ≤ (1 − δ/2) 50/(εδ) ≤ e −25/ε ≤ ε/32. Let A be the event that |U ′ v | ≥ (1/2 + ε/16)n for every v ∈ V R . Since for different u ∈ U v the sets E v,u are disjoint, by Lemma 3.1 we have P[|U ′ v | ≤ (1/2 + ε/16)n] ≤ P[|U ′ v | ≤ (1 − ε/16)|U v |] ≤ e −ε 2 n/10 6 for each v ∈ V R . Therefore, P[A] ≤ ne −ε 2 n/10 6 ≤ e −ε 3 n .B v   ≥ (1 − e 1−δd/8 ) n ≥ e −ε 4 n > P[A]. Therefore, the probability both events A and B occur is strictly positive, implying that there exists some F ⊆ G ′ \ R satisfying the required properties. We have now shown that a.a.s. if the random graph G n,d contains a sparse 3-expander subgraph R after deleting some H ∈ H 1/2−ε (G n,d ), then G ′ = G n,d \ H must also have a sparse subgraph F with the property that R has 'many' boosters with help from F . Our next goal is to prove that some primary edge of these boosters must actually be present in G ′ . Lemma 5.7. For all 0 < ε < 1/10 5 there exist δ, D > 0 such that for D ≤ d ≤ log 2 n the random graph G = G n,d satisfies the following a.a.s. Let S ⊆ [n] with |S| ≤ δn and let R, F ⊆ G be two spanning edge-disjoint subgraphs such that (P1) ∆(R), ∆(F ) ≤ 2δd, (P2) R has ε-many boosters with help from F , such that every secondary edge is vertex-disjoint from S. Then, for any H ∈ H 1/2 (G), the graph G ′ := G\H contains an edge e for which there exists some edge e ′ ∈ E(F ) with the property that {e, e ′ } is a booster for R, and such that V ({e, e ′ })∩ S = ∅. Proof. Let 1/D ≪ δ ≪ ε < 1/10 5 . Let P be the set of all triples (R, F, S) where R and F are edge-disjoint graphs on [n] which satisfy (P1) and (P2) and S ⊆ [n] with |S| ≤ δn. Fix a triple (R, F, S) ∈ P. For every x ∈ [n], let V x be the set of vertices v ∈ [n] \ (S ∪ {x}) for which there exists some edge e ∈ E(F ) such that none of the endvertices of e lies in S and {xv, e} is a booster for R. Let X ′ := {x ∈ [n] : |V x | ≥ (1/2 + 3ε/4)n}. By assumption on the triple (R, F, S) and using Definition 5.5, we must have that |X ′ | ≥ εn. Let X := X ′ \ S. Let f (R, F, S) be the event that x∈X e G\(R+F ) (x, V x ) ≥ (1 + ε)d|X|/2. It follows by Proposition 4.4 that P[f (R, F, S) | R + F ⊆ G] ≤ e −(ε/30) 4 dn . (5.2) It follows that the probability that (R + F ⊆ G) ∧ f (R, F, S) for some triple (R, F, S) ∈ P is at most where the second inequality follows by Proposition 5.1. (R,F,S)∈P P[f (R, F, S) | R + F ⊆ G] P[R + F ⊆ G] We conclude that a.a.s. for all (R, F, S) ∈ P with R + F ⊆ G we have x∈X e G\(R+F ) (x, V x ) − 1 2 d G (x) ≥ (1 + ε) d|X| 2 − d|X| 2 > 0. Hence, there must exist some x ∈ X with e G\(R+F ) (x, V x ) > d/2. Therefore, for any H ∈ H 1/2 (G), there is some vertex x ∈ X such that e G ′ (x, V x ) ≥ e G (x, V x ) − d H (x) > 0. That is, there must be some v ∈ N G ′ (x) ∩ V x . By the definition of V x , there is some e ∈ E(F ) such that {xv, e} is a booster for R. Furthermore, by construction, we have V ({xv, e}) ∩ S = ∅, and this completes the proof of the lemma. Armed with the previous lemmas, we are now in a position to complete the proof of Theorem 1.2. Proof of Theorem 1.2. Let 1/D ≪ δ ≪ ε < 1/10 5 be such that Corollary 5.6 holds for ε, Lemma 5.7 holds for ε/16 and Lemma 4.9 holds for ε. Condition on each of these holding. Let H ∈ H 1/2−ε (G) and let G ′ := G \ H. By Lemma 4.9, there exists a subgraph R ⊆ G ′ which is a spanning 3-expander with ∆(R) ≤ δd. Let R 0 := R. We now proceed recursively as follows: for each i ∈ [n], choose e i,1 , e i,2 ∈ E(G ′ ) such that {e i,1 , e i,2 } is a booster for R i−1 , and let R i := R i−1 + e i,1 + e i,2 . In order to show that there exist such e i,1 , e i,2 for all i ∈ [n], consider the following. Assume that R i−1 satisfies ∆(R i−1 ) ≤ 2δd. Let S i ⊆ [n] be the set of vertices v ∈ [n] with d R i−1 (v) ≥ 2δd−1. For all i ∈ [n] we have |S i | ≤ 2e(R i−1 \R 0 )/(δd−1) ≤ 4n/(δd−1) < δn. By applying Corollary 5.6 with S i , R i−1 playing the roles of S and R, respectively, there exists some subgraph F i ⊆ G ′ \ R i−1 such that R i−1 has (ε/16)-many boosters with help from F i , where each secondary edge is vertex-disjoint from S i . Furthermore, we have ∆(F i ) ≤ 2δd. Therefore, by applying Lemma 5.7, there are some e i,1 , e i,2 ∈ E(G ′ ) such that {e i,1 , e i,2 } is a booster for R i−1 and where V ({e i,1 , e i,2 }) ∩ S i = ∅. It follows that ∆(R i ) ≤ 2δd. By the end of this process, we have added n boosters to R to obtain R n ⊆ G ′ . Therefore R n , and hence G ′ , is Hamiltonian. 6. Graphs of small degree with low resilience In this section we prove Theorem 1.3. For this, we will require a crude bound on the number of edges spanned by any set of n/2 vertices in G n,d . To achieve this, we shall make use of the following result, which follows from a theorem of McKay [28] (see e.g. [38]). We denote by α(G) the size of a maximum independent set in G. Proof. By Theorem 6.1, every set of size n/2 must span at least n/100 edges, as otherwise it would contain an independent set of size n/2 − n/50 > 0.46n. Alternatively, this lemma can be proved directly using a switching argument. In order to prove Theorem 1.3 we will use a switching argument. Given a graph G ∈ G n,d and any integer ℓ ∈ [d], let u ∈ [n] and let Λ + u,ℓ = (e 1 , . . . , e ℓ , f 1 , . . . , f ℓ ) be an ordered set of 2ℓ edges from E(G) such that e i = uv i with v i = v j for all i = j, and {f i : i ∈ [ℓ]} is a set of independent edges such that, for each i ∈ [ℓ], the distance between f i and e i is at least 2. We call Λ + u,ℓ a (u, ℓ)-switching configuration. For each i ∈ [ℓ], choose an orientation of f i and write f i = x i y i , where f i is oriented from x i to y i . Let λ + u,ℓ := {e 1 , . . . , e ℓ , f 1 , . . . , f ℓ }, Λ − u,ℓ := (uy 1 , . . . , uy ℓ , x 1 v 1 , . . . , x ℓ v ℓ ) and λ − u,ℓ := {uy 1 , . . . , uy ℓ , x 1 v 1 , . . . , x ℓ v ℓ }. We say that the graph G ′ := ([n], (E(G) \ λ + u,ℓ ) ∪ λ − u,ℓ ) ∈ G n,d is obtained from G by a u-switching of type ℓ. Observe that, given such a setting, we also have that G is obtained from G ′ by a u-switching of type ℓ, interchanging the roles of Λ + u,ℓ and Λ − u,ℓ . Proof of Theorem 1.3. Fix any odd d > 2. LetĜ n,d ⊆ G n,d be the collection of graphs for which the statement of Lemma 6.2 holds. We have by Lemma 6.2 that |Ĝ n,d | = (1 − o(1))|G n,d |. Let G ′ n,d ⊆Ĝ n,d be the collection of all graphs G ∈Ĝ n,d which are not (d − 1)/2-resilient with respect to Hamiltonicity. Let p := |G ′ n,d |/|Ĝ n,d |. We will prove that p is bounded from below by a positive constant which does not depend on n. . It is then clear that M = G \ H is not Hamiltonian, as it is an unbalanced bipartite graph. Furthermore, we have that ∆(H) ≤ (d − 1)/2, so we conclude that G is not (d−1)/2-resilient with respect to Hamiltonicity and, thus, G ∈ G ′ n,d . (Below we will use that the same conclusion holds if there is any cut M of G such that |A M | < |B M |, d M (x) > d/2 for all x ∈ A M , and d M (y) > d/2 for all y ∈ B M .) Therefore, for every G ∈Ĝ n,d \ G ′ n,d we have that |A M | = |B M | for every maximum cut M of G. For each G ∈Ĝ n,d \ G ′ n,d , fix a maximum cut M G of G which partitions [n] into A G and B G . Then, for each x ∈ A G there exists k ∈ [⌈d/2⌉] such that d M G (x) = ⌊d/2⌋ + k. Let ℓ ∈ [⌈d/2⌉] be such that there exist at least (1 − p)|Ĝ n,d |/d graphs G ∈Ĝ n,d \ G ′ n,d with the property that there are at least n/(2d) vertices x ∈ A G with d M G (x) = ⌊d/2⌋ + ℓ. Let D := ⌊d/2⌋ + ℓ. Denote the collection of all such graphs G by Ω. Theorem 1 . 3 . 13For any odd d > 2, the random graph G n,d is not a.a.s. (d − 1)/2-resilient with respect to Hamiltonicity. Let A ′ := A \ {x} and B ′ := B ∪ {x}. As before, by deleting the edges in G[A ′ ] ∪ G[B ′ ], we obtain a graph which is not Hamiltonian. . Notation. For n ∈ N, we denote [n] := {1, . . . , n}. Given any set S, we denote S (2) := {{s 1 , s 2 } : s 1 , s 2 ∈ S, s 1 = s 2 }. The parameters which appear in hierarchies are chosen from right to left. Lemma 3 . 1 . 31Let X be the sum of n independent Bernoulli random variables and let µ := E[X]. Then, for all 0 consider a set of m vertices labelled as x ij for i ∈ [n] and j ∈ [d i ]. For each i ∈ [n], we call the set {x ij : j ∈ [d i ]} the expanded set of i. Similarly, for any X ⊆ [n], we call the set {x ij : i ∈ X, j ∈ [d i ]} the expanded set of X. Choose uniformly at random a perfect matching M covering the expanded set of [n]. Then, obtain a multigraph ϕ(M ) = ([n], E) by letting E be the following multiset: for each edge e ∈ M , consider its endpoints . Let d ≤ log 2 n be a positive integer and let R be a graph on vertex set [n] with degree sequence d and let A, B ⊆ [n] be any (not necessarily disjoint) sets of vertices such that 2|A| < (1 − δ)n. Then, the random variable e G (A, B) is stochastically dominated by a random variable X ∼ Bin( a∈A d a , |B|/((1 − δ)n − 2|A|)).Proof. Let t := a∈A d a . Let X , A ′ and B ′ be the expanded sets of [n], A and B, respectively. Label the points of X so that all the points in A ′ come first, that is,A ′ = {x 1 , . . . , x t }.Generate a random configuration M ∼ C * n,d following this labelling. Then, e G (A, B) is the number of pairings in M with one endpoint in A ′ and the other in B ′ , and we will estimate the probability that each pairing added to M contributes to e G (A, B). First, note that all pairings added after M t do not contribute to e G (A, B), as they do not have an endpoint in A ′ . For each i ∈ [t], define an indicator random variable X i which takes value 1 if M i = M i−1 and e = x i y ∈ M i \ M i−1 is such that y ∈ B ′ , and 0 otherwise, so that e G (A, B) = i∈[t] X i . Observe that, in the above process, the bound P[X i = 1 | M i = M i−1 ] ≤ |B| (1 − δ)n − 2|A| holds for all i ∈ [t] since at every step of the process there are at most |B|d points available in B ′ and at least (1 − δ)nd − 2|A|d points available in X \ (V (M i−1 ) ∪ {x i }). On the other hand, P[X i = 1 | M i = M i−1 ] = 0, so given M 0 , M 1 , . . . , M i−1 , each X i is stochastically dominated by a Bernoulli random variable Y i with parameter |B|/((1−δ)n−2|A|). By summing over all i ∈ [t], we conclude that e G (A, B) is stochastically dominated by X ∼ Bin(t, |B|/((1 − δ)n − 2|A|)). 4. On the existence of a sparse 3-expander Definition 4.1. An n-vertex graph G is called a 3-expander if it is connected and, for every S ⊆ [n] with |S| ≤ n/400, we have |N G (S)| ≥ 3|S|. Definition 4. 2 . 2Let G = (V, E) and G ′ = (V, E ′ ) be two multigraphs on the same vertex set such that |E| Lemma 4. 3 . 3Let d = (d 1 , . . . , d n ) be a degree sequence with d i ≤ log 2 n for all i ∈ [n], and such that n i=1 d i is even. Let ∆ := max i∈[n] {d i }. Let c > 0 and let X be a random variable on C * n,d such that, for every pair of configurations M ∼ M ′ , we have |X(M ) − X(M ′ )| ≤ c. Then, for all ε > 0,P[|X − E[X]| ≥ εE[X]] ≤ 2e − ε 2 E[X] 2 2∆nc 2 . Proof. Let m := n i=1 d i . Fixany labelling x 1 , . . . , x m of the expanded set of [n]. Let M ∼ C * n,d be generated following this labelling. Let the partial configurations of M be M 0 , . . . , M m . For each i ∈ [m] ∪ {0}, let Y i (M ) := E[X(M ) | M i ] = E[X(M ) | M 0 , . . . , M i ]. It follows that the sequence Y 0 (M ), Y 1 (M ), . . . , Y m (M ) is a Doob martingale, where Y 0 (M ) = E[X] and Y m (M ) = X(M ). We will now show that the differences of this martingale are bounded by c. For any i ∈ [m], if M i = M i−1 , then Y i (M ) = Y i−1 (M ) and there is nothing to prove, so assume that M i = M i−1 , that is, when generating the i-th partial configuration, the i-th point does not lie in any of the previous pairings. For each j ∈ [m] \ (V (M i−1 ) ∪ {i}), let M j be the set of configurations which contain M i−1 as well as ij. It is easy to see that for each k ∈ [m] \ (V (M i−1 ) ∪ {i}) there is a bijection g j,k between M j and M k so that g j,k (M ′ ) ∼ M ′ for all M ′ ∈ M j . Let N := |M j |. Fix j ∈ [m] \ (V (M i−1 ) ∪ {i}) and label the configurations in M j as M j,1 , . . . , M j,N . For all Thus, property (i) in the statement holds with probability 1 − o(1). Similarly, we can show that property (ii) also holds with probability 1 − o(1). a contradiction. The result follows. In order to prove (ii), let S ⊆ [n] with δ 2 n ≤ |S| ≤ n/400. Suppose there exists S ′ ⊆ [n] with |S| ≤ |S ′ | ≤ 3|S| and such that e G (S, S ′ ) > d|S|/5. We have by Lemma 4.5 that 4|S|d/25 ≥ e G (S ∪ S ′ ) ≥ e G (S, S ′ ) > d|S|/5, a contradiction. The result follows. Proposition 5 . 1 . 51Let 1/n ≪ 1/d, δ ≪ 1, where n, d ∈ N, and let G = G n,d . Let R be a family of graphs R on vertex set [n] with e(R) ≤ δdn for all R ∈ R. Then, R∈R P[R ⊆ G] ≤ e 2δdn log(1/δ) . F1: | a∈A (Z a ∩ E(G))| ≥ (1 − ε/4)|A|d.F2: For any U ⊆ B with |U | ≤ (1/2 + ε/8)n we have | u∈U Y u ∩ E(G)| < (1/2 + ε/4)|A|d. p * ≤ 2 n ne −(ε/500) 4 dn R∈R P[R ⊆ G] ≤ 2 n ne −(ε/500) 4 dn e 2δdn log(1/δ) = o(1),where the second inequality follows from Proposition 5.1. This shows the statement in the lemma holds a.a.s.Definition 5.5. Given graphs H and H ′ with V (H) = V (H ′ ) = V and E(H) ∩ E(H ′ ) = ∅,we say H has ε-many boosters with help from H ′ if there are at least ε|V | vertices v ∈ V for which there exists a set U v ⊆ V \ {v} of size at least (1/2 + ε)|V | with the property that for every u ∈ U v there exists e ∈ E(H ′ ) so that {uv, e} is a booster for H. We call uv the primary edge and we call e the secondary edge. n e −(ε/30) 4 dn e 4δdn log 1/(2δ) ≤ e −(ε/50) 4 dn , Theorem 6 . 1 . 61For every fixed d ≥ 3, a.a.s. we have that α(G n,d ) ≤ 0.46n. Lemma 6.2. For every fixed d ≥ 3, a.a.s. we have that e G n,d (A) > n/100 for all A ⊆ [n] with |A| = ⌊n/2⌋. For each G ∈Ĝ n,d , consider a maximum cut M with parts A M and B M , where |A M | ≤ |B M | (thus M = E G (A M , B M )). By abusing notation, we also use M to denote the bipartite graph G[A M , B M ]. Note that for all x ∈ A M we have d M (x) > d/2, as otherwise we could move x from A G to B G to obtain a larger cut; similarly, d M (y) > d/2 for all y ∈ B M . Given G ∈Ĝ n,d , suppose there exists a maximum cut M for G such that |A M | < |B M |. Let H := ([n], E G (A M ) ∪ E G (B M )) Now, let B be the event that ∆(F ) ≤ 2δd. For each v ∈ [n], let B v be the event that d F (v) > 2δd. By Lemma 3.2, we have P[B v ] < e −δd/8 for all v ∈ [n]. Now observe that G ′ \ R is itself a dependency graph for {B v } v∈[n] , and every vertex in this graph has degree at most d. It follows by Lemma 3.4 that P[B] = P  v∈[n] For each G ∈ Ω and for each x ∈ A G such that d M G (x) = D, we consider all possible xswitchings of type D where the (x, D)-switching configuration Λ +x,D = (e 1 , . . . , e D , f 1 , . . . , f D ) satisfies that {e 1 , . . . , e D } = E M G (x, B G ) and {f 1 , . . . , f D } ⊆ E G (A G ). We say that any G ′ ∈ G n,d which can be obtained from G by such an x-switching of type D, is obtained by an outswitching from G, and we call Λ +x,D an out-switching configuration. Let Ω ′ denote the set of all graphs G ′ ∈ G n,d which can be obtained by out-switchings from some graph G ∈ Ω. In particular, note that for each G ′ obtained from G by an out-switching we may define A ′ := A G \ {x} andTo show that Ω ′ is large, we consider an auxiliary bipartite graph Γ with parts Ω and Ω ′ . 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An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. Our approach is based on the method of high-frequency homogenisation (HFH), introduced in the context of scalar out-of-plane elastic waves in[20], here extended to the full Maxwell system for electromagnetic waves in three dimensions. In doing so, a vector treatment of Maxwell's equations is necessary, of which scalar transverse electric (TE) and transverse magnetic (TM) polarised cases are a strict subset. The resulting procedure yields effective dynamic continuum equations, valid even at high frequencies, that govern a scalar function providing long-scale modulation of short-scale Bloch eigensolutions inside the structure. The form of the effective equation changes in the case of non-trivially repeated eigenvalues, leading to degeneracies that are linked to the appearance of Dirac-like points, a case that we explore using the asymptotic method. We examine the low-frequency long-wave limit in the case of dielectric structures, and find the expected result of classical homogenisation theory is captured, in which the usual governing equation is recovered, but now with the permittivity replaced by an effective tensor.The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. Advantage can be taken of the axial invariance to reduce the three-dimensional example such that it permits manageable computations on a twodimensional domain. Results of the asymptotic theory are verified against these numerical simulations by comparing photonic band diagrams and evanescent decay rates for localised modes. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. These waves can be highly localised in the plane, decaying exponentially in directions normal to the array, and at certain frequencies strongly directional dynamic anisotropy is observed. Computationally this is a challenging three-dimensional calculation, which we do here, and then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour. arXiv:1506.02292v1 [math-ph] 7 Jun 2015 tures[34,62], in particular those exhibiting photonic band gaps in one or more directions. Further to this, the rich band structures possessed by photonic crystals (PCs) give rise to a host of novel physical phenomena including, but not limited to, dynamic anisotropy [15], ultrarefraction [3] and light confinement [8] that can also impact upon the design of PCFs[53]. From a numerical point of view, the implementation of Maxwell's equations in such structures can pose considerable difficulty, particularly when the wavelength is of a similar scale to that of the repeating microcell, of which there may be many hundreds or more. In such cases, multiple scattering and interference play a central role. Typical numerical schemes such as finite element methods [62] are highly effective, but the computational cost in tackling the coupled vector system increases rapidly with the number of nodes, which is intimately connected with the dimensionality, scale and operating wavelength of the system. Optimisation and design of a PC or PCF may require multiple simulations across changes in geometry, material parameters and microstructure, so it is attractive to explore complementary effective medium, or homogenisation, schemes [1, 7] that attempt to treat the medium in some averaged sense and use these instead of, or alongside, direct numerical simulation.Asymptotic homogenisation dates back to the French, Italian and Russian schools, and was developed for studying Partial Differential Equations (PDEs) with rapidly oscillating coefficients. The approach of introducing independent fast and slow variables applies naturally to structured media, leading to 'effective medium' models on which there is extensive literature[6,9,47,61]. The effective properties attained using such methodologies are valid within the low-frequency (quasi-static) regime, which is the relevant domain for the rapidly developing theory of sub-wavelength metamaterials, in which a typical application involves tuning these effective properties to desired values as prescribed by transformation optics (see, for example[22,23]). Unfortunately these homogenisation theories often perform poorly near band edges [7] and the limitation of homogenisation theories to long wavelength excitations [1] means that most of the interesting effects associated with PCs, which are dynamic in nature and therefore correspond to high-frequency bands, are not captured.The article [20] described how, by considering perturbations about the standing wave eigenfrequencies of a periodic structure, a long-scale PDE can be derived that governs the propagation of a scalar modulation function, characterising the group behaviour of waves in the structure. Crucially, this approach allows for arbitrarily rapid field variation inside the long-scale envelope, which in turn allows us to break free of the quasi-static regime, relying only on the conditions of Bloch's theorem [10] being satisfied. Accordingly, we take advantage of a result from solid state physics: the dispersive properties of a bulk periodic structure are entirely characterised by the frequency dependence of the Bloch wave vector within the irreducible Brillouin zone[36]. This result holds for general Bravais lattices, and recent work has been done to extend the current theory to such structures[43]. Here, we focus on simple square and cubic lattice arrays for clarity. Since publication of the asymptotic theory in 2010, further work has been done in which its results have been compared against direct numerical calculations, demonstrating that it is capable of capturing dynamic effects unique to high-frequency bands, including all-angle negative refraction, dynamic anisotropy and cloaking near Dirac-like cones [3], as well as reproducing the value of the reflection coefficient in simple scattering problems[35]. This supports the claim that the resulting behaviour can be considered that of an effective medium, and justifies our reference to the theory as high-frequency homogenisation (HFH). Alongside analogous work on discrete lattices [21], HFH was originally developed for the planar Helmholtz equation, which models the frequency-domain response of out-of-plane elastic shear waves, TE or TM polarised electromagnetic waves, or pressure waves in a fluid. Further work has been done to extend the theory to in-plane elasticity, in which the resulting Lamé system poses a vector problem in two dimensions[4,16]; see also[11]for further work in elasticity.The main result here is to extend the theory to the three-dimensional vector formulation of Maxwell's equations. We shall see that HFH leads to an effective PDE for a scalar modulation function f 0 , in which the local dispersive properties of the structure are captured by a frequency-dependent tensor T . We show that in the low-frequency limit, this leads naturally to the expected result of classical homogenisation theory, in which the governing vector equation is recovered with effective material parameters. The theory we present is very general in its scope and we choose to illustrate its use on two relevant examples, the first being that of guided wave propagation in PCFs, and

1506.02292

a1c58b08d2b8cc543835a5ae2776e96b58ad8cd5

The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms June 9, 2015 B Maling Department of Mathematics Imperial College London SW7 2AZLondonU.K D J Colquitt Department of Mathematics Imperial College London SW7 2AZLondonU.K R V Craster Department of Mathematics Imperial College London SW7 2AZLondonU.K The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms June 9, 2015 An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. Our approach is based on the method of high-frequency homogenisation (HFH), introduced in the context of scalar out-of-plane elastic waves in[20], here extended to the full Maxwell system for electromagnetic waves in three dimensions. In doing so, a vector treatment of Maxwell's equations is necessary, of which scalar transverse electric (TE) and transverse magnetic (TM) polarised cases are a strict subset. The resulting procedure yields effective dynamic continuum equations, valid even at high frequencies, that govern a scalar function providing long-scale modulation of short-scale Bloch eigensolutions inside the structure. The form of the effective equation changes in the case of non-trivially repeated eigenvalues, leading to degeneracies that are linked to the appearance of Dirac-like points, a case that we explore using the asymptotic method. We examine the low-frequency long-wave limit in the case of dielectric structures, and find the expected result of classical homogenisation theory is captured, in which the usual governing equation is recovered, but now with the permittivity replaced by an effective tensor.The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. Advantage can be taken of the axial invariance to reduce the three-dimensional example such that it permits manageable computations on a twodimensional domain. Results of the asymptotic theory are verified against these numerical simulations by comparing photonic band diagrams and evanescent decay rates for localised modes. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. These waves can be highly localised in the plane, decaying exponentially in directions normal to the array, and at certain frequencies strongly directional dynamic anisotropy is observed. Computationally this is a challenging three-dimensional calculation, which we do here, and then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour. arXiv:1506.02292v1 [math-ph] 7 Jun 2015 tures[34,62], in particular those exhibiting photonic band gaps in one or more directions. Further to this, the rich band structures possessed by photonic crystals (PCs) give rise to a host of novel physical phenomena including, but not limited to, dynamic anisotropy [15], ultrarefraction [3] and light confinement [8] that can also impact upon the design of PCFs[53]. From a numerical point of view, the implementation of Maxwell's equations in such structures can pose considerable difficulty, particularly when the wavelength is of a similar scale to that of the repeating microcell, of which there may be many hundreds or more. In such cases, multiple scattering and interference play a central role. Typical numerical schemes such as finite element methods [62] are highly effective, but the computational cost in tackling the coupled vector system increases rapidly with the number of nodes, which is intimately connected with the dimensionality, scale and operating wavelength of the system. Optimisation and design of a PC or PCF may require multiple simulations across changes in geometry, material parameters and microstructure, so it is attractive to explore complementary effective medium, or homogenisation, schemes [1, 7] that attempt to treat the medium in some averaged sense and use these instead of, or alongside, direct numerical simulation.Asymptotic homogenisation dates back to the French, Italian and Russian schools, and was developed for studying Partial Differential Equations (PDEs) with rapidly oscillating coefficients. The approach of introducing independent fast and slow variables applies naturally to structured media, leading to 'effective medium' models on which there is extensive literature[6,9,47,61]. The effective properties attained using such methodologies are valid within the low-frequency (quasi-static) regime, which is the relevant domain for the rapidly developing theory of sub-wavelength metamaterials, in which a typical application involves tuning these effective properties to desired values as prescribed by transformation optics (see, for example[22,23]). Unfortunately these homogenisation theories often perform poorly near band edges [7] and the limitation of homogenisation theories to long wavelength excitations [1] means that most of the interesting effects associated with PCs, which are dynamic in nature and therefore correspond to high-frequency bands, are not captured.The article [20] described how, by considering perturbations about the standing wave eigenfrequencies of a periodic structure, a long-scale PDE can be derived that governs the propagation of a scalar modulation function, characterising the group behaviour of waves in the structure. Crucially, this approach allows for arbitrarily rapid field variation inside the long-scale envelope, which in turn allows us to break free of the quasi-static regime, relying only on the conditions of Bloch's theorem [10] being satisfied. Accordingly, we take advantage of a result from solid state physics: the dispersive properties of a bulk periodic structure are entirely characterised by the frequency dependence of the Bloch wave vector within the irreducible Brillouin zone[36]. This result holds for general Bravais lattices, and recent work has been done to extend the current theory to such structures[43]. Here, we focus on simple square and cubic lattice arrays for clarity. Since publication of the asymptotic theory in 2010, further work has been done in which its results have been compared against direct numerical calculations, demonstrating that it is capable of capturing dynamic effects unique to high-frequency bands, including all-angle negative refraction, dynamic anisotropy and cloaking near Dirac-like cones [3], as well as reproducing the value of the reflection coefficient in simple scattering problems[35]. This supports the claim that the resulting behaviour can be considered that of an effective medium, and justifies our reference to the theory as high-frequency homogenisation (HFH). Alongside analogous work on discrete lattices [21], HFH was originally developed for the planar Helmholtz equation, which models the frequency-domain response of out-of-plane elastic shear waves, TE or TM polarised electromagnetic waves, or pressure waves in a fluid. Further work has been done to extend the theory to in-plane elasticity, in which the resulting Lamé system poses a vector problem in two dimensions[4,16]; see also[11]for further work in elasticity.The main result here is to extend the theory to the three-dimensional vector formulation of Maxwell's equations. We shall see that HFH leads to an effective PDE for a scalar modulation function f 0 , in which the local dispersive properties of the structure are captured by a frequency-dependent tensor T . We show that in the low-frequency limit, this leads naturally to the expected result of classical homogenisation theory, in which the governing vector equation is recovered with effective material parameters. The theory we present is very general in its scope and we choose to illustrate its use on two relevant examples, the first being that of guided wave propagation in PCFs, and Introduction The practical implementation of photonic crystal fibres (PCFs) in communications, lasers and other areas of engineering has resulted in much interest being ascribed to the dispersive properties of periodic photonic struc-the second being that of dynamic anisotropy upon a structured metafilm. The first application we consider is that of dielectric structures that are invariant in one spatial direction. The method could also be applied to so-called wire media [38], but we note that dielectric structures are typical models for PCFs and, unlike metallic waveguides, support coupled modes at oblique incidence which are neither polarised in the transverse electric (TE) plane nor the transverse magnetic (TM) plane, and therefore demand a full vector treatment of Maxwell's equations. Compared with fully three-dimensional problems, this example has the advantage that the dependence of the field on the third spatial coordinate, say x 3 , can be explicitly factored out as being proportional to exp(iβx 3 ), reducing the problem to a quasi-two-dimensional one (see figure 4.1), and therefore is a perfect example for testing the theory in a regime where full numerical simulations are tractable. Similar examples have been studied by various authors using numerical simulations [25,26] as well as semi-analytic techniques such as multipole expansions [27], and our current work complements these, adding further quantitative analysis and providing insight via the appealing notion of asymptotic homogenisation. The final application we choose is that of guided waves along a structured metafilm of dielectric spheres. Such structures [31], and metasurfaces [30,44], are of considerable interest as a subclass of electromagnetic metamaterials that have advantages in terms of fabrication. We shall consider guided waves within a planar array of spheres that decay exponentially normal to the plane and, as such, are described as Rayleigh-Bloch waves [16,51,58]. In terms of the three-dimensional Maxwell system, Rayleigh-Bloch waves guided along a linear array of spheres was examined in [40]. Due to the computational effort required there have been very limited studies of the waves that can exist within planar array of spherical inclusions, with [57] focussing on scalar acoustics only, and multipole methods being used by [55]. See also [31]; Rayleigh-Bloch waves are identified but only in limited directions. Here we access the full Brillouin zone numerically and identify frequencies for which marked dynamic anisotropy occurs, where the effective medium becomes locally hyperbolic, and as a result energy is localised and directed along characteristic directions. In simpler two-dimensional situations star-shaped highly directional wave motion at specific frequencies for structured media has emerged in experiments and theory in optics [15,18], and is most strikingly seen in discrete mass-spring lattice systems [5,17,18,37]. Here for the full vector Maxwell system the three-dimensional simulations are substantial and require many hours of computer time, whereas the asymptotic technique identifies these features, provides physical interpretation and finds quantitatively comparable results rapidly. The article is structured as follows: we begin by setting the scene in terms of the governing equations and necessary details regarding periodicity through reciprocal lattices, Brillouin zones and Bloch waves in section 2. Given these preliminaries we then move on to developing the asymptotic theory in section 3, and use this as an opportunity to clarify some details regarding Dirac-like points (section 3.1). Furthermore, we confirm that the general theory provided does indeed retrieve the quasi-static classical homogenised result (section 3.2). Applications to PCFs and then to dynamically anisotropic Rayleigh-Bloch waves on a metafilm are given in sections 4, 5, in which we use the asymptotic theory to complement full numerical simulations. Finally, some concluding remarks are drawn together in section 6. Governing equations and reciprocal lattice For time harmonic excitations proportional to exp(−iωt), and in the absence of sources, Maxwell's equations in linear media are given by [33]: ∇ · D = 0, ∇ × E − iωB = 0, ∇ · B = 0, ∇ × H + iωD = 0,(2.1) where E and B are the electric and magnetic fields respectively and are related to the displacement field D and magnetizing field H through the constitutive relations D = E, B = µH. (2.2) The parameters = r 0 and µ = µ r µ 0 are the permittivity and permeability respectively. Our convention is to assume that the fields are complex vectors in C 3 and the measurable fields are obtained by taking the real part. We assume that and µ are piecewise smooth, spatially varying real parameters that are exactly periodic in each direction. Combining the curl equations in (2.1) with the constitutive relations (2.2) yields a decoupled equation satisfied by the magnetizing field: − ∇ × ( −1 ∇ × H) + µω 2 H = 0, (2.3) along with the condition that the component of both E and H parallel to a discontinuity in or µ must be continuous. In practice, both and µ are usually piecewise constant, and (2.3) has components that are Helmholtz equations coupled through boundary conditions between adjacent phases. We note that the magnetic formulation (2.3) turns out to be more convenient than the symmetric equation for E in the case of dielectrics, because the assumption that µ is constant ensures that ∇ · H is identically zero, which is useful for the elimination of spurious modes that may arise in the numerical scheme [25]. Perfect conductors can also be included in the structure, imposing a vanishing tangential electric field component at the boundary, corresponding to a vanishing curl of the magnetic field parallel to the boundary. This notion is meaningful only in the microwave region of the spectrum, which is significantly lower than the plasma frequency of many metals; such cases are of physical interest as this is the relevant regime for telecommunications. The underlying periodicity of the medium that we seek to homogenise is an important ingredient of the asymptotic theory, and has a bearing on the numerical simulations we perform, and some notation is necessary which we provide here; this is available in standard texts [12,36] amongst others. Suppose we have a Cartesian co-ordinate system with basis vectorsx i for i = 1, 2, 3. We assume that the periodicity of the structure is that of a simple cubic array, given explicitly by (x) = (x + 2l[mx 1 + nx 2 + px 3 ]) for m, n, p ∈ Z, and similarly for µ(x). The primitive lattice vectors 2l{x i } form an orthonormal set, and the elementary cell C can be chosen as any cube of side 2l oriented in accordance with these. Also central to our analysis will be the first Brillouin zone (shown in figure 2.1 for a simple cubic lattice, along with the analogue for a square lattice in two dimensions), defined in terms of the reciprocal lattice vectors k i for i = 1, 2, 3, where k i ·2l{x j } = 2πδ ij [13]. The relevance of reciprocal space here stems from Bloch's theorem, which states that for an infinite structure with direct lattice vectors {d}, the electric field components are quasi-periodic such that H(x+d) = H(x) exp(ik·d), and similarly for E, where in this context the reciprocal vector k is called the Bloch wave vector [10]. In particular, if the Bloch wave vector corresponds to a reciprocal lattice vector, the field has the periodicity of the direct lattice, resulting in a standing wave solution. In fact, standing wave solutions are found whenever the Bloch wave vector implies periodicity or anti-periodicity in each spatial direction, and it is these points about which the following asymptotic perturbation scheme is applied. The asymptotic procedure To adopt a two-scale approach we define the micro-scale position vector ξ = x/l, which is restricted such that inside the elementary cell C it takes values ξ i ∈ [−1, 1] for i = 1, 2, 3. Clearly, variation in and µ are functions of these co-ordinates only. From Bloch theory, we know that the three-dimensional structure admits standing wave solutions at discrete eigenfrequencies [36], corresponding to solutions with Bloch wave vectors at the vertices Standing waves occur at each of the lettered points. (a) (b) κ 2 κ 1 X Γ M κ 1 κ 2 κ 3 R M Γ X of the irreducible Brillouin zone. Perturbing the frequency slightly results in long wavelength modulation of the Bloch modes, and with this in mind we introduce a macro-scale position vector X = ηx/l relative to some fixed origin, with 0 < η 1. Provided that the frequency perturbation is not too great, there is a natural separation of scales, with X representing the long-scale response of the structure. In the following theory, we are interested in the vanishing limit of η. As is typical in homogenisation theories, the disparity of the length scales associated with ξ and X allows us to treat them as independent variables, so the partial derivative operators are expanded using the chain rule as ∂ xi = (∂ ξi + η∂ Xi )/l. We label the dimensionless standing wave eigenfrequencies Ω 0 = ω 0 l/c, with c = 1/ √ µ 0 0 the speed of light in vacuum, corresponding to solutions satisfying periodic and/or antiperiodic boundary conditions on the cell level, explicitly given by X 3 X 1 C : ξ 3 ξ 2 ξ 1 D 1 D 2 2lH| ξj =−1 = ±H| ξj =+1 , −1 r ∇ × H| ξj =−1 = ± −1 r ∇ × H| ξj =+1 j = 1, 2, 3, (3.1) where ± is chosen depending on the vertex of the irreducible Brillouin zone. Following [20], we seek solutions in the form of an asymptotic ansatz using the small parameter η: H = ∞ i=0 η i H i , Ω 2 = ∞ i=0 η i Ω 2 i ,(3.2) which leads to a hierarchy of equations to be solved upwards from the lowest order in η, with periodic/antiperiodic boundary conditions to be applied on the short scale at each level. The leading order system poses an eigenvalue problem on the short scale only: O(1) : −∇ ξ × −1 r (ξ)∇ ξ × H 0 (X, ξ) + µ r (ξ)Ω 2 0 H 0 (X, ξ) = 0, (3.3) subject to short-scale periodicity conditions resulting from the two-scale expansion of (3.1), and the following continuity conditions at any phase interfaces: −1 r ∇ ξ × H 0 ∂D1,2 × n = 0, H 0 ∂D1,2 × n = 0, (3.4) where [·] denotes a jump discontinuity and n is normal to the interface ∂D 1,2 between two adjacent phases. For certain simple cases, it may be possible to construct analytic or semi-analytic solutions to this problem (for example, using multipole expansions for spherical or cylindrical inclusions, analogously to the methods used in [27,40]). More often, however, the problem is solved numerically using a finite element package, say [41], which amounts to minimising the residue of the following weak equation: −1 r ∇ ξ × H 0 · ∇ ξ × H * 0 + δ(∇ ξ · µ r H 0 )(∇ ξ · µ r H * 0 ) − µ r Ω 2 0 H 0 · H * 0 dV = 0 ,(3.5) where H 0 are the weight functions, δ is a small positive constant and · * denotes complex conjugation. Although for Ω 0 = 0 it is usually unnecessary to further impose the divergence-free condition (which follows from taking the divergence of (3.3)), we follow the approach of [25] and include the second weak term in (3.5) to penalise spurious solutions that lie on the lowest branch and do not automatically satisfy this condition. This approach alleviates the restriction on geometry or choice of materials. We note that, as we are studying the vanishing limit η, the asymptotic hierarchy is robust even for high-contrast media, a subject that is of great interest in acoustic metamaterials, and approached in this context by Zhikov and others (see [56,60]). In practice, a higher contrast will limit the frequency range over which we can expect our results to be reliable. Returning to the current problem, the general solution to (3.3) is a linear combination of independent vector solutions whose coefficients must be allowed to vary on the long scale. Explicitly H 0i (X, ξ) = f (r) 0 (X)h (r) 0i (ξ, Ω 0 ) for i = 1, 2, 3, and we sum over r = 1, 2, ..., p for an eigenvalue with multiplicity p. Each term is the product of a long-scale scalar modulation function f (r) 0 (X) with a vector short-scale h (r) 0i (ξ, Ω 0 ) that is a known Bloch eigensolution of (3.3); the aim is to find PDEs that the long-scale functions satisfy. Note that the separated-scale component of functions are distinguished by lower-case lettering; this convention is followed for the remainder of this paper. We now move to the next order in the hierarchy, resulting in an inhomogeneous version of the leading order equation: O(η) : −∇ ξ × −1 r ∇ ξ × H 1 + µ r Ω 2 0 H 1 = ∇ ξ × −1 r ∇ X × H 0 + −1 r ∇ X × ∇ ξ × H 0 − µ r Ω 2 1 H 0 . (3.6) with periodicity conditions as before and continuity conditions given by −1 r (∇ ξ × H 1 + ∇ X × H 0 ) ∂D1,2 × n = 0, H 1 ∂D1,2 × n = 0, (3.7) at the phase interfaces. Before attempting to solve this, we derive a compatibility condition (see appendix A), whose implication depends on the nature of the eigenvalue in question. The result resembles a p-component Dirac equation in long-scale position space: P nr j ∂f (r) 0 ∂X j + Ω 2 1 Q nr f (r) 0 = 0, (3.8) where P nr j = iΩ 0 C e 0 (n) * × h (r) 0 + e 0 (r) × h (n) * 0 j dV, Q nr = C µ r h (n) * 0 · h (r) 0 dV. Since the operator in (3.3) is Hermitian, we can always choose the eigenvectors h (n) 0 to be orthonormal such that Q nr = δ nr , and this can be achieved using the Gram-Schmidt procedure. For fixed n, P nn j is proportional to the j-component of the complex Poynting flux e 0 (n) × h (n) * 0 through the cell, which is necessarily zero for standing wave solutions satisfying the boundary conditions (3.4). This observation is easily proved from the fact that with (anti-)periodic boundary conditions it is always possible to normalise the fields such that either e 0 (n) or h (n) 0 is entirely real and the other is imaginary. The value of P nr j for n = r depends on the nature of the degeneracy under consideration; the first and most frequent type encountered is an essential degeneracy, which results from the invariance of the system (i.e. the elementary cell and boundary conditions) under a particular symmetry transformation. For cubic and square arrays, such degeneracies can be shown by symmetry to yield P nr j = 0. The second and more interesting case is that of an accidental degeneracy, which occurs if we continuously tune the parameters of the structure until two otherwise distinct eigenvalues occur at the same frequency. Such degeneracies do not result from a symmetry group of the cell, and in general P nr j = 0 for n = r. Such cases are dealt with in section 3.1. We deduce from (3.8) that for an eigenvalue with P nr j = 0 for every n and r then Ω 1 = 0 for non-trivial solutions. This results in locally quadratic dispersion bands passing through essentially degenerate eigenvalues (which is not necessarily the case for other Bravais lattices, as noted in [14]). With this in place, we are in a position to solve (3.6). The solution consists of a complimentary part, which has the same short scale form as the leading order solution, and a particular solution modulated by first order partial derivatives of f 0 . The most general form thus has components given by H 1i (X, ξ) = f (r) 1 (X)h (r) 0i (ξ, Ω 0 ) + f (r) 0,Xj (X)h (r) 1ij (ξ, Ω 0 ). Exploiting the linearity of the system and the independence of the disparate position variables, (3.6) then reduces to 3p separate systems of 3 coupled equations corresponding to each permutation of j, r. These are solved numerically using finite elements. We next move onto the O(η 2 ) system: O(η 2 ) : −∇ ξ × −1 r ∇ ξ × H 2 + µ r Ω 2 0 H 2 = ∇ ξ × −1 r ∇ X × H 1 + −1 r ∇ X × ∇ ξ × H 1 + −1 r ∇ X × ∇ X × H 0 − µ r Ω 2 2 H 0 , (3.9) with periodicity conditions as before and phase boundary conditions: −1 r (∇ ξ × H 2 + ∇ X × H 1 ) ∂D1,2 × n = 0, H 2 ∂D1,2 × n = 0. (3.10) Here we derive a second compatibility condition (see appendix B), analogous to that of the previous order. The result is an effective PDE satisfied by the modulation function f 0 (X), posed on the long-scale. For each independent eigenfunction at Ω 0 , f 0 is governed by T ij ∂ 2 f 0 ∂X i ∂X j + Ω 2 2 f 0 = 0 ⇐⇒ T ij ∂ 2 f 0 ∂x i ∂x j + (ω 2 − ω 2 0 ) c 2 f 0 = 0, (3.11) where the components of the tensor T ij are formed from integrals over the cell involving the leading and first order solutions and are given in appendix B. The equation on the left makes clear the crucial point that the short scale is completely absent, having been integrated out and encapsulated by constant tensor T ij , whereas the equation on the right has been rewritten in terms of dimensional variables, and makes clear that the the long-scale position variable X is an artefact of the asymptotic method. In a continuum setting, (3.11) permits Bloch wave solutions of the form f 0 = exp(iκ · X/η) = exp(iκ · x/l), where the dimensionless quantity κ = (k − k 0 )l is proportional to the difference between the Bloch vector at frequency Ω and that at Ω 0 , corresponding to the vertex of the irreducible Brillouin zone. Substituting this into (3.11) leads to locally quadratic behaviour of the dispersion bands as Ω 2 2 = T ij κiκj η 2 =⇒ Ω = Ω 0 + T ij κiκj 2Ω0 , where for the second relationship we have used the binomial theorem to expand the ansatz (3.2(b)) for small κ. This serves as a useful verification of the asymptotic method, giving expressions for the local behaviour of the dispersion curves, as shown in figure 3.2 for the case of perfectlyconducting spherical inclusions. The low-frequency limit, as well as the case of so-called Dirac-like points differ from this, both yielding locally linear dispersion and discussed in the following two sections. Γ X M Γ R X M R Γ Ω 0 0.5 1 1.5 2 2.5 (a) (b) (c) 2l 0.79l R M X Dirac-like points The unusual properties assciated with Dirac points in graphene [46] have prompted a surge of interest in the study of Dirac-like points in photonics [3,14]. Such points are characterised by linear crossing of dispersion branches at Brillouin zone vertices, and in terms of the asymptotic theory are associated with non-trivial degeneracies of (3.3). In figure 3.2, a so-called accidental degeneracy is induced by choosing the radius of the spherical inclusion such that two eigenvalues coincide at R point. In contrast, the other degeneracies observed in the band diagram are robust under the change of radius or material parameters, and can only be removed by breaking the symmetry of the cell; these are called essential degeneracies. In the case of an accidental degeneracy (or incidentally for an essential degeneracy in certain non-cubic/square lattices), we find that the quantity P nr j appearing in equation (3.8) contains non-zero elements. As a result, the dispersion is governed to leading order by (3.8), which results, unusually, in the appearance of locally linear dispersion at non-zero frequencies at a Brilouin zone vertex. It has been noted by various authors [3,32], that unusual wave guiding effects, as well as perfect transmission and cloaking can be observed in some instances around Dirac-like points. Such behaviour is considered in detail by Chan et al in [14], where it is demonstrated using an effective medium theory [59] that under certain conditions the dispersion can be mapped to that of a zero refractive index material. This is shown to occur if one can induce an accidental degeneracy between two triply degenerate eigenvalues at Γ point, which is then demonstrated to be possible in a core-shell structure with perfectly conducting inclusions. The local dispersion in such cases is characterised by the crossing of two linear bands, with relatively flat quadratic bands passing through the middle, as seen in figure 3.3. [14]. Here the air-filled elementary cell (b) contains a spherical shell of permittivity 12 surrounding a perfectly conducting core. The effective equation governing the (repeated) linear branches is 0.054f 0,X 1 X 1 + 0.054f 0,X 2 X 2 + 0.054f 0,X 3 X 3 + Ω 4 1 f0 = 0, resulting from (3.12). In order to extract the relevant asymptotics in the case of a Dirac-like point, we differentiate (3.8) once with respect to X i , and then by self-substitution obtain D nr ij ∂ 2 f (r) 0 ∂X i ∂X j + Ω 4 1 f (n) 0 = 0,(3.12) where D nr ij = −C np j C pr i and C nr j = (Q −1 ) np P pr j . Substituting the Bloch-wave solution f (r) 0 (X) =f (r) 0 exp(iκ · X/η) for fixed small κ leads to a homogeneous matrix equation that can be solved numerically to give the linear asymptotics, as has been done in figure 3.2, but in general it is not possible to obtain separate equations governing the evolution of the distinct modes. This is consistent with the assertion in [14] that in general no effective medium description can be applied satisfactorily at Dirac-like points. Under certain circumstances, however, it does turn out to be possible; one such example being that of figure 3.3, with (3.12) decoupling to four identical isotropic equations: 0.054f 0,X1X1 +0.054f 0,X2X2 +0.054f 0,X3X3 +Ω 4 1 f 0 = 0, governing the linear bands, with a final pair yielding Ω 1 = 0, and hence with quadratic dispersion governed by (3.11). For the modes with linear dispersion, we deduce an effective Helmholtz equation for the Bloch wave, written in dimensional quantities as ∇ 2 f 0 + n 2 eff 0 µ 0 (ω 2 − ω 2 0 )f 0 = 0, with n 2 eff = 0 µ 0 (ω 2 − ω 2 0 )l 2 /0.054 → 0 as ω → ω 0 . 3.2 The low-frequency long-wave limit in dielectric media One surprising feature of HFH, when applied to a vector problem, is that the effective PDEs or systems of PDEs are scalar equations that do not mirror the form of the governing vector system of (2.3). This observation leads to an apparent inconsistency with classical homogenisation theory in the low-frequency regime, where the result is a vector equation resembling (2.3) but with effective tensors playing the role of and µ [29]. In fact, using the current formulation, the effective permittivity and permeability are embedded in the system of equations (B.2), which correctly describes the low-frequency linear asymptotics, but the two tensors cannot in general be extracted individually. However, in the commonly-occurring case of purely dielectric media, to which we now specify, the effective permeability µ r ≡ 1, and the effective permittivity tensor can then be extracted, as we now proceed to show. Classical homogenisation theory is quasi-static and limited to dealing with situations where the harmonic frequency is sufficiently low that the field on the cell level is characterised by a small perturbation to an otherwise static field. This leads to dispersionless straight lines emanating from the Γ point at zero frequency such as those seen in figure 3.2. Formally, this occurs when Ω 2 = O(η 2 ), corresponding to the substitution Ω 0 = 0 into the leading order system, along with the divergence-free condition ∇ ξ · H 0 = 0, which is no longer automatically satisfied by solutions of (3.3). Following the HFH procedure of section 3, we deduce that the leading order eigensolution is a constant vector field, and Ω 0 = 0 is thus to be treated as a repeated eigenvalue with multiplicity p = 3, corresponding to the three spatial directions. After some algebra, the resulting system of coupled effective PDEs can indeed be written as a single vector equation, which is the familiar result from classical homogenisation theory given in the context of dielectric media in [28]: − ∇ × ( −1,hom r ∇ × H 0 ) + ω 2 0 µ 0 H 0 = 0,(3.13) where −1,hom ij = −1 δ ij +˜ −1 ij , and H 0 is the leading order magnetic field. Here · denotes a cell-averaged quantity, and the correction tensor˜ −1 ij has components depending on the geometry of the cell. To prove this, we note that in the quasi-static limit Ω 0 = 0, the leading order solutions can be chosen without loss of generality to be constant vectors oriented along each of the three co-ordinate axes so that h for some secondrank tensor −1,hom kl , which is equivalent to the statement that the tensor T contains the following symmetries: T nr ij = T ij nr = −T nj ir = −T ir nj . Substituting the constant fields into (B.2) we find the explicit form for T as T nr ij = C −1 r δ ij δ nr − δ jn δ ir + ∂ i h (r) 1nj − ∂ n h (r) 1ij dV. (3.14) It is straightforward to show that the first two terms in the integrand satisfy the necessary symmetries, which are inherited directly from the −∇ X × ∇ X × (·) operator in (B.1). It is also trivial to note that the remaining two terms are antisymmetric in n and i. The only non-trivial symmetry that remains is antisymmetry of the second two terms with respect to j and r. In order to prove this, we consider the inhomogeneous problem (3. 1ij leads to the following coupled problems for i, j, r ∈ {1, 2, 3}: ∂ k ( −1 r ∂ k h (r) 1ij ) − ∂ k ( −1 r ∂ i h (r) 1kj ) = δ ij ∂ r −1 r − δ ir ∂ j −1 r ,(3.15) subject to boundary conditions at any phase interface given by −1 r ∂ k h (r) 1ij n k − −1 r ∂ i h (r) 1kj n k D1,2 = −1 r D1,2 (δ ij n r − δ ir n j ), (3.16) where n i is the component of the unit normal n in the x i -direction and [·] denotes a jump discontinuity. The two-scale expansion of the divergence-free condition ∇ ξ · H 1 = −∇ X · H 0 must also be imposed, which gives f (r) 0,Xj h (r) 1ij,ξi = −f (r) 0,Xj δ jr . In the case that j = r, this condition is homogeneous, and because the right hand sides of both (3.15) and (3.16) are antisymmetric in j, r, we deduce that h (r) 1ij = −h (j) 1ir . On the other hand, for j = r the divergence-free condition yields h (r) 1ij,ξi = −1. Integrating this over the cell C and applying the divergence theorem and periodicity leads to a contradiction "0 = 1" that can only be avoided if f (r) 0,Xj δ jr = 0. We have hence derived the long-scale divergence-free condition ∇ X · H 0 = 0, and with this in place we can show for j = r that h (r) 1,ij = 0. In doing so, T is deduced to contain the correct symmetries, and we derive an expression for the effective permittivity as follows: T nr ij = ε nik ε jrl −1,hom kl =⇒ −1,hom kl = 1 (2!) 2 ε nik ε jrl T nr ij . (3.17) For each permutation of k, l, we only need evaluate one component of T . It is natural to split the resulting inverse permittivity tensor into a sum of two parts, the first resulting from the Kronecker delta terms in the integrand of T nr ij , given by −1 δ ij , and the second from the two remaining terms, which we refer to as the correction tensor Application to PCFs We now turn our attention to the case of dielectric cylinders of infinite length and constant permittivity c aligned in the x 3 -direction and embedded in a matrix phase with permittivity m . This model is appropriate for both holeytype PCFs, in which the cylinders are air holes in a dielectric background, as well as ARROW-type PCFs, in which the cylinders have a higher refractive index than the background. As discussed in section 1, the x 3 -dependence of the solution can be factored out as ∝ exp(iβx 3 ) where β is the propagation constant of the radiation. For a fixed value of β, the resulting Bloch wave structure is projected onto the (x 1 , x 2 ) plane and the effective PDEs will govern propagation in these directions. Tuning β, along with the geometric and material properties of the cell, can then be used to induce particular features, such as partial stop-bands or Dirac-like points in the transverse plane. In figure 4.2, we show the band diagrams for one such configuration of air holes in a dielectric background at two different values of β chosen such that a transverse stop-band and a Dirac-like point are exhibited. In order to obtain the quasi-planar analogues of (3.8) and (3.11), we simply make the substitutions ∂/∂X 3 → 0 and ∂/∂ξ 3 → iβ, and the corresponding integrals become surface integrals over the two-dimensional cell. It is also straightforward to check for β = 0 that our theory reduces to that of planar HFH, as published in [2,20]. , β has been increased to 3/l, causing the higher-frequency bands to separate from the lowest two, so the PCF exhibits a partial band gap in the transverse plane. Note that different scales have been used on the vertical axes. Transverse stop bands and localised defect modes So far we have focused on the application of HFH to propagating 'slow-modes' inside a bulk periodic structure, and the corresponding modes in the quasi-planar case propagate in the transverse plane as well as in the axial direction. Alternatively, we may consider a structure exhibiting a transverse stop band at a particular value of β. If we introduce a finite defect of some kind into the structure, it may be possible to excite a mode whose frequency lies within the stop band, and hence decays evanescently in the surrounding medium. If the frequency lies in the vicinity of one of the eigenfrequencies Ω 0 , the exponentially decay is then governed by (3.11). In figure 4.3, we show one such mode, where a defect is induced by removing one air cylinder from a large array of cells corresponding to those of fig 4.2(b). Such a model is highly relevant for real-life applications of PCFs, and one can envisage the extension of such work giving rise to the notion of 'effective fibres'. 4.2(b). The dominant effective equation is 1.01f 0,X 1 X 1 + 0.098f 0,X 2 X 2 + Ω 2 2 f0 = 0, where Ω0 = 1.707, corresponding to the third eigenvalue at X, which leads to directed leakage in the x1-direction, and hence also in the x2-direction due to the symmetry point at the opposite vertex of the first Brillouin zone. The decay of the mode in these directions is then very well approximated by an exponential form exp(−αx) (the red dashed line in (b)), where the decay rate α = 0.24/l follows from the PDE. Application to Rayleigh-Bloch modes For various wave systems, it is well known that periodic arrays of defects in otherwise homogeneous media can support localised modes. Such modes exhibit Bloch wave quasi-periodicity in the array, and decay evanescently in the surrounding medium, reminiscent of Rayleigh waves at the surface of solids. It is therefore natural to describe them as Rayleigh-Bloch modes, and their existence is ubiquitous in systems where the dimensionality of the array is at least one less than that of the wave system in question. Rayleigh-Bloch waves are distinguished from "pure" surface waves (e.g. Rayleigh waves, Lamb waves, etc.) in that they can propagate along surfaces that exhibit some material or geometric periodicity and which, in the absence of this periodicity, do not support surface waves. In this sense they are similar to spoof surface plasmons [49]. In two-and three-dimensional systems, linear arrays may be employed as diffraction gratings, and have been studied in the context of water waves and acoustics as solutions to the Helmholtz equation [39,52,57], surface plasmons [42,45], as well as coupled elastic waves [16]. In three-dimensional systems, such as the vector Maxwell system, Rayleigh-Bloch modes can exist for linear or planar arrays [40], the latter allowing for the possibility of a range of in-plane effects to be observed. The asymptotic technique we have developed is particularly well-suited to studying electromagnetic Rayleigh-Bloch systems, as the corresponding full numerical computations can be prohibitively demanding, particularly for fully coupled three-dimensional problems of the type examined here. In order to assess and demonstrate the efficacy of the asymptotic technique developed in the preceding sections, we consider an infinite doubly-periodic planar array of dielectric spheres embedded in free space, as depicted in figure 5.1. Once again, the theory is largely unaltered from the three-dimensional case; we simply make the substitution ∂/∂X 3 → 0 in (3.8) and (3.11), and extend the cell in the x 3 direction as shown in figure 5.2(b). The corresponding photonic band diagram is shown in 5.2(a). Planar dynamic anisotropy & hyperbolic metafilms The band diagram of figure 5.2 and has several interesting features including a planar band-gap as well as flat bands associated with so-called slow waves. A particularly fascinating feature is the change in curvature of the lowest branch at lattice point X in the reciprocal space, shown in the inset of figure 5.2(a) along with the corresponding local asymptotics. Such saddle points are associated with dynamic anisotropy [17,19,48] and hyperbolic materials [50,54]; this type of media, when excited by an appropriate multipole source, direct the electromagnetic energy along clearly defined characteristic lines. The existence of these saddle points for the system under consideration suggests the interesting possibility of dynamic anisotropy in a Rayleigh-Bloch wave setting, creating an effective hyperbolic metasurface. Using the asymptotic theory developed earlier, we first obtain the leading order solution for the field when the array is excited by a dipole source at a frequency close to that of the saddle point on the dispersion surface (Ω 0 = 0.8071). This involves solving the leading and first order cell problems as outlined in section 3, and the solvability condition for the second-order problem then generates a PDE, of the same form as (3.11), for the long-scale field: T ij ∂ 2 f 0 ∂x i ∂x j + (ω 2 − ω 2 0 ) c 2 f 0 = 0, (5.1) where T is now a rank-2 tensor. In this case, we find that T = diag(−0.19, 0.11) is diagonal and T 11 T 22 < 0, yielding a hyperbolic PDE on the long-scale. The leading order asymptotic field for the forced problem is then a superposition of the solution to this PDE, whose forcing must take into account the symmetry of the cell problem, along with its symmetrical counterpart excited at point Y in the Brillouin zone. The result is most striking when the electric field is plotted, as in figure 5.3a, where since the hyperbolic mode is primarily out-of-plane electric, For comparison the above problem is also treated, purely numerically, using finite elements. In particular, a 37×37 planar array of spheres is considered; the finite cluster is surrounded by regions of perfectly matched layers in order to simulate an infinite domain. Moreover, the computational cost of the full finite element simulation is lessened by making full use of the available symmetries, thereby reducing the computational domain by a factor of 8. The array is then excited, at the centre of the array, by a dipole source of unitary magnitude and oriented along the x 3 -axis. The E 3 component of the electric field from the full numerical simulation is shown in figure 5.3b and should be compared with the asymptotic field shown in figure 5.3a. For comparison, we also plot the field along the line (x, y, z) = (x, x, 0) and passing through the dipole source (see figure 5.3c). Finally, figure 5.4 shows the E 3 component of the electric field both in the plane of the array and, over two slices, perpendicular to the array. The figure illustrates the decay of the field in the direction perpendicular to the array in addition to the dynamic anisotropy exhibited in the plane of the array. The novel exhibition of dynamic anisotropy on a structured metafilm is an exciting effect with several potential applications in the guiding of surface waves; this effect allows not only the confinement of waves to a surface or interface, but also the control of their propagation within the metafilm itself. These effects occur, by necessity, at frequencies where the wavelength is comparable to the size of the microstructure where traditional (long-wave) homogenisation theories are no longer valid. However, as demonstrated by figure 5.3, the asymptotic homogenisation scheme developed here is capable of accurately and conveniently describing such effects. Although we have not carried out a comprehensive analysis of the computational cost of the asymptotic method, it is illuminating to consider the following: The full finite element simulation, using all the available symmetries, required approximately 16.8 million degrees of freedom and took 11.5 hours to solve on a dedicated machine using 77GB of RAM and 16 processor cores running at 2.5GHz. In comparison, the numerical element of the asymptotic scheme used around 830 thousand degrees of freedom, 3.7GB of RAM and required 2.7 minutes to solve on a laptop with 2 cores running at 2.5GHz. Concluding remarks We have demonstrated that in the vicinity of Brillouin zone vertices for a periodic structure, photonic modes governed by the vector Maxwell system give rise to second-order scalar PDEs, whose solutions govern the propagation, or decay, of waves inside the structure. In some cases, these PDEs can be used straightforwardly to predict qualitatively and quantitatively the long-scale behaviour of a periodic photonic structure, which has been demonstrated in the context of a forced problem in a planar PC, as well as an eigenvalue problem for a localised mode in a PCF. A particularly interesting physical example was considered in section 5 wherein the asymptotic homogenisation theory was applied to a fully coupled three-dimensional problem for the Maxwell system. Using a planar array of dielectric spheres we were able to create an effective hyperbolic metafilm which supports surface waves that also exhibit dynamic anisotropy over the surface. One can envisage many applications where the ability to guide electromagnetic waves along a surface whilst also controlling the propagation over the surface itself would be useful. A natural extension of this work is to consider more general Bravais geometries that are not necessarily square/cubic. In fact, such work has recently been completed for the simpler case of TE-polarised radiation [43], and it is clear that the majority of the theory presented here carries through mostly unaltered. A second extension would be to consider structures with a wider range of material properties, for example those experiencing dielectric loss, or perhaps more interestingly real metals, whose frequency-dependent permittivity can be described by the Drude model. B The O(η 2 ) compatibility condition: effective PDEs We now derive a compatibility condition from (3.9) following a similar methodology to that of appendix A; we begin by taking the scalar product of (3.9) with h (n) * 0 , subtract from the scalar product of (3.3) * with H 2 /f (n) 0 , and integrate the resulting equation over the elementary cell C. The left hand side vanishes like the first order equivalent, leaving 0 = C − −1 r ∇ ξ × h (n) * 0 · ∇ X × H 1 − −1 r h (n) * 0 · ∇ X × ∇ ξ × H 1 − −1 r h (n) * 0 · ∇ X × ∇ X × H 0 + µ r Ω 2 2 h (n) * 0 · H 0 dV. (B.1) Using the cyclic property of the scalar triple product, it is straightforward to show that the contribution of the complementary part of H 1 in the above equation is zero. Changing to tensor notation, and making use of the contracted epsilon identity ε ijk ε klm = δ il δ jm − δ im δ jl , we are led to the following system of effective PDEs: T nr ij ∂ 2 f (r) 0 ∂X i ∂X j + Ω 2 2 Q nr f (r) 0 = 0 ⇐⇒ T nr ij ∂ 2 f (r) 0 ∂X i ∂X j + Ω 2 2 f (n) 0 = 0, (B.2) where T nr ij = (Q −1 ) nqT qr ij , with Q nr = C µ r h In the case of a distinct eigenvalue, the above system reduces to a single equation with n = r = 1. For a repeated eigenvalue, the substitution f (r) 0 (X) =f (r) 0 exp(iκ · X/η) yields a homogeneous matrix equation whose solution gives the quadratic asymptotics, but in fact the independent Bloch modes are non-interacting in the sense that they can be decoupled as: M npT pq ij (M −1 ) qr ∂ 2 f (r) 0 ∂X i ∂X j + Ω 2 2 f (n) 0 = 0, (B.4) where M is matrix of eigenvectors shared by the the matrices T ij , and henceT ij are all diagonal matrices. Premultiplying by M −1 , we are then left with the decoupled system 0 . Note that the tildes have been omitted in equation (3.11) as it is assumed that the decoupling is done automatically. Figure 2 . 1 : 21First Brillouin zones for (a) simple square and (b) simple cubic arrays. The triangle in (a), or tetrahedron of (b), whose vertices are labelled are the irreducible Brillouin zones, defined as the first Brillouin zones reduced by the point symmetries of the cell, in this case that of the square, cube respectively. Figure 3 . 1 : 31Disparate co-ordinate systems in a two-phase triply-periodic structure. The micro-scale co-ordinates (ξ1,ξ2,ξ3) capture the field variation over a single elementary cell C = D1 ∪ D2, whilst the macro-scale co-ordinates (X1,X2,X3) encode the long-scale behaviour of the medium. Figure 3 . 2 : 32(a) Photonic band diagram for a cubic array of perfectly-conducting spheres of radius r = 0.79l in a background of air, as shown in (b). The band structure is plotted around the edges of the tetrahedral irreducible Brillouin zone (c). Solid black curves are from FEM calculation, and red dashed curves are asymptotics from equation(3.11). Note the Dirac-like point at R, which occurs only at this particular value of the radius. Figure 3 . 3 : 33(a) Dirac-like dispersion in a core-shell structure as demonstrated in T . We need to access the algebra of appendix B and compare equation(3.13) to the HFH result for repeated eigenvalues, (B.2): the two are identical if we can write T nr ij = ε nik ε jrl −1,hom kl Putting all this together we recover (3.13) and hence the quasi-static homogenisation of Maxwell's equations. Figure 4 . 1 : 41Reduction of a typical fibre-like structure to two dimensions. As shown in (a), dielectric cylinders of permittivity c are embedded in a homogeneous matrix phase with permittivity m . The resulting problem is projected onto the (x1, x2) plane, and the angle of propagation depends on the ratio of the parameter β and the planar Bloch wave vector, which characterises the phase shift across an irreducible cell (the dashed square in (b)) Figure 4 . 2 : 42Transverse photonic band diagrams for the lowest 5 bands for a square array of cylindrical air holes of radius r = 0.75l, where 2l is the pitch of the array, in a dielectric background with relative permittivity r = 6. In (a), the parameter β = 0.89/d has been tuned to induce an accidental degeneracy, resulting in a Dirac-like point at M . In (b) Figure 4 . 3 : 43Magnetic field norm for a localised defect mode at Ω = 1.690, induced by removing one air cylinder from a large array of cells (of which a portion are shown here), corresponding to those of fig Figure 5 . 1 : 51A Doubly-periodic planar square array of dielectric spheres. For the purpose of computation, the spheres have radii of 0.8l, where the pitch of the array is 2l. The spheres are embedded in free space and have a relativity permittivity r = 20. Figure 5 . 2 : 52(a) In-plane photonic band diagram for a square array of high-permittivity ( r = 20) dielectric spheres in vacuum, as shown in figure 5.1. The radius of the spheres is 0.8l, where 2l is the pitch of the array, corresponding to the elementary cell (b). Here the cell is extended in the ξ3-direction, and perfectly conducting boundary conditions are applied at its top and bottom of the cell. The inset of (a) shows an enlarged portion of the diagram in which the local behaviour is governed by the hyperbolic effective equation −0.19f 0,X 1 X 1 + 0.11f 0,X 2 X 2 + Ω 2 2 f0 = 0, leading to the asymptotics shown by red dots. hence we only include the E 3 -component of the field. Here, the non-dimensional angular frequency of excitation is Ω = 0.80705, which lies close to the resonant frequency of Ω 0 = 0.8071. Both the cell and long-scale problems were solved using the commercial finite element package Comsol Multiphysics R . Figure 5 . 3 : 53The z-component of the electric field for the planar array obtained from (a) the asymptotic analysis and (b) the full finite element simulations. In both cases the colour scale is linear running for minimum (blue), through zero (green) to maximal (red). Part (c) shows the z-component of the electric field plotted along a line passing through the points x = (0, 0, 0) and(1, 1, 0). The solid curve represents the field from the full finite element simulation, whereas the dashed curve corresponds to the leading order solution from the asymptotic analysis. The non-dimensional angular frequency of excitation is Ω = 0.80705, which is close to the resonant frequency of Ω0 = 0.8071. Figure 5 . 4 : 54Slices through the array showing the E3 component of the electric field and illustrating the decay of the surface wave in the direction perpendicularto the array together with the in-plane dynamic anisotropy. Once again, the colour scale is linear running for minimum (blue), through zero (green) to maximal (red). We can write the general leading order solution as a linear combination of these decoupled modes H 0 =f 6): substituting in the constant leading order solutions, along with the candidate particular solution H 1i = f(r) 0,Xj h (r) AcknowledgementsThe authors thank the EPSRC for support through research grants (EP/I018948/1, EP/L024926/1, EP/J009636/1) and Mathematics Platform grant (EP/I019111/1).A The O(η) compatibility conditionFor each n ∈ {1, 2, ...p}, we first take the scalar product of (3.6) with h (n) * 0 and subtract from the scalar product of (3.3which is simply the divergence of the quantity −1Integrating this over the elementary cell C and applying the divergence theorem leaves us with an integral over the surface of the cell which vanishes by periodicity/antiperiodicity of H 1 and h (n) 0 , along with a non-vanishing contribution at the phase boundaries given bywhere again square brackets denote a jump discontinuity. 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Backaction-evading measurements of mechanical motion can achieve precision below the zero-point uncertainty and quantum squeezing, which makes them a resource for quantum metrology and quantum information processing. We provide an exact expression for the conditional state of an optomechanical system in a two-tone backaction-evading measurement beyond the standard adiabatic approximation and perform extensive numerical simulations to go beyond the usual rotating-wave approximation. We predict the simultaneous presence of conditional mechanical squeezing, intra-cavity squeezing, and optomechanical entanglement. We further apply an analogous analysis to the multimode optomechanical system of two mechanical and one cavity mode and find conditional mechanical Einstein-Podolski-Rosen entanglement and genuinely tripartite optomechanical entanglement. Our analysis is of direct relevance for state-of-the-art optomechanical experiments that have entered the backaction-dominated regime.

10.1103/physrevlett.123.093602

1903.05901

952984788dbb840ef23a34cff90783e25d0fe276

Conditional dynamics of optomechanical two-tone backaction-evading measurements 14 Mar 2019 Matteo Brunelli Cavendish Laboratory University of Cambridge CB3 0HECambridgeUnited Kingdom Daniel Malz Max-Planck-Institut für Quantenoptik Hans-Kopfermann-Strasse 1D-85748GarchingGermany Andreas Nunnenkamp Cavendish Laboratory University of Cambridge CB3 0HECambridgeUnited Kingdom Conditional dynamics of optomechanical two-tone backaction-evading measurements 14 Mar 2019(Dated: March 15, 2019)2 10 -4 10 -2 Backaction-evading measurements of mechanical motion can achieve precision below the zero-point uncertainty and quantum squeezing, which makes them a resource for quantum metrology and quantum information processing. We provide an exact expression for the conditional state of an optomechanical system in a two-tone backaction-evading measurement beyond the standard adiabatic approximation and perform extensive numerical simulations to go beyond the usual rotating-wave approximation. We predict the simultaneous presence of conditional mechanical squeezing, intra-cavity squeezing, and optomechanical entanglement. We further apply an analogous analysis to the multimode optomechanical system of two mechanical and one cavity mode and find conditional mechanical Einstein-Podolski-Rosen entanglement and genuinely tripartite optomechanical entanglement. Our analysis is of direct relevance for state-of-the-art optomechanical experiments that have entered the backaction-dominated regime. Introduction.-The standard quantum limit (SQL) is the precision limit that arises from the fundamental trade-off between the information extractable from a measurement and the associated backaction when continuously monitoring the mechanical motion [1,2]. Backaction-evading (BAE) measurements bypass this limit by restricting the measurement to a single quadrature of motion [3][4][5]. One way to implement this is to parametrically couple the mechanical motion to a cavity driven on both mechanical sidebands [3,6]. BAE measurements have been demonstrated in optomechanics, both in the microwave [7,8] and in the optical domain [9], with sensitivities approaching the SQL. BAE measurements have been exploited to generate spin squeezing in light-controlled atomic ensembles [10]. They have also been extended to collective observables of two modes [11,12], with implementations proposed in an optomechanical system [13] (partially realized in [14]) and in an atomic medium coupled to a nanomechanical resonator [15,16] (realized in [17]). Recent experimental advances have allowed to access the conditional dynamics and real-time feedback of weakly monitored optomechanical systems at the quantum limit [18][19][20][21]. In BAE measurements, continuous monitoring would enable uncertainties below the SQL and the generation of mechanical squeezing, conditional on the measurement record [22][23][24]. Surprisingly, the current literature only considers an approximate description of such process, based on the intracavity field adiabatically following the mechanical motion [25]. With state-of-the-art cavity optomechanics experiments operating in the backaction-dominated regime [26], this description has become inadequate and, as we shall show, it fails to address the quantum features present in the joint conditional dynamics of the cavity mode and mechanical resonator. In this Letter we present an exact treatment of the conditional dynamics of BAE measurements. We give an analytical expression for the steady-state covariance matrix and find that mechanical (single-mode) squeezing is maximal for intermediate sideband parameters. For small sideband parameters, i.e., a very good cavity, the measurement is inefficient, while in the bad-cavity limit, the measurement of the photons leav- ing the cavity only weakly affects the state of mechanical oscillator. We then numerically go beyond the rotating-wave approximation (RWA). Excitingly, we uncover conditional mechanical squeezing, intra-cavity squeezing, and optomechanical entanglement. All of these have been missed by taking the adiabatic approximation. We finally extend our analysis to two mechanical modes coupled to a common cavity. We demonstrate both conditional generation of mechanical Einstein-Podolski-Rosen (EPR) as well as genuine tripartite optomechanical entanglement. Our study provides a substantial improvement in the description of weakly monitored optomechanical systems (as well as parametrically coupled superconducting circuits [27][28][29]) and opens novel avenues for measurement-based quantum control of mechanical motion. Optomechanical conditional dynamics.-We consider a standard optomechanical system where a mechanical oscillator of frequency ω m modulates the frequency of a cavity mode of frequency ω c [30]. The Hamiltonian is given by ( = 1) Mechanical squeezing (in dB) for g = 0.01ωm (red), g = 0.05ωm (yellow) and g = 0.3ωm (cyan) as predicted by Eq. (5). Other parameters are γ = 10 −4 ωm,n = 10, η = 1. Solid black lines represent the prediction of the adiabatic solution σ 2 H = ω câ †â +ω mb †b −g 0â †â (b+b † )+E(t)â † +E * (t)â ,(1) Xm,ad while dashed lines that of a slow cavity σ 2 Xm,slow . For each curve, the part to the left of the black dot (g = κ) is in the strong-coupling regime. single-photon coupling strength, and the cavity is driven on both mechanical sidebands ω c ± ω m with the same strength, i.e., the driving field reads E(t) = 2|E|e −iωct cos ω m t. After linearizing the equations of motion and moving to an interaction picture with respect to the free mechanical and cavity evolution, we obtain the interaction Hamiltonian H I (t) = −gX c X m (1 + cos 2ω m t) +P m sin 2ω m t , (2) with coupling strength g ≡ g 0 |E|/ ω 2 m + κ 2 /4, cavity decay rate κ, and dimensionless quadraturesX c = (â +â † )/ √ 2, X m = (b +b † )/ √ 2, andP m = i(b † −b)/ √ 2. The Hamil-tonianĤ I (t) consists of a time-independent part,Ĥ QND = −gX cXm , and an oscillating partĤ CR (t). In the good-cavity limit κ ω m , the latter term can be neglected and the interaction takes a manifestly QND form [6]. We also include system-environment interactions with the photonic and the mechanical environment [31]. In a quantum noise picture, both environments consist of a collection of uncorrelated modes that interact with the system at time t and are otherwise uncoupled; this assumption both gives rise to a Markovian environment and provides a monitoring channel. After interacting with the system, we assume that the photonic modes of the environment undergo a homodyne measurement of the phase quadrature [23] [see Fig. 1 (a)]. Given the (bi)linear nature of both the interaction and the measurement and given a Gaussian initial state, the state of the optomechanical systemˆ is exhaustively described in terms of the mean vectorx = Tr [ˆ x] and covariance matrix (CM) σ = 1 2 Tr ˆ {x −x, (x −x) T } , where we have grouped the system quadratures into the vector x = (X c ,P c ,X m ,P m ) T [32]. The conditional evolution of the continuously monitored system is then described by the following set of equations [33,34] dx = Axdt − (σB − N )dW t ,(3)σ = Aσ + σA T + D − (σB − N )(σB − N ) T ,(4) where A = A(t) is the drift matrix, D the diffusion matrix, B and N account for the reduction of uncertainty and added noise due to the measurement process; W t is a vector of independent Wiener processes (dW j dW k = δ jk dt); see SM for details [31]. We notice that the stochastic evolution, consequence of the measurement-induced disturbance, is confined to the first moments. Therefore, at any time the conditional state is represented by a Gaussian state whose covariance matrix evolves deterministically according to Eq. (4). This will represent the main tool of our analysis. Mechanical squeezing beyond adiabatic approximation.-We start by studying the conditional dynamics of a two-tone BAE measurement within the RWA, namely when Eq. (2) reduces to the perfect QND interactionĤ QND = −gX cXm . The steady-state conditional CM (4) can be obtained analytically (cf. SM [31]). Here, we will focus on the properties of the variances of the two mechanical quadratures σ 2 Xm = γ 2 + κ 2 + 2ζ 16g 2 ηκ ζ + γ 2 − γ γ 2 + κ 2 + 2ζ ,(5)σ 2 Pm =n + 1 2 + 2g 2 γ(γ + κ) ,(6) where ζ = γκ[16g 2 η(1 + 2n) + γκ],n is the thermal occupancy of the mechanical bath and 0 ≤ η ≤ 1 is the quantum efficiency of the measurement. These exact expressions are the first central result of our work. We note that for η → 0 no measurement is recorded and Eq. (5) reduces to the unconditional variance σ 2 Xm →n + 1 2 , which is consistent with the fact thatX m is a conserved quantity and the initial thermal variance thus remains unaffected. On the other hand, the acquisition of information via the measurement (η > 0) reduces the variance, eventually resulting in mechanical squeezing σ 2 Xm < 1 2 . We show the degree of mechanical squeezing [expressed in −10 log 10 (2σ 2 Xm ) Decibel (dB)] in Fig. 2, as a function of the sideband parameter κ/ω m . It is instructive to distinguish the regimes of a slow cavity (small κ/ω m ) and that of a fast cavity (large κ/ω m ). For a fast cavity κ ω m , we obtain the well-known adiabatic result σ 2 Xm,ad = √ 1+4ηC(1+2n)−1 4ηC , where we introduced the cooperativity C = 4g 2 /κγ. In this regime, decreasing κ leads to a larger cooperativity, so information is extracted faster from the mechanical resonator. This expression can also be obtained by adiabatically eliminating the cavity mode and considering the resulting effective measurement of the mechanical quadraturê X m [35]; this approach was put forward in Refs. [22,25] and has become the standard tool for describing the conditional evolution of weakly monitored systems [6,13,24,26,36]. The adiabatic approximation (solid lines in Fig. 2) clearly becomes inaccurate in the good-cavity, strong-coupling limit. For example, for g = κ = 10 −2 ω m , σ 2 Xm,ad overestimates the actually amount of squeezing by approximately a factor of two (cf. Fig. 2). Physically, as κ decreases, the measurement rate starts to be limited by the rate at which photons leave the cavity and the adiabatic solution fails to account for this effect. To describe this regime correctly, we express Eq. (5) in terms of C and keep only the leading term in the expansion C 1, which yields σ 2 Xm,slow = (1+2n) 3/4 (Cη) 1/4 γ/κ shown as dashed lines in Fig. 2. The rationale is that, for fixed large cooperativity C, increasing κ will increase the measurement rate, which in turn will reduce the variance σ 2 Xm . Our exact solution (5) interpolates between these two limits. An approximate condition for optimal squeezing is obtained from the intersection of the two straight lines in Fig. 2 κ opt = 4g 2/3 [ηγ(1 + 2n)] 1/3 .(7) This gives the optimal value of the sideband parameter, which both depends on the rate at which information is transferred to the cavity mode and the rate at which thermal decoherence influences the measurement. Turning our attention to (6), we notice the variance σ 2 Pm is not affected by the measurement (independent of η). Heating of the phase quadrature is an unavoidable consequence of measurement backaction and entails that highly pure squeezed states are not accessible via continuous monitoring. Furthermore, having access to the full CM we can consider the conditional state of modeâ (which is never squeezed) and the steady-state correlations. Due to the measurement, conditional entanglement between modesâ andb can be established which has been missed in previous analyses of this setting. Effects of counter-rotating terms.-We now explore the effect of the counter-rotating (CR) terms appearing in Eq. (2). For the unconditional dynamics, corrections to the RWA have been studied in Ref. [37]. As the drift matrix is explicitly timedependent, we numerically integrate the equations of motion (4) and consider the long-time limit, when the system settles in a time-periodic steady state. If the effect ofĤ CR (t) is nonnegligible, the ideal QND regime is perturbed and we expect a reduction of mechanical squeezing. This expectation is indeed confirmed by inspecting Fig. 3 (a). However, such a reduction is accompanied by the emergence of two novel features: (i) the stabilization of optomechanical entanglement to considerably larger values [panels (b), (d)] and (ii) the appearance of squeezing in the cavity quadratureX c [panel (c)]. In particular, we see that the presence of CR terms can have a dramatic effect on entanglement, which survives in the steady state, as opposed to the typical entanglement 'sudden death' predicted by RWA [38]. Furthermore, the RWA solution entirely misses intra-cavity squeezing [31]. We thus see that corrections to RWA can lead to qualitatively different quantum features, which is a second major result of the present work. In contrast to unconditional two-tone BAE measurements, where CR terms are always detrimental to quantum correlations [6,37], we find under continuous monitoring quantum correlations can be stronger in their presence. We expect this to be a general feature of continuously monitored systems. Remarkably, in the strong-coupling regime we observe the joint presence of conditional optical squeezing, mechanical squeezing, and entanglement. This unusual set of properties has been predicted for the ground state of a pair of bosonic modes in the ultra-strong coupling regime [39,40] and observed in analog quantum simulation of that model [41]. The presence of monitoring could make the same phenomenology accessible without such stringent experimental requirements. Conditional entanglement in a three-mode optomechanical system.-We now consider two mechanical resonators of frequency ω m,1 and ω m,2 coupled to a common cavity mode, as sketched in Fig. 1 (b). While the two mechanical oscillators are not directly coupled, the measurement of the output cavity field can induce conditional EPR-like entanglement between them [13,15,42]. Following Ref. [13], we introduce the mean and the relative mechanical frequency, respectively, defined as ω = (ω m,1 + ω m,2 )/2, Ω = (ω m,1 − ω m,2 )/2 (we assume ω m,1 > ω m,2 without loss of generality) and the collective EPR mechanical variableŝ X ± = (X m,1 ±X m,2 )/ √ 2 ,P ± = (P m,1 ±P m,2 )/ √ 2 ,(8) that satisfy [X ± ,P ± ] = i, [X ± ,P ∓ ] = 0. In terms ofX + and P − , all-mechanical entanglement is certified by the violation of Duan's inequality σ 2 X+ + σ 2 P− ≥ 1 [43]. Amplitude modulation of a resonant drive at the mean mechanical frequency ω results in the Hamiltonian H I (t) = Ω(X +X− +P +P− ) − √ 2gX cX+ +Ĥ CR . (9) In the limit ω κ, CR terms can be dropped and Eq. (9) becomes a perfect two-mode QND interaction [12,13]. This is due to the fact thatĤ QND =Ĥ I (t) −Ĥ CR (t) couplesX + andP − in the same way as for simple harmonic motion, so that the interaction with the cavity turns into a joint continuous measurement of bothX + andP − . SinceX + andP − commute, they can be simultaneously squeezed by the measurement, while the backaction is confined to the conjugate quadraturesP + andX − [12]. If their combined uncertainties are reduced below twice the zero-point level, the measurement induces conditional mechanical entanglement, in the form of two-mode squeezing. In Fig. 4 (a) we quantify the amount two-mode squeezing through the violation of Duan's bound (expressed in Decibel). We observe a trade-off which can be physically understood as in the single-mode case [cf. Fig. 2], although a simple analytic expression [like Eq. (5)] is no longer available due to the intricacy of the coupled Riccati equations (4). The effects due to CR terms in Eq. (9), responsible for the reduction of the entanglement and the appearance of cavity squeezing for g > κ, are akin to our findings for the single-mode case [cf. Fig. 3 (a), (c)]. We compare our result with the prediction derived in the adiabatic limit (dotted curves, see Ref. [13] for the expressions), which is only accurate for γ Ω, g κ ω. In particular, decreasing the coupling, the adiabatic approximation predicts a constant amount of entanglement, only shifted towards smaller sideband parameters. This prediction can fail dramatically (see red curve), while our theory correctly quantifies mechanical entanglement in the experimentally relevant good-cavity limit. Finally, we study the full conditional dynamics of the threemode optomechanical system, described by Eq. (4), with the appropriate expressions given in the SM [31]. We can determine the separable/entangled nature of the system with respect to all the possible bipartitions, i.e. (â|b 1b2 ), (b 1 |âb 2 ) and (b 2 |âb 1 ), leading to the notion of k-biseparable states [44]. In particular, there are states that are entangled for any bipartition of the modes [45]; these states are called fully inseparable and possess genuine tripartite entanglement. In Fig. 4 (b), (c) we show the inseparability structure induced by the two-mode QND measurement. We find ample regions where genuinely (a) Mechanical two-mode squeezing (in dB) assuming the RWA (dashed curves), beyond the RWA (lighter shaded areas) and in the adiabatic limit (dotted darker curves). The curves are for g = 0.01ω (red, which is zero), g = 0.05ω (yellow) and g = 0.3ω (cyan). As in Fig. 3 the dotted gray curve shows the average two-mode squeezing (taken over 2π/ω). In the inset the conditional cavity squeezing is shown. Other parameters are Ω = 0.1ω, γ = 10 −4 ω,n = 10, η = 1. (b) Inseparability structure of the conditional three-mode optomechanical system. In particular, the orange region indicates genuinely tripartite entanglement and the shaded region marks the presence of mechanical two-mode squeezing. Other parameters as in (a). (c) Same as (b) except forn = 100. tripartite entanglement and mechanical two-mode squeezing (marked by the shaded area) coexist, which survive even for large thermal occupation. Tripartite entanglement in optomechanical devices has been considered in Refs. [46,47], however not under continuous monitoring and for a pair of cavity modes and a single mechanical resonator. Most remarkably, our study shows that continuous monitoring can induce nonclassical features at every 'layer' of the three-mode system: at the single-mode level, the cavity field is squeezed [cf. inset panel (a)]; the two-mode mechanical state is entangled and the optomechanical system as a whole displays genuine multipartite entanglement [ Fig. 4 (b), (c)]. Conclusions.-We provided a description of the conditional dynamics of single-and two-mode BAE measurement beyond the adiabatic limit, which was missing from previous studies. Our results are needed to correctly describe state-ofthe-art experiments and implement quantum feedback. They open new prospects for the generation and characterization of measurement-based squeezing and entanglement, as well as quantum correlations in many-mode systems. Beyond cavity Supplementary Material: Conditional dynamics of optomechanical two-tone backaction-evading measurements DETAILS ABOUT THE OPTOMECHANICAL CONDITIONAL DYNAMICS Two-mode optomechanical system In the following we provide the explicit expressions of the terms appearing in Eq. (4), necessary to quantify the conditional dynamics of the continuously monitored system, together with their derivation. We will employ the phase-space formalism, which is particularly convenient for our problem, and in particular we will follow closely the treatment of Ref. [34]. We start by rewriting the linearized optomechanical Hamiltonian Eq. (2) in terms of the quadrature vectorx = (X c ,P c ,X m ,P m ) T , which takes the formĤ I (t) = 1 2x T Sx, with the matrix S given by S =       0 0 −g(1 + cos 2ω m t) −g sin 2ω m t 0 0 0 0 −g(1 + cos 2ω m t) 0 0 0 −g sin 2ω m t 0 0 0       . (S1) The system-bath couplingĤ diss is modeled by an energy-preserving interaction between each of the system modes and the excitations of two distinct baths, namelyĤ diss = i √ κ(â †ξ c −âξ † c ) + i √ γ(b †ξ m −bξ † m ) ,(S2) which is valid in the weak-coupling limit. For the mechanical system, the limit γ m ω m is also understood, where the damping mechanism of quantum Brownian motion reduces to standard quantum-optical dissipation. The environmental modesξ c,m =ξ c,m (t) are labeled by time and provide a microscopic description of a white-noise process. In terms of quadrature operators the latter condition is expressed by {x b (t),x b (t )} = σ b δ(t − t ), where we defined the vector x b (t) = (X ξc (t),P ξc (t),X ξm (t),P ξm (t)) T , each quadrature operator being defined analogously to system quadratures, and σ b = diag 1 2 , 1 2 ,n + 1 2 ,n + 1 2 ,(S3) withn the thermal occupation of the mechanical bath. Similarly to the optomechanical coupling H I , the bilinear interaction (S2) can be written asĤ diss =x T Cx b ,(S4) where the matrix C is given by C = √ κ ω −1 ⊕ √ γ ω −1 , and we introduced the symplectic form ω = 0 1 −1 0 . With these ingredients at hand, the drift (A) and diffusion (D) matrices appearing in Eq. (4) can be expressed as [33,34] A = ΩS + 1 2 ΩCΩC T ,(S5)D = ΩCσ b C T Ω T ,(S6) with Ω = ω ⊕ ω. Their explicit expression reads A =        − κ 2 0 0 0 0 − κ 2 g(1 + cos 2ω m t) g sin 2ω m t −g sin 2ω m t 0 − γ 2 0 g(1 + cos 2ω m t) 0 0 − γ 2        ,(S7)D = diag κ 2 , κ 2 , n + 1 2 γ, n + 1 2 γ .(S8) We also need to incorporate the measurement process into the dynamical evolution. We consider the case of continuous monitoring of the output cavity field via homodyne detection. This measurement can be described by a projection onto a pure squeezed state, which is modeled by the following covariance matrix σ meas = 1 2 R θ diag(r, r −1 ) R T θ ,(S9) where R θ is a rotation matrix. In particular, homodyne detection of the optical phase quadrature is recovered in the limit r → 0 and θ = π/2. It is also desirable to account for non-unit efficiency of the detection process, which is modeled by a beam splitter of transmissivity √ η prior to the detection, and gives σ η meas = 1 η σ meas + 1 − η 2η 1 . (S10) The matrices B and N describing the effect of the measurement on the environment in Eq. (4) are given by B = CΩ(σ b + σ η meas ) − 1 2 , N = ΩCσ b (σ b + σ η meas ) − 1 2 . (S11) We point out that, since we are interested in the case where only the photonic modes undergo monitoring, the correct way of evaluating Eq. (S11) is to take the covariance matrix of a bipartite measurement [i.e., Eq. (S10) for both optical and mechanical modes] and then taking the limit of vanishing efficiency on the mechanical modes, which corresponds to no monitoring of the mechanical environment. Three-mode optomechanical system In the case of a three-mode optomechanical system, the expressions entering the conditional evolution of the covariance matrix (4) can be easily deduced following the construction outlined above. In particular, the quadrature vector is now given bŷ x = (X c ,P c ,X m,1 ,P m,1 ,X m,2 ,P m,2 ) T and the expression of the optomechanical interaction (9) in terms of the quadratures readsĤ I (t) = Ω 2 j=1,2 (−1) j+1 (X 2 m,j +P 2 m,j ) − g j=1,2X c [X m,j (1 + cos 2ωt) +P m,j sin 2ωt] . (S12) For simplicity, in our study we consider the case of equal single-photon optomechanical couplings, equal mechanical damping rates, and same occupancies of the baths. For non-degenerate mechanical modes, these conditions entail adjusting the local temperatures of the baths to achieve the same occupancy. However, we stress that our analysis can be easily extended to the case of asymmetric couplings and/or damping rates to describe experimental inaccuracies. The expressions entering Eq. (4) are given by A =             − κ 2 0 0 0 0 0 0 − κ 2 g(1 + cos 2ωt) g sin 2ωt g(1 + cos 2ωt) g sin 2ωt −g sin 2ωt 0 − γ 2 Ω 0 0 g(1 + cos 2ωt) 0 −Ω − γ 2 0 0 −g sin 2ωt 0 0 0 − γ 2 −Ω g(1 + cos 2ωt) 0 0 0 Ω − γ 2             ,(S13)D = diag κ 2 , κ 2 , n + 1 2 γ, n + 1 2 γ, n + 1 2 γ, n + 1 2 γ ,(S14)σ b = diag 1 2 , 1 2 ,n + 1 2 ,n + 1 2 ,n + 1 2 ,n + 1 2 ,(S15)C = √ κ ω −1 ⊕ √ γ ω −1 ⊕ √ γ ω −1 ,(S16)Ω = ω ⊕ ω ⊕ ω .(S17) EXPRESSION OF THE STEADY-STATE CONDITIONAL COVARIANCE MATRIX The matrix equation (4) can be solved exactly at the steady state. Besides the expression of the conditional mechanical variances σ 3,3 ≡ σ 2 Xm and σ 4,4 ≡ σ 2 Pm , respectively given in Eq. (5) and (6), the other elements of the covariance matrix read σ 1,1 ≡ σ 2 Xc = 1 2 , σ 2,2 ≡ σ 2 Pc = 1 4ηκ γ 2 + κ 2 + 2ζ + κ(2η − 1) − γ ,(S18)σ 1,4 = g γ + κ , σ 2,3 = 1 8gηκ ζ + γ 2 − γ γ 2 + κ 2 + 2ζ ,(S19) while σ 1,2 = σ 1,3 = σ 2,4 = σ 3,4 = 0. We recall that we set ζ = γκ[16g 2 η(1 + 2n) + γκ]. One can check that the optical phase quadrature is never squeezed, and that steady-state optomechanical entanglement can be present for suitable values of the parameters. PERTURBATIVE SOLUTION FOR THE EFFECT OF COUNTERROTATING TERMS In order to gain some analytical understanding of the long-time behavior of the covariance matrix associated to Eq. (2), we expand the latter in Fourier components [48] σ(t) = n exp(in2ω m t)σ n , retaining only the leading-order contribution σ ±1 for simplicity. In principle, truncating at sufficiently high order yields a set of algebraic equations that capture the steady-state covariance matrix, but as we already have a numerical method, we instead aim to obtain simple closed-form solutions and only perform second order perturbation theory in H CR . Given the solution in RWA σ 0 (see previous section), we find σ 1 , which fulfills 0 = −2iω m σ 1 + A 0 σ 1 + σ 1 A T 0 + A 1 σ 0 + σ 0 A T 1 − σ 0 BB T σ 1 − σ 1 BB T σ 0 + σ 1 BN + N B T σ 1 ,(S20) where we have also introduced Fourier components of the coupling matrix A(t) = n exp(in2ω m t)A n . This equation is linear in σ 1 , which means that it can readily be obtained from the RWA solution for σ 0 (note that σ −1 = σ † 1 ). The n = 0 Fourier components of the covariance matrix cause oscillating variances associated to a periodic steady state, which is the reason why in Fig. 3 the squeezing corresponds to a shaded area rather than a single value. Physically, A ±1 in the above expression are a modulated coupling of the quadratures. To first order, they are a source term for the oscillating variances. The coupling between the quadratures is not QND, such that information about the previously unmonitored quadrature P m now enters the cavity via A ±1 . The last four terms in Eq. (S20) entail that, as a result of mixing of oscillating and stationary parts, the cavity output and thus the conditioning due to the measurement also oscillates. To second order in the counterrotating terms, they affect the stationary part of the covariance matrix as σ 0 + σ 0,correction , with the correction given by σ 0,correction = A −1 σ 1 + σ 1 A T −1 + A 1 σ −1 + σ −1 A T 1 − σ −1 BB T σ 1 − σ 1 BB T σ −1 .(S21) Again we can distinguish two types of contributions. The terms containing A ±1 arise due to the unitary dynamics induced through the CR terms, whereas the terms containing B are a result of the measurement. Deep in the backaction-dominated regime, the correction to the variance of the squeezed quadrature arises entirely from the dynamical part and reads σ 2 Xm,correction = κ 2ω m |χ c (2ω m )| 2 g 2 1 2 + O(γ/ω m ) ,(S22) where χ c (ω) = (κ/2 − iω) −1 is the cavity susceptibility. This contribution can be interpreted as measurement backaction (or shot noise) from the cavity entering the squeezed mechanical quadrature due to the CR terms. The fact that it results from cavity sidebands off resonance is captured by the cavity susceptibility evaluated at the position of the next-order sidebands at 2ω m . On the other hand, the absence of the measurement efficiency clearly indicates that this is a dynamical effect. This is the dominant leading-order source of squeezing loss. We can also look at the correction to the anti-squeezed quadrature, which to lowest order in γ/ω m is σ 2 Pm,correction = −η κ 2ω m |χ c (2ω m )| 2 g 2 1 2 (C + 2n + 1) 2 + O(γ/ω m ), where for convencience we have kept both g 2 and C, which adds slight inconsistencies in the expansion for low γ. Comparison to the full second-order solution obtained from Eq. (S21) shows that Eq. (S23) is indeed a very good approximation. There is a FIG. 1 . 1(a) Backaction-evading (BAE) measurement of a single mechanical quadrature. An optomechanical cavity (â) is driven on the lower and upper mechanical (b) sideband and is continuously monitored via the output homodyne current. Mechanical squeezing, optical squeezing, and entanglement can be generated conditional on the measurement record. (b) If two mechanical modesb1 andb2 are considered instead (as indicated in the dashed boxes), a two-mode BAE measurement can be realized. whereâ (b) describes the cavity (mechanical) mode, g 0 is the arXiv:1903.05901v1[quant-ph] FIG. 3 . 3(a) Conditional mechanical squeezing (in dB) assuming the RWA [Eq.(5)] (dashed dark curves) and beyond the RWA (lighter shaded areas). The curves are for g = 0.01ωm (red), g = 0.05ωm (yellow) and g = 0.3ωm (cyan); when present, the dotted curve shows the mean squeezing (averaged over one mechanical period) and the shaded area extends between the minimum and maximum value of squeezing.Solid black lines represent the prediction of the adiabatic solution σ 2 Xm,ad . (b) Conditional optomechanical entanglement (measured by the logarithmic negativity) for the same coupling values as panel (a); the vertical dashed line corresponds to κ = 0.05ωm and in the inset we show the temporal evolution of entanglement along this cut for the case g = 0.05ωm. (c) Conditional cavity squeezing for the same coupling values as panel (a). (d) Zoom-in of panel (b). In all panels other parameters are: γ = 10 −4 ωm,n = 10, η = 1. FIG. 4. (a) Mechanical two-mode squeezing (in dB) assuming the RWA (dashed curves), beyond the RWA (lighter shaded areas) and in the adiabatic limit (dotted darker curves). The curves are for g = 0.01ω (red, which is zero), g = 0.05ω (yellow) and g = 0.3ω (cyan). As in Fig. 3 the dotted gray curve shows the average two-mode squeezing (taken over 2π/ω). In the inset the conditional cavity squeezing is shown. Other parameters are Ω = 0.1ω, γ = 10 −4 ω,n = 10, η = 1. (b) Inseparability structure of the conditional three-mode optomechanical system. In particular, the orange region indicates genuinely tripartite entanglement and the shaded region marks the presence of mechanical two-mode squeezing. Other parameters as in (a). (c) Same as (b) except forn = 100. Acknowledgments, We thank A. Schliesser for discussions at an early stage of the project. M. B. thanks F. Albarelli, M. Genoni, and A. Serafini for useful discussions. D. M. acknowledges support by the Horizon 2020 ERC Advanced Grant QUENOCOBA. 742102grant agreementAcknowledgments.-We thank A. Schliesser for discus- sions at an early stage of the project. M. B. thanks F. Albarelli, M. Genoni, and A. 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Serafini, New Journal of Physics 17, 073019 (2015). . D Malz, A Nunnenkamp, 10.1103/PhysRevA.94.053820Phys. Rev. A. 9453820D. Malz and A. Nunnenkamp, Phys. Rev. A 94, 053820 (2016). . T Yu, J H Eberly, 10.1126/science.1167343Science. 323598T. Yu and J. H. Eberly, Science 323, 598 (2009). . C Ciuti, I Carusotto, 10.1103/PhysRevA.74.033811Phys. Rev. A. 7433811C. Ciuti and I. Carusotto, Phys. Rev. A 74, 033811 (2006). . C Ciuti, I Carusotto, 10.1103/PhysRevA.74.033811Phys. Rev. A. 7433811C. Ciuti and I. Carusotto, Phys. Rev. A 74, 033811 (2006). . D Marković, S Jezouin, Q Ficheux, S Fedortchenko, S Felicetti, T Coudreau, P Milman, Z Leghtas, B Huard, 10.1103/PhysRevLett.121.040505Phys. Rev. Lett. 12140505D. Marković, S. Jezouin, Q. Ficheux, S. Fedortchenko, S. Fe- licetti, T. Coudreau, P. Milman, Z. Leghtas, and B. Huard, Phys. Rev. Lett. 121, 040505 (2018). . E S Polzik, K Hammerer, 10.1002/andp.201400099Annalen der Physik. 52715E. S. Polzik and K. Hammerer, Annalen der Physik 527, A15 (2015). . L.-M Duan, G Giedke, J I Cirac, P Zoller, 10.1103/PhysRevLett.84.2722Phys. Rev. Lett. 842722L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000). . G Giedke, B Kraus, M Lewenstein, J I Cirac, 10.1103/PhysRevA.64.052303Phys. Rev. A. 6452303G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, Phys. Rev. A 64, 052303 (2001). On the other hand, we recall that 3-biseparable states can be either separable or bound entangled states. On the other hand, we recall that 3-biseparable states can be either separable or bound entangled states. . C Genes, A Mari, P Tombesi, D Vitali, 10.1103/PhysRevA.78.032316Phys. Rev. A. 7832316C. Genes, A. Mari, P. Tombesi, and D. Vitali, Phys. Rev. A 78, 032316 (2008). . Y.-D Wang, S Chesi, A A Clerk, 10.1103/PhysRevA.91.013807Phys. Rev. A. 9113807Y.-D. Wang, S. Chesi, and A. A. Clerk, Phys. Rev. A 91, 013807 (2015). . A Mari, J Eisert, 10.1103/PhysRevLett.103.213603Physical Review Letters. 103213603A. Mari and J. Eisert, Physical Review Letters 103, 213603 (2009). as both result from a coupling to the cavity sideband at 2ω m . Interestingly, the correction to the anti-squeezed quadrature is negative, which means that the variance is decreased. Physically, the CR terms lead to some coupling of the anti-squeezed quadrature into the optical phase quadrature, such that the measurement reduces the uncertainty in P m. striking similarity between the lowest-order correction to the mechanical quadratures Eqs. (S22) and (S23). This conclusion is supported by the fact that the whole expression is proportional to the measurement efficiency. As this reduction is larger in magnitude than the correction to the variance of X m , the mechanical state overall is purified, a conclusion that is borne out by our numerical simulationsstriking similarity between the lowest-order correction to the mechanical quadratures Eqs. (S22) and (S23), as both result from a coupling to the cavity sideband at 2ω m . Interestingly, the correction to the anti-squeezed quadrature is negative, which means that the variance is decreased. Physically, the CR terms lead to some coupling of the anti-squeezed quadrature into the optical phase quadrature, such that the measurement reduces the uncertainty in P m . This conclusion is supported by the fact that the whole expression is proportional to the measurement efficiency. As this reduction is larger in magnitude than the correction to the variance of X m , the mechanical state overall is purified, a conclusion that is borne out by our numerical simulations.

Based on the (3+1)-dimensional hydrodynamic model, the spacetime evolution of hot and dense nuclear matter produced in noncentral relativistic heavy-ion collisions is discussed. The elliptic flow parameter v 2 is obtained by Fourier analysis of the azimuthal distribution of pions and protons which are emitted from the freeze-out hypersurface. As a function of rapidity, the pion and proton elliptic flow parameters both have a peak at midrapidity. *

nucl-th/9904082

4188eb207303a3e09682ae7de29fbd02b5bed10a

Elliptic Flow Based on a Relativistic Hydrodynamic Model Apr 1999 November 5, 2017 Tetsufumi Hirano [email protected] Department of Physics Waseda University 169-8555TokyoJapan Elliptic Flow Based on a Relativistic Hydrodynamic Model Apr 1999 November 5, 2017arXiv:nucl-th/9904082v1 30 Based on the (3+1)-dimensional hydrodynamic model, the spacetime evolution of hot and dense nuclear matter produced in noncentral relativistic heavy-ion collisions is discussed. The elliptic flow parameter v 2 is obtained by Fourier analysis of the azimuthal distribution of pions and protons which are emitted from the freeze-out hypersurface. As a function of rapidity, the pion and proton elliptic flow parameters both have a peak at midrapidity. * One of the main goals in relativistic heavy-ion physics is the creation of a quark-gluon plasma (QGP) and the determination of its equation of state (EoS) [1]. It is therefore very important to study collective flow in non-central collisions, such as directed or elliptic flow [2]. Recently experimental data concerning collective flow in semicentral collisions at SPS energies has been reported [3,4,5]. This data should be analysed using various models. Some groups [6,7,8] have used their microscopic transport models to analyse the collective flow obtained by the NA49 Collaboration [3]. In this paper we investigate collective flow, especially elliptic flow, in terms of a relativistic hydrodynamic model. In non-central collisions elliptic flow arises due to the fact that the spatial overlap region of two colliding nuclei in the transverse plane has an "almond shape". That is, the hydrodynamical flow becomes larger along the short axis than along the long axis because the pressure gradient is larger in that direction. Therefore this spatial anisotropy causes the nuclear matter to also have momentum anisotropy. Consequently, the azimuthal distribution may carry information about the pressure of the nuclear matter produced in the early stage of the heavy-ion collisions [9]. The relativistic hydrodynamical equations for a perfect fluid represent energy-momentum conservation ∂ µ T µν = 0,(1)T µν = (E + P )u µ u ν − P g µν(2) and baryon density conservation ∂ µ n µ B = 0,(3)n µ B = n B u µ ,(4) where E, P , n B and u µ are, respectively, the energy density, pressure, baryon density and local four velocity. We numerically solve these equations without assuming cylindrical symmetry [10,11] by specifying the model EoS and we obtain the space-time dependent thermodynamical variables and the four velocity. We use the following models of the EoS with a phase transition. Hagedorn's statistical bootstrap model [12] with Hagedorn temperature T H = 155 MeV is employed for the hadronic phase. We directly use the integral representation of the solution of the bootstrap equation [13] instead of using the very famous hadronic mass spectrum, exp(m/T H ), which is the asymptotic solution of this equation. It is well known that this model has a limited temperature range, i.e., the energy density and pressure diverge at T H . This singularity, however, disappears when an exclude volume approximation [14] (with a Bag constant B 1 4 = 230 MeV) is associated with the Hagedorn model. In the QGP phase, we use massless free u, d and s-quarks and the gluon gas model for simplicity. The two equations of state are matched by imposing Gibbs' condition for phase equilibrium. Consequently we obtain a first order phase transition model which has a critical temperature T C = 159 MeV and a mixed phase pressure of P mix = 70.9 MeV/fm 3 at zero baryon density. We mention our numerical algorithm for the relativistic hydrodynamic model. It is known that the Piecewise Parabolic Method (PPM) [15] is very robust scheme for the non-relativistic gas equation with a shock wave. We have extended the PPM scheme of Eulerian hydrodynamics to the relativistic hydrodynamical equation. Note that this is a higher order extension of the piecewise linear method [16]. Assuming non-central Pb+Pb collisions at SPS energy, we choose very simple formulas for the initial condition at the initial (or passage) time t 0 = 2r 0 /(γv) ∼ 1.4 fm (r 0 , γ and v are, respectively, the nuclear radius, Lorentz factor and the velocity of a spectator in the center of mass system) E(x, y, z) = E 1 (z)θ(z 0 − z)θ(z +z 0 )ρ(r p )ρ(r t ),(5)n B (x, y, z) = n B1 (z)θ(z 0 − z)θ(z +z 0 )ρ(r p )ρ(r t ),(6)v z (x, y, z) = v 0 tanh(z/z 0 ) × θ(z 0 − z)θ(z +z 0 )ρ(r p )ρ(r t ),(7) where θ(z) is the step function, ρ(r) is the Woods-Saxon parameterization in the transverse direction, [17] and the z dependence of the baryon density n B1 (z) is taken from Ref. [18] ρ(r) = 1 exp r−r 0 δr + 1 ,(8)E 1 (z) is Bjorken's solutionE 1 (z) = E 0 ×   t 2 0 − z 2 t 0   − 4 3 ,(9)n B1 (z) = κ × 0.17 t 2 0 − z 2 t 0 .(10) See also Fig. 1. We have employed Bjorken's longitudinal solution just as an initial condition. This is in contrast to Ref. [9,19], in which Bjorken's boost-invariant solution was used as an assumption The initial condition is in the region with slanting lines. b is the impact parameter vector. r p and r t are respectively the distances from the center of the projectile and the target nucleus in the transverse plane. and the hydrodynamical equation was numerically solved only in the transverse plane. At relativistic energies the Lorentz-contracted spectators leave the interaction region after ∼ 1 fm, we therefore assume the hydrodynamical description is valid only in the overlap region and neglect the interaction between the spectators and the fluid. Therefore we can say that our model gives a good description only in the vicinity of the midrapidity region and fails to reproduce directed flow at present. It may be possible to treat this problem if we use a hadronic cascade model for both spectators and particles emitted from the freeze-out hypersurface, together with the hydrodynamic model. There are four initial (and adjustable) parameters in our hydrodynamic model: 1) the energy density at z = 0, E 0 = 2500 MeV/fm 3 , 2) the factor in the baryon density distribution κ = 2.5, 3) the initial longitudinal factor ε = 0.9 and 4) the "diffuseness parameter" δ r = 0.3 fm. In the present analysis we select these values 'by hand', i.e., we guess them. These parameters, however, should be chosen so as to reproduce the experimental data for the (pseudo-)rapidity and the transverse momentum distribution. To make our analysis more quantitative, we need this experimental data. We would like the experimental group to analyze the centrality dependence of the hadron spectra, especially, the (pseudo-)rapidity distribution. For this reason we wish to emphasize that our numerical results presented below are only preliminary. Figure 2 shows our numerical results for the temporal behavior of the pressure (left column) and the baryonic flow (right column) at z = 0 in the non-central Pb+Pb collision with impact parameter b = 7 fm at SPS energy. Initially almost all matter in this plane is in the QGP phase and there is no transverse flow anywhere by definition. At t = t 0 + 0.5 fm we see the shell structure corresponding to the mixed phase with the same pressure ∼ 70 MeV/fm 3 , and the initial pressure gradient gives the baryons transverse flow. The QGP phase disappears at t = t 0 + 1.0 fm and after that the mixed phase occupies the central region. There is still no transverse flow near the origin due to the absence of a pressure gradient. At about t = t 0 + 5.0 fm all the nuclear matter initially in the QGP phase has gone through the phase transition and is in the hadronic phase. We can see from these figures that the shape of the nuclear matter is changing from almond (top figure on page 5) to round (bottom figure on page 6), and the elliptic flow reduces the initial geometric deformation. The numerical results of the hydrodynamical simulation give us the momentum distribution through the Cooper-Frye formula [20] with freeze-out temperature T f = 140 MeV. The elliptic flow parameter v 2 , as a function of rapidity y, is obtained from the momentum distribution v 2 (y) = p x p t 2 − p y p t 2 = 2π 0 dφ cos(2φ) p + p − p t dp t E d 3 N dp 3 2π 0 dφ p + p − p t dp t E d 3 N dp 3 .(11) Before calculating v 2 in non-central collisions with impact parameter b = 7 fm, we checked the numerical error in our hydrodynamic model in central collisions. Since there is no special direction in the transverse plane for head-on collisions, ideally the elliptic flow vanishes in the infinite particle limit. Performing the numerical simulation with b = 0 fm, we obtain the value of v 2 as less than 10 −1 percent, therefore we can safely neglect the numerical error. Note that the numerical error in the energy and baryon density conservation of the fluid is less than one percent in our analysis. Figure 3 shows our results for the rapidity dependence of elliptic flow for pions in different transverse momentum regions. These results show that elliptic flow rises with transverse momentum p t [21] and has a peak at midrapidity. This seems to be in contrast with the 0.5< p t <1.0GeV 1.0< p t <1.5GeV 1.5< p t <2.0 experimental data obtained by the NA49 Collaboration [3]. Their data appears to be slightly peaked at medium-high rapidity. Our results for v 2 for protons are shown in Fig. 4. We see the same behavior as for the pion case. We obtain a larger v 2 for protons than for pions because we are integrating over a larger transverse momentum region. Since the initial parameters in our hydrodynamic model have been chosen by hand, we would like readers to not take these results quantitatively. In summary, we reported our preliminary analysis of elliptic flow in non-central heavy-ion collisions using the hydrodynamic model. We numerically simulated the hydrodynamic model without assuming cylindrical symmetry or Bjorken's boost-invariant solution, using the extended version of the Piecewise Parabolic Method which is known as a robust scheme for the non-relativistic gas equation with a shock wave. We presented the temporal behavior of high temperature and high density nuclear matter produced in Pb+Pb collisions with b = 7 fm at SPS energy. Our preliminary results showed that the elliptic flow parameter v 2 has a peak at midrapidity for both pions and protons and increases with transverse momentum. Since there are some ambiguities in the initial parameters of our hydrodynamical model, we should fix these parameter using experimental data for the rapidity distribution in non-central collisions. If we regard the hydrodynamical model as a predictive one, we can choose initial parameters using results from a parton cascade model, such as VNI [22]. The study of these issues is a future work. The author is much indebted to Prof. I. Ohba, Prof. H. Nakazato, Dr. Y. Yamanaka and Prof. S. Muroya for their helpful comments, and to Dr. H. Nakamura, Dr. C. Nonaka and Dr. S. Nishimura for many interesting discussions. The numerical calculations were performed on workstations of the Waseda Univ. high-energy physics group. Figure 1 : 1Schematic view of the initial geometry in the center of mass system. The left figure shows the reaction plane and the right the transverse plane. Figure 2 : 2Time evolution of pressure and baryon flow in the transverse plane. Left: The pressure contours. Right: The baryon flow velocity vector (n B v x , n B v y ). Figure 3 : 3Rapidity dependence of elliptic flow for pion. Four curves correspond to the different transverse momentum regions. The midrapidity is 2.92. Figure 4 : 4Rapidity dependence of elliptic flow for proton. Three curves correspond to the different transverse momentum regions. Note that the integral region of transverse momentum is larger than for pions. Quark Matter '97. See, Nucl. Phys. 638See, for example, Quark Matter '97, Nucl. Phys. A638 (1998). . See J.-Y For A Review, Ollitrault, Nucl. Phys. 638195For a review, see J.-Y. Ollitrault, Nucl. Phys. A638 (1998) 195c. . H Appelshäuser, NA49 CollaborationPhys. Rev. Lett. 804136H. Appelshäuser et al. (NA49 Collaboration), Phys. Rev. Lett. 80 (1998) 4136. . S Nishimura, WA98 CollaborationNucl. Phys. 638459S. Nishimura et al. (WA98 Collaboration), Nucl. Phys. A638 (1998) 459c. . F Ceretto, CERES CollaborationNucl. Phys. 638467F. Ceretto et al. (CERES Collaboration), Nucl. Phys. A638 (1998) 467c. . H Liu, S Panitkin, N Xu, Phys. Rev. 59348H. Liu, S. Panitkin and N. Xu, Phys. Rev. C59 (1999) 348. . H Heiselberg, A.-M Levy, nucl-th/9812034H. Heiselberg and A.-M. Levy, nucl-th/9812034. . S Soff, nucl-th/9903061S. Soff et al., nucl-th/9903061. . J.-Y Ollitrault, Phys. Rev. 46229J.-Y. Ollitrault, Phys. Rev. D46 (1992) 229. . D H Rischke, Nucl. Phys. 595346D. H. Rischke et al., Nucl. Phys. A595 (1995) 346. Note that to my knowledge there is only one scheme to simulate the non-central heavy-ion collisions which uses Lagrangian hydrodynamics. C Nonaka, these proceedingsNote that to my knowledge there is only one scheme to simu- late the non-central heavy-ion collisions which uses Lagrangian hydrodynamics: C. Nonaka et al., these proceedings. R Hagedorn ; See Also, R Hagedorn, J Rafelski, Statistical Mechanics of Quarks and Hadrons. H. SatzAmsterdam3237R. Hagedorn, Suppl. Nuovo. Cim. 3 (1965) 147; see also R. Hage- dorn and J. Rafelski, in Statistical Mechanics of Quarks and Hadrons, edited by H. Satz (1981) p. 237, North Holland, Ams- terdam; Hot Hadronic Matter, Theory and Experiment. R Hagedorn, J. Letessier, H. H. Gutbrod and J. RafelskiPlenum Press13New YorkR. Hagedorn, in Hot Hadronic Matter, Theory and Ex- periment, edited by J. Letessier, H. H. Gutbrod and J. Rafelski (1995) p. 13, Plenum Press, New York. . R Hagedorn, J Rafelski, Commun. Math. Phys. 83563R. Hagedorn and J. Rafelski, Commun. Math. Phys. 83 (1982) 563. . J I Kapusta, K A Olive, Nucl. Phys. 408478J. I. Kapusta and K. A. Olive, Nucl. Phys. A408 (1983) 478. . P Colella, P R Woodward, J. Comput. Phys. 54174P. Colella and P. R. Woodward, J. Comput. Phys. 54 (1984) 174. . See, V Example, Schneider, J. Comput. Phys. 10592See, for example, V. Schneider et al., J. Comput. Phys. 105 (1993) 92. . J D Bjorken, Phys. Rev. 27140J. D. Bjorken, Phys. Rev. D27 (1983) 140. . J Sollfrank, Phys. Rev. 55392J. Sollfrank et al., Phys. Rev. C55 (1997) 392. . D Teaney, E V Shuryak, nucl-th/9904006D. Teaney and E. V. Shuryak, nucl-th/9904006. Although there is a well-known problem in this formula when it is applied to the space-like freeze-out hypersurface, we use this formula for simplicity. F Cooper, G Frye, Phys. Rev. 10186F. Cooper and G. Frye, Phys. Rev. D10 (1974) 186. Although there is a well-known problem in this formula when it is applied to the space-like freeze-out hypersurface, we use this formula for simplicity. . P Danielewicz, Phys. Rev. 51716P. Danielewicz, Phys. Rev. C51 (1995) 716. . See, K Example, Geiger, Phys. Rev. 464965See, for example, K. Geiger, Phys. Rev. D46 (1992) 4965.

We introduce the Schubert form a 3-bridge link diagram, as a generalization of the Schubert normal form of a 3-bridge link. It consists of a set of six positive integers, written as (p/n, q/m, s/l), with some conditions and it is based on the concept of 3-butterfly. Using the Schubert normal form of a 3-bridge link diagram, we give two presentations of the 3-bridge link group. These presentations are given by concrete formulas that depend on the integers {p, n, q, m, s, l} . The construction is a generalization of the form the link group presentation of the 2-bridge link p/q depends on the integers p and q.

10.1142/s0218216518500293

1703.00041

00148c147e778737872775fae2cf16a99dadd44d

The Schubert normal form of a 3-bridge link and the 3-bridge link group 28 Feb 2017 November 2016 Margarita Toro [email protected] Universidad Nacional de Colombia MedellínColombia Mauricio Rivera [email protected] Universidad Nacional de Colombia MedellínColombia The Schubert normal form of a 3-bridge link and the 3-bridge link group 28 Feb 2017 November 2016 We introduce the Schubert form a 3-bridge link diagram, as a generalization of the Schubert normal form of a 3-bridge link. It consists of a set of six positive integers, written as (p/n, q/m, s/l), with some conditions and it is based on the concept of 3-butterfly. Using the Schubert normal form of a 3-bridge link diagram, we give two presentations of the 3-bridge link group. These presentations are given by concrete formulas that depend on the integers {p, n, q, m, s, l} . The construction is a generalization of the form the link group presentation of the 2-bridge link p/q depends on the integers p and q. Introduction In [5] it was introduced the butterfly presentation of a link diagram as a generalization of the 2-bridge Schubert's notation. Moreover, the particular concept of a 3-butterfly was implemented in order to study 3-bridge links and to obtain a codification of a 3-bridge link diagram. Here, for our purpose, we do not need all the machinery of the butterfly construction presented in [6], so we will take a different approach. We will describe the construction of the codification by a direct and combinatorial approach, using the ideas in [3], where Ferri constructed the crystallization of the double cover of S 3 , with a link as ramification set. For any link diagram L there is a strong relation between crystallization of the double cover of S 3 , with L as the ramification set, and the 3-butterfly associated to L, that we will explain in [12]. For any n-bridge link diagram the construction of an n-butterfly is possible, see [6], but in this paper we want to be specific and we will work only with 3-bridge link diagrams. To a 3-bridge diagram we associate a 3-butterfly that is described by a set of six positive integers {p, n, q, m, s, l}, with some restrictions, and then we define the Schubert form of the link diagram as (p/n, q/m, s/l) , for geometrical reasons that will be explained in Section 1. As each 3-bridge link admits infinitely many different link diagrams, the Schubert normal form for a link L is defined by taking the minimum among all 3-bridge link diagrams of L according to a lexicographical type of order, see [5]. For the purpose of this paper we only need the Schubert form of the link diagram, but in further research and in the compilation of link tables, it will be interesting to consider the Schubert normal form of a link. In this paper we find formulas for the over and under presentation of the 3-bridge link represented by the Schubert form (p/n, q/m, s/l), that depends on the integers {p, n, q, m, s, l}. The formula for the under presentation of the 3-bridge link is a natural extension of the formula for the presentation of the 2-bridge link p/q, that depends on the integers p and q. The paper is organized as follows: in Section 1 we describe the construction of a 3-butterfly associated to a 3-bridge link diagram L and introduce the Schubert form of L, that consists of a set of 6 positive integers, (p, n, q, m, s, l), that captures the relevant information of the 3-butterfly and, therefore, of the 3-bridge diagram L. In Section 3 we describe a canonical diagram associate to a 3-butterfly (p, n, q, m, s, l), in a similar way to the canonical diagram of a 2-bridge link, see [13]. In Section 5 we will give an orientation to this canonical diagram. In Section 4 we define two permutations, γ and φ, associated to the Schubert form (p/n, q/m, s/l), and study the composition µ = γφ. The cyclic structure of µ is the key point in the rest of the paper. A variation of the permutation µ is very useful for the construction of a Gauss code for the link diagram, and, from there we can find the Dowker code and we are able to compute link invariants, such as the link group, the Seifert matrix, the Alexander, Jones and HOMFLY polynomials. In Section 6 we present our main result, Theorems 14, 15 and 18, that give explicit presentations for the knot group π (L) of a 3-bridge link L. These presentation are described by clear algorithms, that are easy to program in a computer and depend on the integers in the Schubert form of the link diagram. In the last section we propose a special family of links, (p/n, p/n, p/n), that have a strong symmetry that is reflected in the group presentation. This symmetry could be exploited in the study of the representations into SL(2, C) of the link group. Some authors allow that any n-bridge link diagram can be consider as a k-bridge link diagram, for any k > n, by considering bridges without any undercrossings, see [9]. We neither allow this situation nor consider a split link diagram with more than 3 components, as the one in Fig. 3a, as a 3-bridge link diagram We work with 3-bridge link diagrams as in [1] and [8]. Remark on notation: In [11] the author uses subindexes and denote a butterfly by (M 1 , N 1 , M 2 , N 2 , M 3 , N 3 ). In this paper we avoid the use of subindexes in the Schubert form, and prefer to assign a different role to each integer, in that way we reach simpler formulas. Description of the 3-butterfly of a 3-bridge link diagram Let L be a 3-bridge link such that the projection on the xy plane is a 3-bridge diagram D. Let a, b and c the bridge projections. Draw an ellipse around each of the bridges, in such a way that they are disjoint, and they have the bridges as principal axes. Each ellipse will intercept the diagram D in an even number of points, that will be the vertices. We denote by P, Q and S the ellipses around the bridges a, b and c, respectively, and let 2p (resp. 2q and 2s) the number of intersections of P (resp. Q and S) with the diagram D. Following [6], the ellipses P, Q, S are called butterflies. Take the graph R 1 formed by the butterflies P, Q, S, the vertices and the bridges. In each butterfly we have the bridge, that divides each butterfly in two halves, that will be the wings. The reflection along the bridges inside the butterflies is called γ. The segment of the underarcs that are inside the butterflies are forgotten, but they can be recovered with the reflection γ. The edges outside the butterflies will give the information on how the butterflies intercept to each other, see Fig. 1a. Each one of these edges connect two vertices of two butterflies, we identify these vertices, to form a set that will be the vertices of our graph. The identification will give an involution on the vertex of the graph, that we call φ. We define the 3-butterfly as the graph R = R 1 /φ, formed with the vertices of R 1 identified by the involution φ. We draw the graph R in any of the three forms shown in Fig. 1. If we consider that the diagram L is in S 2 = ∂B, then the graph R define a polygonalization of S 2 formed by three polygons, as shown in Fig. 2. Compare this simple construction with the formal one given in [6] and [5]. When we identify the butterflies, there will appear two new points, that will be denoted 0 and * . These two points are fundamental, but they are not considered vertices of the 3-butterfly. There can be only two basic forms for the graph R, that are determined by the way the butterflies P, Q and S intersect. Type II: Two of the polygons do not intersect, see Fig. 2.b. When a 3bridge link diagram produces a type II butterfly, there is a wave move, see [9] that allows us to construct a new 3-bridge diagram with lower crossing number. So, we work only with type I butterflies, such that there are no wave moves. In order to obtain a canonical way to describe a 3-butterfly, we will always assume that p ≥ q ≥ s ≥ 2,(1) the condition s ≥ 2 is to ensure that each bridge has at least one crossing. By rotating the plane and interchanging the points 0 and * , we can always obtain a 3-butterfly diagram with P at the top, Q to the left and S to the right, and we read it in the counterclockwise direction, P Q S, as shown in Fig. 1. Let |P ∩ Q| = t be the number of vertices between P and Q, v = |Q ∩ S| and w = |P ∩ S|. As each butterfly intersects the other two, then t, v and w are positive integers that satisfy 2p = t + w, 2q = t + v, and 2s = v + w,therefore t = p + q − s, v = q + s − p, w = p + s − q. (2) As we will only consider the link diagrams with v ≥ 1, then p + 1 ≤ s + q.(3) So the integers p, q and s satisfy (1) and (3). Reciprocally, if we have integers satisfying (1) and (3) we can construct the butterflies P, Q and S. Now let us describe the positions of the bridges. We orient clockwise each butterfly. From the point 0, following the orientation, we count the number of vertices between 0 and the vertex in which the bridge begins. We call n, m and l the initial points of the bridges in P, Q and S, respectively. Clearly 1 ≤ n ≤ p, 1 ≤ m ≤ q, 1 ≤ l ≤ s.(4) We are working only with link diagrams with exactly three bridges and not only two or one, this impose conditions on the integers n, m and l. In [5] they found the conditions given in the following theorem. (5,1,5,2,5,1) Figure 3: Diagrams associated to a non reduced butterfly and to two Schubert forms Theorem 1 Every 3-butterfly defines a unique set of integers {p, m, q, n, s, l} such that (4/2,4/1,3/1) (4/1,4/2,3/1) a b cp ≥ q ≥ s ≥ 2, 1 ≤ n ≤ p, 1 ≤ m ≤ q, 1 ≤ l ≤ s, p + 1 ≤ s + q (5) n + m = q + 1, n + l = p + 1, if m > q + s − p then n + m = 2q + 1 and n + m = 2q − p + 1 if l < p − s then n + l = p − s + 1 if m ≤ q + s − p then m + l = s + 1. Reciprocally, if a set of integers {p, m, q, n, s, l} satisfies conditions (5) then it defines a 3-butterfly. Note that we may think that inside the butterfly P (resp. Q, S) the bridge a make a (n/p) π rotation, (resp. (m/q) π, (l/s) π). For this geometrical reason we want to use the notation (p/n, q/m, s/l) instead of {p, m, q, n, s, l}, but p/n is not considered as a rational number. The conditions on the integers {p, n, q, m, s, l} impose in (5) define a 3butterfly and a link diagram L, but it is possible that L is a split link with some trivial components, as the diagram associated to {5, 1, 5, 2, 5, 1} shows, see Fig. 3. Definition 2 We say that (p, n, q, m, s, l) is a 3-butterfly if the set of integers {p, n, q, m, s, l} satisfies the conditions (5). We say that a 3-butterfly is reduced if the associated diagram is a 3-bridge diagram without any trivial components. We say that (p/n, q/m, s/l) is a Schubert form of a 3-bridge link if the 3-butterfly (p, n, q, m, s, l) is reduced. We need to be careful with the relative order of the numbers in (p/n, q/m, s/l). Example 3 If we change the order, it is possible that we get different Schubert forms. The Schubert form (4/2, 4/1, 3/1) represents a knot and (4/1, 4/2, 3/1) represents a two component link. See Fig. 3. Algorithm to draw a canonical 3-bridge link diagram We associate to each 3-butterfly (p, n, q, m, s, l) a canonical diagram, in a similar way as the canonical diagram of a 2-bridge link is associated to p/q, see [7]. We draw the three bridges as three segments: bridge a as a vertical segment; bridge b as a segment forming a 120 0 angle with the bridge a and bridge c as a segment forming a 240 0 angle with the bridge a. We divide the bridge a in p segments, and we fix two points in each division, one to the left and one to the right, except at the extreme points, where there is only one. Label them with A = {a 0 , a 1 , · · · , a 2p−1 }, in a counterclockwise sense, so the extreme bridges are labeled a 0 and a p . For the bridge b we repeat the process, but we divide the bridge in q segments and label the points with B = {b 0 , · · · , b 2q−1 }. For the bridge c the number of segments is s and the labels are C = {c 0 , · · · , c 2s−1 }. The subscripts of A (resp. B and C) are taken mod (2p), (resp. mod (2q) and mod (2s)). To draw the link diagram we need to join, with appropriate arcs, the points a i , b j and c k , i ∈ Z 2p , j ∈ Z 2q , and k ∈ Z 2s , according to the rules given by permutations φ and γ. There are t = p + q − s arcs between the a and b bridges, namely a n−1 b m , a n−2 b m+1 , a n−3 b m+2 , . . . , a n−j b m+j−1 , . . . , a n−t b m+t−1 , likewise there are v = q + s − p arcs between the b and c bridges, that are b m−1 c l , b m−2 c l+1 , b m−3 c l+2 , . . . , b m−j c l+j−1 , . . . , b m−v c l+v−1 , and, finally, w = p + s − q arcs between the c and a bridges, c l−1 a n , c l−2 a n+1 , c l−3 a n+2 , . . . , c l−j a n+j−1 , . . . , c l−w a n+w−1 . It is enough to know how to construct the first arc between each pair of bridges, and the rest of the arcs are "parallel " arcs to them, see In the rest of this paper we will refer to the diagram described as the link canonical diagram associated to the Schubert form (p/n, q/m, s/l). Notice that if (p/n, q/m, s/l) does not satisfy the conditions in Theorem 1, we still may use this algorithm to draw a link diagram. Lemma 4 If the Schubert form (p/n, q/m, s/l) is reduced, the 3-bridge diagram has p + q + s − 3 crossings. Permutations associated to a Schubert form The conditions for a 3-butterfly {p, n, q, m, s, l} to be reduced can not be described using simple conditions on the integers in a similar way as the conditions to be a 3-butterfly given in (5). Now we need to go deeper and study in detail the permutations φ and γ. Given a set {p, n, q, m, s, l} that satisfies (5) we construct explicitly the associated 3-butterfly and then we draw the 3-bridge diagram. Define the 3-butterfly by labelling the vertices of each of the butterflies: P have vertices labeled by A = {a 0 , · · · a i , · · · , a 2p−1 }, i ∈ Z 2p ; Q with vertices B = {b 0 , · · · , b j , · · · , b 2q−1 }, j ∈ Z 2q ; and S with vertices C = {c 0 , · · · , c l , · · · , c 2s−1 }, l ∈ Z 2s . The bridge ends are labeled by a 0 and a p in P (resp. by b 0 and b q in Q and c 0 and c s in S). See Fig. 5. We have the permutations γ and φ on the set A∪B ∪C. The permutation γ is the reflection along the bridges. The permutation φ is determined by the identification of the vertices of two butterflies, so in the 3-butterfly each vertex has two labels. The proofs of the following lemmas are straightforward computations. Lemma 5 The function defined in the set A ∪ B ∪ C by γ (a i ) = a 2p−i , 0 ≤ i < 2p, γ (b j ) = b 2q−j , 0 ≤ j < 2q, γ (c h ) = b 2s−h , 0 ≤ h < 2s(6) is an order 2 permutation. The set of fixed points is E = {a 0 , a p, b 0 , b q , c 0 , c s } .(7) The set E = {a 0 , a p, b 0 , b q , c 0 , c s } corresponds to the endpoints of the bridges. It will play an important role in the rest of the paper. Lemma 6 The map φ : A ∪ B ∪ C → A ∪ B ∪ C defined by a n−i ←→ b m+i−1, if 1 ≤ i ≤ t, a n+j ←→ c l−j−1, if 0 ≤ j ≤ w − 1, b m−h ←→ c l+h−1, if 1 ≤ h ≤ v,(8) is an order 2 permutation, where t = p+q −s, v = q +s−p and w = p+s−q. Note that φ does not have fixed points, and among the bicycles in φ there is no a bicycle in the set F = {(a 0 , b 0 ) , (a 0 , b q ) , (a 0 , c 0 ) , (a 0 , c s ) , (b 0 , a p ) , (b 0 , c 0 ) ,(9)(b 0 , c s ) , (c 0 , a p ) , (c 0 , b q ) , (a p , b q ) , (a p , c s ) , (b q , c s )} The construction of φ is well defined for any polygonalization of S 2 with 3 polygons, even if they do not satisfy the conditions of Theorem 1. In fact, in terms of the permutation φ, we can rewrite Theorem 1 as follows. Theorem 7 A set {p, n, q, m, s, l}, with p ≥ q ≥ s ≥ 2, 1 ≤ n ≤ p, 1 ≤ m ≤ q, 1 ≤ l ≤ s describes a 3-butterfly if and only if the associated permutation φ does not have any of the bicycles in the set F . We study the cyclic decomposition of µ = φγ. The orbit of a vertex v will be denoted by O µ (v). For a cycle τ = (z 1 z 2 · · · z k ) we will use the same symbol to refer to the cycle, to its orbit {z 1 , z 2 , · · · , z k } and to the word z 1 z 2 · · · z k . The length of τ will be denoted by |τ |, τ (x) will denote the cycle that contains x and for a function Γ, Γ (τ ) will be the word (set) formed by applying Γ to each element in τ . Theorem 8 Let (p, n, q, m, s, l) be a 3-butterfly, γ and φ be its associated permutations, given in Lemmas 5 and 6 and let µ = φγ. (p/n, q/m, s/l) is a Schubert form for a 3-bridge link if and only if µ is the product of three disjoint cycles, µ = τ 1 τ 2 τ 3 such that, for i=1,2,3, |τ i ∩ E| = 2, where E is given in (7). Proof. Let µ = φγ associated to the 3-butterfly (p, n, q, m, s, l). The 3butterfly (p, n, q, m, s, l) defines a Schubert form (p/n, q/m, s/l) if and only if the 3-butterfly is reduced. Suppose that the butterfly is reduced. The orbit of a p under µ, O µ (a p ) will describe a path that follows the underarc with initial point in a p , so eventually it will arrive to the endpoint of the underarc, say e = µ k (a p ), e ∈ E, E defined in (7). Then µ (e) = φγ (e) = φ (e) = φφγµ k−1 (a p ) = γµ k−1 (a p ), so the orbit will go back to the same underarc, in opposite direction. We take τ 1 as the cycle formed by the orbit of a p and τ 1 ∩ E will contain exactly two vertices. We repeat the same process with the other vertices in E. Since the butterfly is reduced, all the vertices will be crossed by one of the underarcs, so we have only three orbits. Reciprocally, if the butterfly is not reduced, there will be a component whose vertex will not be in the orbit of any of the elements in E. See Fig. 3a. From now on we will assume that the permutation µ associated to the Schubert form (p/n, q/m, s/l) is the product of three disjoint cycles, µ = τ 1 τ 2 τ 3 . The cyclic decomposition of µ allows us to determine the number of components of the associated link diagram. Theorem 9 (Classification) Let (p/n, q/m, s/l) be a Schubert form, γ and φ its associated permutations given in 5 and 6, µ = φγ. The 3-bridge link diagram L represented by (p/n, q/m, s/l) satisfies: (i) L is a knot if and only if a p / ∈ O µ (a 0 ) , b q / ∈ O µ (b 0 ) and c s / ∈ O µ (c 0 ). (ii) L is a two component link if and only if one, and only one, of the following conditions holds: a p ∈ O µ (a 0 ) , b q ∈ O µ (b 0 ) or c s ∈ O µ (c 0 ). (iii) L is a three component link if and only if a p ∈ O µ (a 0 ) , b q ∈ O µ (b 0 ) and c s ∈ O µ (c 0 ). Proof. Take the cyclic decomposition of µ = τ 1 τ 2 τ 3 and study each one of the cycles, using the interpretation given in the proof of Theorem 8. Orientation of the canonical 3-bridge link diagram (p/n, q/m, s/l) Until now we have not considered the orientation of the link L, but in order to find a group presentation for the link group π (L) we will give an orientation to the canonical diagram described in Section 3. Let µ = φγ = τ 1 τ 2 τ 3 , we study in detail these cycles. In each cycle τ i we have two special vertices, that are the fixed points of γ and form the set E defined in (7). Each cycle describes a path around one of the link diagram underarcs, see Fig. 6.b, so one of this special vertices corresponds to the arc initial point, denoted I i ; and the other one to the arc endpoint, denoted F i . So we consider that when we follow the link, we travel it in the order τ 1 , τ 2 and τ 3 and the bridge a in the direction from a 0 to a p . Definition 10 We define δ a (resp. δ b , δ c ), the direction in which we travel the bridge a (resp. b, c) as: δ a = 1 and δ b = 1, if we go from b 0 to b q −1, if we go from b q to b 0 , δ c = 1, if we go from c 0 to c s −1, if we go from c s to c 0 When the condition (i) in Theorem 9 is satisfied, the Schubert form corresponds to a knot diagram, hence the orientation of bridge a is enough to determine the knot orientation. We take τ 1 as the cycle that contains a p and τ 3 as the cycle that contains a 0 . In the link case we need to determine the orientation of each component. If L is a 3-component link, the condition (iii) in Theorem 9 is satisfied and we orient each component by δ a = δ b = δ c = 1, τ 1 contains a p , τ 2 contains b q and τ 3 contains c s . If L is a 2-component link the condition (ii) in Theorem 9 holds, again we take τ 1 as the cycle that contains a p , and τ 3 the cycle that corresponds to the other component. Lemma 11 a. If L is a knot, Table 1 contains all possibilities for the endpoints of the cycles τ 1 , τ 2 , τ 3 and the knot orientation. b. If L is a 2-component link, Table 2 contains all possibilities for the endpoints of the cycles τ 1 , τ 2 , τ 3 and the link orientation. Table 1 Table 2 To avoid the lack of uniqueness in the cycles, we always write the cycle τ i as an ordered set with initial point I i , but to simplify notation we keep the cycle notation. In general this will not generate confusion in our work. I 1 F 1 I 2 F 2 I 3 F 3 δ b δ c a p b 0 b q c 0 c s a 0 1 1 a p b 0 b q c s c 0 a 0 1 −1 a p b q b 0 c 0 c s a 0 −1 1 a p b q b 0 c s c 0 a 0 −1 −1 a p c 0 c s b 0 b q a 0 1 1 a p c 0 c s b q b 0 a 0 −1 1 a p c s c 0 b 0 b q a 0 1 −1 a p c s c 0 b q b 0 a 0 −1 −1 I 1 F 1 I 2 F 2 I 3 F 3 δ b δ c a p b 0 b q a 0 c s c 0 1 1 a p b q b 0 a 0 c s c 0 −1 1 a p c 0 c s a 0 b q b 0 1 1 a p c s c 0 a 0 b q b 0 1 −1 a p a 0 b q c 0 c s b 0 1 1 a p a 0 b q c s c 0 b 0 1 −1 Lemma 12 Let µ be the permutation associated to the Schubert form (p/n, q/m, s/l), µ = τ 1 τ 2 τ 3 , for i=1,2,3 we have: (i) Each cycle τ i is even, with order |τ i | greater than 4. (ii) τ i = {I i , z 1 , · · · , z k , F i , γ (z k ) , · · · , γ (z 1 )} , for z j ∈ A ∪ B ∪ C, j = 1, · · · , k, k ≥ 1. (iii) τ |τ i |/2 i contains the transposition (I i , F i ). Proof. By condition (9) we get µ (I i ) = τ i (I i ) = F i , so the length of the cycle τ i is greater than 3. Then, there exists z j ∈ A ∪ B ∪ C, j = 1, · · · , k ≥ 1 such that z 1 = µ (I i ) , z 2 = µ (z 1 ) , · · · , z k = µ (z k−1 ) and F i = µ (z k ) = φγ (z k ) , this yields µ (F i ) = φγ (F i ) = φ (F i ) = φ (φγ (z k )) = γ (z k ) . Now, µ (γ (z k )) = φγ (γ (z k )) = φ (z k ) = φ (φγ (z k−1 )) = γ (z k−1 ) , and then, for j = k, · · · , 2, we get µ (γ (z j )) = φγ (γ (z j )) = φ (z j ) = φ (φγ (z j−1 )) = γ (z j−1 ) . The relevant information on each cycle τ i is contained in the first part of the cycle, we define the initial segment of τ i as τ i = {I i , z 1 , · · · , z k }(10) We may summarize the results up to now in an algorithm that allows us to find the cycles τ 1 , τ 2 , τ 3 , the set E and therefore the directions δ b and δ c associated to a Schubert form. Let σ be the permutation in A ∪ B ∪ C defined by σ = (a 0 a p ) (b 0 b q ) (c 0 c s ) . This permutation corresponds to "travel the bridges" in the diagram. Algorithm 13 Given a Schubert form (p/n, q/m, s/l) the following algorithm finds the orientation of the associated link diagram. It provides the cycles τ 1 , τ 2 , τ 3 such that µ = τ 1 τ 2 τ 3 , where the cycle τ i has the form given in Lemma 12. 1. Take I 1 = a p , τ 1 = O µ (I 1 ) and F 1 = τ |τ 1 |/2 1 (I 1 ) . 2. If F 1 = I 1 take I 2 = b q else take I 2 = σ (F 1 ) . 3. Take τ 2 = O µ (I 2 ) and F 2 = τ |τ 2 |/2 2 (I 2 ) . 4. If σ (F 2 ) / ∈ {I 1 , F 1 , I 2 , F 2 } then take I 3 = σ (F 2 ) else take I 3 as the unique element in {b q , c s } − {I 1 , F 1 , I 2 , F 2 }. Take τ 3 = O µ (I 3 ) and F 3 = τ |τ 3 |/2 3 (I 3 ). Presentation of the 3-bridge link group Let L be the link diagram with Schubert form (p/n, q/m, s/l). We have an explicit way to find the over and under presentations of the link group of L, see [1] and [2]. This method requires to use the link diagram. We will use this method, but we will replace the explicit use of the diagram by an algorithm that uses the permutations φ, γ and µ and some new functions defined on the set A∪B ∪C. As the description of the over and under presentations requires an oriented link diagram, we will always refer to the standard link diagram and orientation described in Section 3. It is important to remark that we need the diagram only to explain the construction, but the algorithm to find the presentations do not require to draw the link diagram, it depends only on the permutations φ, γ and µ = φγ. As φ, γ depend only of the Schubert form (p/n, q/m, s/l), the presentation of the link group will depend only on the integers {p, n, q, m, s, l}. The algorithm is efficient and easy to implement in a software such as Mathematica. Over presentation of the 3-bridge link (p/n, q/m, s/l) We take meridians around the bridges as group generators, and label them by the same name as the bridges, so we have generators a, b and c, see Fig. 6a. We find the relators by traveling the frontier of a neighborhood of the underarcs, as shown in Fig. 6b, so these paths are precisely the orbits τ i , i = 1, 2, 3. Each relator is a word in a, b and c constructed with the convention that each time we cross the bridge a (resp. b, c) we write a ±1 (resp. b ±1 , c ±1 ) depending of the sign of the crossing, given by the convention + -. We replace this graphic process by defining a function Γ that "forgets the index but remembers the direction". Consider Γ : A∪B ∪C → {a ±1 , b ±1 , b ±1 } defined by Γ (a i ) = a if 0 < i ≤ p a −1 otherwise , Γ (b i ) = b δ b if 0 < i ≤ q b −δ b otherwise. ,(11)Γ (c i ) = c δc if 0 < i ≤ s c −δc otherwise. The relators are r 1 = Γ (τ 1 ) , r 2 = Γ (τ 2 ) and r 3 = Γ (τ 3 ), where Γ (τ i ) means the word obtained when we apply Γ to each element in the orbit τ i . Thus we have proved the following proposition. Proposition 14 The link group of the link L given by the Schubert form (p/n, q/m, s/l) admits a presentation given by π (L) = a, b, c | Γ (τ 1 ) , Γ (τ 2 ) , Γ (τ 3 ) were µ = τ 1 τ 2 τ 3 is the associated permutation and Γ is given in (11). By the symmetry of the cycles described in Lemma 12, we may rewrite the relators as the relations. When the Schubert form (p/n, q/m, s/l) defines a knot, using the information in Table 1 we find that r 1 : aw a = w a b, r 2 : bw b = w b c, r 3 : cw c = w c a, in the first four cases, or r 1 : aw a = w a c, r 2 : cw c = w c b, r 3 : bw b = w b a, in the last four cases. For the case when it is a link we have similar relations. At this moment we have lost the geometrical meaning of the generators, so we may rename the generators, if necessary, and unify the two cases, so we have the following proposition. Proposition 15 The link L given by the Schubert form (p/n, q/m, s/l) admits a presentation given by i. a, b, c | aw 1 = w 1 b, bw 2 = w 2 c, cw 3 = w 3 a if L is a knot, ii. a, b, c | aw 1 = w 1 b, bw 2 = w 2 a, cw 3 = w 3 c if L is a 2-component link, iii. a, b, c | aw 1 = w 1 a, bw 2 = w 2 b, cw 3 = w 3 c if L is a 3-component link, were w i is a word in a, b, c given by w i = Γ ( τ i ) were Γ is defined in (11) and τ i is defined in (10). We know that in the case of knots, one of the relations is redundant, but for practical reasons we prefer to have all of them and, in particular computations, we omit the longest relation, that we do not know in advance which one will be. This contrasts with the under presentation that we will introduce in the next section, in which we know the lengths of the words involved. Lemma 16 The sum of the lengths of the words w 1 , w 2 and w 3 is p+q+s−3. Lemma 17 When L is a knot, the peripheral system of the group is given by a, l with l = w 1 w 2 w 3 a −k and k is the exponent sum of the word w 1 w 2 w 3 . Proof. By direct computation we have al = aw 1 w 2 w 3 a −k = w 1 bw 2 w 3 a −k = w 1 w 2 cw 3 a −k = w 1 w 2 w 3 aa −k = la Under presentation of the 3-bridge link (p/n, q/m, s/l) The under presentation is the dual presentation of the over presentation. Those dual presentations play a central role in the proofs of properties of the knot group and the Alexander polynomial of knots, see [2]. By studying these presentations we find a similar algorithm to the known one to find the 2-bridge link group, that has an explicit formula depending of p and q. Of course, we need a more elaborate algorithm. We take as generators of π (L) the meridians around the underarcs, see 5c. Again, the key point is to use the cyclic decomposition of µ. We call a (resp. b and c) the generators corresponding to the underarc described by τ 1 (resp. τ 2 and τ 3 ). The relations are given by traveling the boundary of each butterfly, that describe simple closed paths around the bridges. So the first path is given by {a 0 , · · · , a 2p−1 }, the second by {b 0 , · · · , a 2q−1 } and the third by {c 0 , · · · , c 2s−1 }. The graphical procedure to find the relators is: Each time we cross the link we encounter one of the vertices in the set A ∪ B ∪ C, we identify the underarc, say x, and the sign of the crossing, sg, and write x sg , with x ∈ {a, b, c} and sg = ±1. Again, this procedure will be established by defining a function ρ, similar to the one defined in (11), that identifies the underarc that contains the vertex and the direction of the crossing. Let ρ : A ∪ B ∪ C → {a ±1 , b ±1 , c ±1 } defined by If x ∈ E, ρ (x) =                a if x ∈ τ 1 a −1 if x / ∈ τ 1 b δ b if x ∈ τ 2 b −δ b if x / ∈ τ 2 c δc if x ∈ τ 3 c −δs if x / ∈ τ 3 . If x / ∈ E, ρ (x) =                a if x ∈ τ 1 a −1 if γ (x) ∈ τ 1 b δ b if x ∈ τ 2 b −δ b if γ (x) ∈ τ 2 c δc if x ∈ τ 3 c −δs if γ (x) ∈ τ 3 The relators are s a = ρ (a 0 a 1 · · · a 2p−1 ) = ρ (a 0 ) ρ (a 1 ) · · · ρ (a 2p−1 ) , s b = ρ (b 0 b 1 · · · b 2p−1 ) = ρ (b 0 ) ρ (b 1 ) · · · ρ (b 2q−1 ) , s c = ρ (b 0 b 1 · · · b 2p−1 ) = ρ (c 0 ) ρ (c 1 ) · · · ρ (c 2s−1 ) . For the symmetry of the functions and the cycles given in Lemma 12, we have that if γ (x) = x, ρ (γ (x)) = ρ (x) −1 , therefore if we take the words Note that the lengths of the words u a , u b and u c are p − 1, q − 1 and s − 1, respectively. In this case it is not possible to change the variable names, because we want to have the information about the word lengths. Now the peripheral system is given by a, l ′ where l ′ = u a u b u c f −e , were e is the exponent sum of the word u a u b u c . We have proved the following theorem: u a = ρ (a 1 ) · · · ρ (a p−1 ) , u b = ρ (b 1 ) · · · ρ (b q−1 ) , u c = ρ (c 1 ) · · · ρ (c s−1 ) ,(12) Theorem 18 The link L given by the butterfly (p/n, q/m, s/l) admits a presentation given by: i. If L is a knot a, b, c | cu a = u a a, au b = u b b, bu c = u c c , or a, b, c | bu a = u a a, au c = u c c, cu b = u b a. , ii. If L is a 3-component link a, b, c | au a = u a a, bu b = u b b, cu c = u c c , with u a , u b and u c words of length p − 1, q − 1 and s − 1, respectively, defined by (12). If L is a 2-component link there are six possible combination for the presentation, that are the natural variations of a, b, c | au a = u a a, cu b = u b b, bu c = u c c . Note: This construction does not depend on the fact that p ≥ q ≥ s, nor that we are working with type I butterfly. So we may use it in a more general way. However, if we take the Schubert form (p/n, q/m, s/l) we know that in the knot case one of the relations is redundant, and in this presentation we know that the longest is the first one, so usually that is the one we eliminate. 7 Special family: (p/n, p/n, p/n) In general we do not have an exact pattern for a 3-bridge link group, as the one we encounter for 2-bridge links, see [8], but there are families of 3-bridge links with a very regular pattern for the fundamental group. One of them is the family of links with Schubert form (p/n, p/n, p/n), for integers 1 ≤ n ≤ p. This family contains: Borromean rings (5/2, 5/2, 5/2), the pretzel link P (p, p, p), that corresponds to (2p/p, 2p/p, 2p/p); the toroidal knot T (3, p) and its mirror image T (3, −p), that corresponds to (p/1, p/1, p/1) and to (p/p, p/p, p/p), respectively. The standard diagrams of the links in this family have symmetries of order 2 and 3. Proposition 19 For the link L with Schubert form (p/n, p/n, p/n) there exists a word w (x, y, z) in the variables x, y and z, such that if w a = w (a, b, c) , w b = w (b, c, a) and w c = w (c, a, b) then: i. If L is a knot, the knot group has the presentation a, b, c | aw a = w a b, bw b = w b c, cw c = w c a . 2. If L is a link, it has 3 components and the link group has the presentation a, b, c | aw a = w a a, bw b = w b b, cw c = w c c . Proof. Study the symmetry of the link diagram. Figure 1 : 1Graph for 3-butterflies. Figure 2 : 2Types of 3-butterflies.Type I: The three butterflies intersect in the two points 0 and * , seeFig. 2.a. Figure 4 : 4Drawing a canonical 3-bridge diagram. Figure 5 : 5Orientation and labels of a 3-butterfly. Figure 6 : 6Generators for the over presentation in a. and the under presentation in c. Path around an underarc in b. the relators become the relations cu a = u a a, au b = u b b, bu c = u c c or bu a = u a a, au c = u c c, cu b = u b a. Example 20 1. The Borromean rings have Schubert normal form (5/2, 5/2, 5/2) and w (x, y, z). Example 20 1. The Borromean rings have Schubert normal form (5/2, 5/2, 5/2) and w (x, y, z) In general, the toroidal link (p/1, p/1, p/1) is a 3-component link if p ≡ 1 mod 3 and it is a knot in the other cases. The knot 8 19 in Rolfsen´s table has Schubert normal form. /1) and w (x, y, z) = zyx. Note that it is the toroidal knot T (3, 4). and the word w in Proposition 19 is w (x, y, z) =The knot 8 19 in Rolfsen´s table has Schubert normal form (4/1, 4/1, 4/1) and w (x, y, z) = zyx. Note that it is the toroidal knot T (3, 4). In general, the toroidal link (p/1, p/1, p/1) is a 3-component link if p ≡ 1 mod 3 and it is a knot in the other cases; and the word w in Proposition 19 is w (x, y, z) = /3, 6/3, 6/3) and w (x, y, z) = z −1 xz −1 yz −1. Rolfsen´s table has Schubert normal form. Note that it is the Pretzel knot (3, 3, 3The knot 9 35 in Rolfsen´s table has Schubert normal form (6/3, 6/3, 6/3) and w (x, y, z) = z −1 xz −1 yz −1 . Note that it is the Pretzel knot (3, 3, 3). . G Burde, H Zieschang, Knots, Walter de GruyterNew York, NYG. Burde and H. Zieschang, Knots, Walter de Gruyter, New York, NY (1985). Introduction To Knot Theory. R Crowell, R Fox, Blaisdell Publishing CompanyNew York, NYR. Crowell and R. Fox, Introduction To Knot Theory, Blaisdell Publish- ing Company, New York, NY, 1963. Crystallizations of 2-fold branched coverings of S 3. M Ferri, Proc. Amer. Math. Soc. 73M. Ferri, Crystallizations of 2-fold branched coverings of S 3 , Proc. Amer. Math. Soc. 73 (1979), 277-276. Representing 3-manifolds by triangulations of S 3. H M Hilden, J M Montesinos, D M Tejada, M M Toro, Revista Colombiana de Matemáticas. 392H. M. Hilden, J. M. Montesinos, D. M. Tejada and M. M. Toro. Rep- resenting 3-manifolds by triangulations of S 3 . Revista Colombiana de Matemáticas, Vol 39, No 2 (2005), 63-86. H M Hilden, J M Montesinos, D M Tejada, M M Toro, arXiv:1203.2045v1A new representation of links: Butterflies. H. M. Hilden, J. M. Montesinos, D. M. Tejada and M. M. Toro, A new representation of links: Butterflies. arXiv:1203.2045v1. On the classification of 3-bridge links. H M Hilden, J M Montesinos, D M Tejada, M M Toro, Revista Colombiana de Matemáticas. 462H. M. Hilden, J. M. Montesinos, D. M. Tejada and M. M. Toro, On the classification of 3-bridge links, Revista Colombiana de Matemáticas, Vol. 46, No 2 (2012), 113-144. A survey of knot theory. A Kawauchi, Basel Berlin. BirkhäuserKawauchi, A. A survey of knot theory. Birkhäuser, Basel Berlin, (1996). K Murasugi, Knot theory and its applications. BostonK. Murasugi, Knot theory and its applications, Birkhäuser, Boston, 1996. The minimun crossing of 3-bridge links. S Negami, Osaka J. Math. 210S. Negami, The minimun crossing of 3-bridge links, Osaka J. Math. 21, N 0 3 (1984), 477-487. . L Knot Neuwirth, Groups, Westview PressBoulder, ColoradoNeuwirth, L. Knot Groups. Westview Press, Boulder, Colorado, 1969. Enlaces de tres puentes, Tesis doctoral. M Rivera, Medellín, ColombiaUniversidad Nacional de ColombiaM. Rivera, Enlaces de tres puentes, Tesis doctoral, Universidad Nacional de Colombia, Medellín, Colombia, 2016. Crystallizations of genus 2 manifolds and butterfly presentations of 3-bridge links, preprint. M Rivera, M Toro, M. Rivera and M. Toro, Crystallizations of genus 2 manifolds and but- terfly presentations of 3-bridge links, preprint, 2016. Knoten mit zwei Brücken. H Schubert, Math. Z. 65H. Schubert, Knoten mit zwei Brücken, Math. Z. 65 (1956), 133-170.

We summarize properties of the spatial sign covariance matrix and especially look at the relationship between its eigenvalues and those of the shape matrix of an elliptical distribution. The explicit relationship known in the bivariate case was used to construct the spatial sign correlation coefficient, which is a non-parametric and robust estimator for the correlation coefficient within the elliptical model. We consider a multivariate generalization, which we call the multivariate spatial sign correlation matrix.

10.17713/ajs.v46i3-4.667

1606.02274

1e626a60cba465efa005c6058fcc6f4ac8a0d771

The spatial sign covariance matrix and its application for robust correlation estimation 7 Jun 2016 A Dürre R Fried D Vogel Fakultät Statistik Institute for Complex Systems and Mathematical Biology Technische Universität Dortmund 44221DortmundGermany University of Aberdeen AB24 3UEAberdeenUnited Kingdom The spatial sign covariance matrix and its application for robust correlation estimation 7 Jun 2016 We summarize properties of the spatial sign covariance matrix and especially look at the relationship between its eigenvalues and those of the shape matrix of an elliptical distribution. The explicit relationship known in the bivariate case was used to construct the spatial sign correlation coefficient, which is a non-parametric and robust estimator for the correlation coefficient within the elliptical model. We consider a multivariate generalization, which we call the multivariate spatial sign correlation matrix. Introduction Let X 1 , . . . , X n denote a sample of independent p dimensional random variables from a distribution F and s : R p → R p with s(x) = x/|x| for x = 0 and s(0) = 0 the spatial sign, then S n (t n , X 1 , . . . , X n ) = 1 n n i=1 s(X i − t n )s(X i − t n ) T denotes the empirical spatial sign covariance matrix (SSCM) with location t n . The canonical choice for the location estimator t n is the spatial median µ n = argmin µ∈R p n i=1 ||X i − µ||. Beside its nice robustness properties like an asymptotic breakdown-point of 1/2, it has (under regularity conditions, see [12]) the advantageous feature that it centres the spatial signs, i.e., 1 n n i=1 s(X i − µ n ) = 0, so that S n (µ n , X 1 , . . . , X n ) is indeed the empirical covariance matrix of the spatial signs of the data. If t n is (strongly) consistent for a location t ∈ R, it was shown in [5] that under mild conditions on F the empirical SSCM is a (strongly) consistent estimator for its population counterpart S(X) = E(s(X − t)s(X − t) T ). There are some nice results if F is within the class of continuous elliptical distributions, which means that F possesses a density of the form f (x) = det(V ) − 1 2 g((x − µ)V −1 (x − µ) ) for a location µ ∈ R p , a symmetric and positive definite shape matrix V ∈ R p×p and a function g : R → R, which is often called the elliptical generator. Prominent members of the elliptical family are the multivariate normal distribution and elliptical t-distributions (e.g. [2], p. 208). If second moments exists, then µ is the expectation of X ∼ F , and V a multiple of the covariance matrix. The shape matrix V is unique only up to a multiplicative constant. In the following, we consider the trace-normalized shape matrix V 0 = V /tr(V ), which is convenient since S(X) also has trace 1. If F is elliptical, then S(X) and V share the same eigenvectors and the respective eigenvalues have the same ordering. For this reason, the SSCM has been proposed for robust principal component analysis (e.g. [13,15]). In the present article, we study the eigenvalues of the SSCM. Eigenvalues of the SSCM Let λ 1 ≥ . . . ≥ λ p ≥ 0 denote the eigenvalues of V 0 and δ 1 ≥ . . . ≥ δ p ≥ 0 those of S(X). Explicit formulae that relate the δ i to the λ i are only known for p = 2 (see [19,3]), namely δ i = √ λ i √ λ 1 + √ λ 2 , i = 1, 2.(1) Assuming λ 2 > 0, we have δ 1 /δ 2 = λ 1 /λ 2 ≤ λ 1 /λ 2 , thus the eigenvalues of the SSCM are closer together than those of the corresponding shape matrix. It is shown in [8] that this holds true for arbitrary p > 2, so λ i /λ j ≥ δ i /δ j for 1 ≤ i < j ≤ p(2) as long as λ j > 0. There is no explicit map between the eigenvalues known for p > 2. Dürre et al. [8] give a representation of δ i as one-dimensional integral, which permits fast and accurate numerical evaluations for arbitrary p, δ i = λ i 2 ∞ 0 1 (1 + λ i x) p j=1 (1 + λ j x) 1 2 dx, i = 1, . . . , p.(3) We use this formula (implemented in R [17] in the package sscor [9]) to get an impression how the eigenvalues of S(X) look like in comparison to those of V 0 . We first look at of equidistantly spaced eigenvalues λ i = 2i p(p + 1) , i = 1, . . . , p, Eigenvalues of S(X) Eigenvalues of S(X) for different p = 3, 11, 101. The magnitude of the eigenvalues necessarily decreases as p increases, since p i=1 λ i = p i=1 δ i = 1 per definition of V 0 and S(X). As one can see in Figure 1, the eigenvalues of S(X) and V 0 approach each other for increasing p. In fact the maximal absolute difference for p = 101 is roughly 2 · 10 −4 . In the second scenario, we take p − 1 equidistantly spaced eigenvalues and one eigenvalue 5 times larger than the rest, i.e., • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0.• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •λ i =    i p((p+1)/2+5)−5 i = 1, . . . , p − 1, 5(p−1) p((p+1)/2+5)−5 i = p. This models the case where the dependence is mainly driven by one principle component. As one can see in Figure 2, the distance between the two largest eigenvalues is smaller for S(X) than for V 0 . This is not surprising in light of (2). Thus in general, the eigenvalues of the SSCM are less separated than those of V 0 , which is one reason why the use of the SSCM for robust principal component analysis has been questioned (e.g. [1,14]). However, the differences appear to be generally small in higher dimensions. Estimation of the correlation matrix Equation (1) can be used to derive an estimator for the correlation coefficient based on the empirical SSCM: the spatial sign correlation coefficient ρ n ( [6]). Under mild regularity assumptions this estimator is consistent under elliptical distributions and asymptotically normal with variance ASV(ρ n ) = (1 − ρ 2 ) 2 + 1 2 (a + a −1 )(1 − ρ 2 ) 3/2 ,(4) where a = v 11 /v 22 is the ratio of the marginal scales and ρ = v 12 / √ v 11 v 22 is the generalized correlation coefficient, which coincides with the usual moment correlation coefficient if second moments exists. Equation (4) indicates that the variance of ρ n is minimal for a = 1, but can get arbitrarily large if a tends to infinity or 0. Therefore a two-step procedure has been proposed, the two-stage spatial sign correlation ρ σ,n , which first normalizes the data by a robust scale estimator, e.g., the median absolute deviation (mad), and then computes the spatial sign correlation of the transformed data. Under mild conditions (see [7]), this two-step procedure yields an asymptotic variance of ASV(ρ σ,n ) = (1 − ρ 2 ) 2 + (1 − ρ 2 ) 3/2 ,(5) which equals that of ρ n for the favourable case of a = 1. Since (5) only depends on the parameter ρ, the two-stage spatial sign correlation coefficient is very suitable to construct robust and non-parametric confidence intervals for the correlation coefficient under ellipticity. It turns out that these intervals are quite accurate even for rather small sample sizes of n = 10 and in fact more accurate then those based on the sample moment correlation coefficient [7]. One can construct an estimator of the correlation matrix R by filling the off-diagonal positions of the matrix estimate with the bivariate spatial sign correlation coefficients of all pairs of variables. This was proposed in [6]. Equation (3) allows an alternative approach: First standardize the data by a robust scale estimator and compute the SSCM of the transformed data. Then apply a singular value decomposition S n (t n , X 1 , . . . , X n ) =Û∆Û T , where∆ contains the ordered eigenvaluesδ 1 ≥ . . . ≥δ p . One obtains estimateŝ λ 1 , . . . ,λ p by inverting (3). Although theoretical results are yet to be established, we found in our simulations that the following fix point algorithm λ (0) i = δ i , i = 1, . . . , p, λ (k+1) i = 2δ i ∞ 0 1 (1 +λ (k) i x) p j=1 (1 +λ (k) j x) 1 2 dx, −1 , i = 1, . . . , p, k = 1, 2, . . . λ (k+1) i =λ (k+1) i p j=1λ j (k+1) −1 , i = 1, . . . , p, k = 1, 2, . . . works reliably and converges fast. LetΛ denote the diagonal matrix containingλ 1 , . . . ,λ p , thenV =ÛΛÛ T is a suitable estimator for for the shape of the standardized data and R withr ij =v ij / v iivjj an estimator for the correlation matrix, which we call the multivariate spatial sign correlation matrix. Contrary to the pairwise approach, the multivariate spatial sign correlation matrix is positive semi-definite by construction. Theoretical properties of the new estimator are not straightforward to establish. By a small simulation study we want to get an impression of its efficiency. We compare the variances of the moment correlation, the pairwise as well as the multivariate spatial sign correlation under several elliptical distributions: normal, Laplace and t distributions with 5 and 10 degrees of freedom. The latter three generate heavier tails than the normal distribution. The Laplace distribution is obtained by the elliptical generator g(x) = c p exp(− |x|/2), where c p is the appropriate integration constant depending on p (e.g. [2], p. 209). We take the identity matrix as shape matrix and compare the variances of an offdiagonal element of the matrix estimates for different dimensions p = 2, 3, 5, 10, 50 and sample sizes n = 100, 1000. We use the R packages mvtnorm [10] and MNM [16] for the data generation. The results based on 10000 runs are summarized in Table 1. Except for the moment correlation at the t 5 distribution, the results for n = 100 and n = 1000 are very similar. Note that the variance of the moment correlation decreases at the Laplace distribution as the dimension p increases, but not so for the other distributions considered. The lower dimensional marginals of the Laplace distribution are, contrary to the normal and the t-distributions, not Laplace distributed (see [11]), and the kurtosis of the one-dimensional marginals of the Laplace distribution in fact decreases as p increases. Equation (5) yields an asymptotic variance of 2 for the pairwise spatial sign correlation matrix elements regardless of the specific elliptical generator, which can also be observed in the simulation results. The moment correlation is twice as efficient under normality, but has a higher variance at heavy tailed distributions. For uncorrelated t 5 distributed random variables, the spatial sign correlation outperforms the moment correlation. Looking at the multivariate spatial sign correlation, we see a strong increase of efficiency for larger p. For p = 50 the variance is comparable to that of the moment correlation. Since the asymptotic variance of the SSCM does not depend on the elliptical generator, this is expected to also hold for the multivariate spatial sign correlation, and we find this confirmed by the simulations. The multivariate spatial sign correlation is more efficient than the moment correlation even under slightly heavier tails for moderately large p. Table 1: Simulated variances (multiplied by √ n) of one off-diagonal element of the correlation matrix estimate based on the moment correlation (cor), the pairwise spatial sign correlation (sscor pairwise) and the multivariate spatial sign correlation matrix (sscor multivariate) for spherical normal (N), t 5 , t 10 , and Laplace (L) distribution, several dimensions p and sample sizes n = 100, 1000. An increase of efficiency for larger p is not uncommon for robust scatter estimators. It can be observed amongst others for M-estimators, the Tyler shape matrix, the MCD, and S-estimators (e.g. [4,18]). All of these are affine equivariant estimators, requiring n > p. This is not necessary for the spatial sign correlation matrix. One may expect that the efficiency gain for large p is at the expense of robustness, in particular a larger maximum bias curve. Further research will be necessary to thoroughly explore the robustness properties and efficiency of the multivariate spatial sign correlation estimator. Figure 1 : 1Eigenvalues of the SSCM wrt the corresponding eigenvalues of the shape matrix in the equidistant setting p = 3 (left), p = 11 (centre) and p = 101 (right). Figure 2 : 2Eigenvalues of the SSCM wrt the corresponding eigenvalues of shape matrix in the setting of one large eigenvalue for p = 3 (left), p = 11 (centre) and p = 101 (right). Robust functional principal components: A projection-pursuit approach. J L Bali, G Boente, D E Tyler, J L Wang, The Annals of Statistics. 39Bali J.L., Boente G., Tyler D.E., Wang J.L. (2011). Robust functional principal components: A projection-pursuit approach. The Annals of Statistics. Vol. 39, pp. 2852-2882. Theory of Multivariate Statistics. M Bilodeau, D Brenner, SpringerNew YorkBilodeau M., Brenner D. (1999). Theory of Multivariate Statistics. Springer, New York. The k-step spatial sign covariance matrix. C Croux, C Dehon, A Yadine, Advances in data analysis and classification. 4Croux C., Dehon C., Yadine, A. (2010). The k-step spatial sign covariance matrix. Advances in data analysis and classification. Vol. 4, pp. 137-150. Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. C Croux, G Haesbroeck, Journal of Multivariate Analysis. 71Croux C., Haesbroeck G. (1999). Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. Journal of Multivariate Analysis. Vol. 71, pp. 161-190. The spatial sign covariance matrix with unknown location. A Dürre, D Vogel, D E Tyler, Journal of Multivariate Analysis. 130Dürre A., Vogel D., Tyler D.E. (2014). The spatial sign covariance matrix with unknown location. Journal of Multivariate Analysis. Vol. 130, pp. 107-117. Spatial sign correlation. A Dürre, D Vogel, R Fried, Journal of Multivariate Analysis. 135Dürre A., Vogel D., Fried R. (2015). Spatial sign correlation. Journal of Multivari- ate Analysis. Vol. 135, pp. 89-105. Asymptotics of the two-stage spatial sign correlation. A Dürre, D Vogel, Journal of Multivariate Analysis. 144Dürre A., Vogel, D. (2016). Asymptotics of the two-stage spatial sign correlation. Journal of Multivariate Analysis. Vol. 144, pp. 54-67. On the eigenvalues of the spatial sign covariance matrix in more than two dimensions. A Dürre, D E Tyler, D Vogel, Statistics & Probability Letters. 111Dürre A., Tyler D.E., Vogel, D. (2016). On the eigenvalues of the spatial sign covariance matrix in more than two dimensions. Statistics & Probability Letters. Vol. 111, pp. 80-85. sscor: Robust Correlation Estimation and Testing Based on Spatial Signs. A Dürre, D Vogel, R package version 0.2Dürre A., Vogel D. (2016). sscor: Robust Correlation Estimation and Testing Based on Spatial Signs. R package version 0.2. A Genz, F Bretz, T Miwa, X Mi, F Leisch, F Scheipl, B Bornkamp, M Maechler, T Hothorn, mvtnorm: Multivariate Normal and t Distributions. R package version 1.0.5Genz A, Bretz F., Miwa T., Mi X., Leisch F., Scheipl F., Bornkamp B., Maech- ler M., Hothorn T. (2016), mvtnorm: Multivariate Normal and t Distributions. R package version 1.0.5. Consistency property of elliptic probability density functions. Y Kano, Journal of Multivariate Analysis. 51Kano Y. (1994). Consistency property of elliptic probability density functions, Journal of Multivariate Analysis. Vol. 51, pp. 139-147. Statistical Data Analysis Based on the L 1 -Norm and Related Methods. J H B Kemperman, The median of a finite measure on a Banach spaceKemperman J. H. B. (1987). The median of a finite measure on a Banach space. Statistical Data Analysis Based on the L 1 -Norm and Related Methods. pp. 217- 230. Robust principal component analysis for functional data. N Locantore, J S Marron, D G Simpson, N Tripoli, J T Zhang, K L Cohen, Test. 8Locantore N., Marron J.S., Simpson D.G., Tripoli N., Zhang J.T., Co- hen K.L. (1999). Robust principal component analysis for functional data. Test. Vol. 8, pp. 1-73. The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions. A F Magyar, D E Tyler, Biometrika. 101Magyar A.F., Tyler D.E. (2014). The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions. Biometrika. Vol. 101, pp. 673-688. Some robust estimates of principal components. J I Marden, Statistics & probability letters. 43Marden, J.I. (1999). Some robust estimates of principal components. Statistics & probability letters. Vol. 43, pp. 349-359. Multivariate L 1 methods: the package MNM. K Nordhausen, H Oja, Journal of Statistical Software. 43Nordhausen K., Oja H. (2011), Multivariate L 1 methods: the package MNM. Journal of Statistical Software. Vol. 43, pp. 1-28. R: A Language and Environment for Statistical Computing. R Development Core TeamR Development Core Team (2016). R: A Language and Environment for Statistical Computing. Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices. S Taskinen, C Croux, A Kankainen, E Ollila, H Oja, Journal of Multivariate Analysis. 97Taskinen S., Croux C., Kankainen A., Ollila E., Oja H. (2006). Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices. Journal of Multivariate Analysis. 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This work investigates the problem of testing whether d functional random variables are jointly independent using a modified estimator of the d-variable Hilbert Schmidt Indepedence Criterion (dHSIC) which generalizes HSIC for the case where d ≥ 2. We then get asymptotic normality of this estimator both under joint independence hypothesis and under the alternative hypothesis. A simulation study shows good performance of the proposed test on finite sample.AMS 1991 subject classifications: 62E20, 46E22.

2208.06940

50c206440a7fe5c91d802f867ed01a9849e74d14

Kernel-based method for joint independence of functional variables Aug 2022 Terence Kevin URMI Université des Sciences et Techniques de Masuku FrancevilleGabon Manfoumbi Djonguet URMI Université des Sciences et Techniques de Masuku FrancevilleGabon Guy Martial [email protected]. URMI Université des Sciences et Techniques de Masuku FrancevilleGabon Nkiet URMI Université des Sciences et Techniques de Masuku FrancevilleGabon Kernel-based method for joint independence of functional variables Aug 2022hypothesis testingjoint independencereproducing kernel Hilbert spacefunctional data analysisasymptotic distribution This work investigates the problem of testing whether d functional random variables are jointly independent using a modified estimator of the d-variable Hilbert Schmidt Indepedence Criterion (dHSIC) which generalizes HSIC for the case where d ≥ 2. We then get asymptotic normality of this estimator both under joint independence hypothesis and under the alternative hypothesis. A simulation study shows good performance of the proposed test on finite sample.AMS 1991 subject classifications: 62E20, 46E22. INTRODUCTION The question of relationships between variables has always been a major concern in statistical analysis. This is why methods providing answers to this type of problems, such as regression methods or independence tests, occupy an important place in the statistical literature. However, there exists just a few works that deal with independence testing in the context of functional data analysis, despite the importance of this field over the last two decades. Kokoszka et al. (2008) and Aghoukeng Jiofack and Nkiet (2010) introduced non-correlation tests between two functional variables, whereas Górecki et al. (2020), Lai et al. (2021) and Lauman et al. (2021) recently proposed tests for independence. Kernel-based methods, that use the distance between the kernel embeddings of probability measures in a reproducing kernel Hilbert space (RKHS), were also introduced in Gretton et al. (2005Gretton et al. ( , 2007. For the case where more than two variables are considered, pairwise independence tests can be obtained by perfoming the aforementioned tests on each pair, but it is not sufficient for some statistical problems such as those arising in causal inference where models often assume the existence of jointly independent noise variables, or in independent component analysis. So, it can be more relevant to consider testing for joint independence of several random variables rather than testing pairwise independence. Recently, Pfister et al. (2018) proposed kernel-based methods for testing for joint independence of d random variables valued into metric spaces, with d ≥ 2. For doing that, they introduced the d-variable Hilbert-Schmidt independence criterion (dHSIC) as a measure of joint independence that extends the Hilbert-Schmidt independence criterion (HSIC) of Gretton et al. (2005) and contains it as a special case. For testing for joint independence they considered, as test statistic, a consistent estimator of dHSIC and derived its asymptotic distribution under null hypothesis. However, as it is the case for most measures introduced in kernel-based methods such as maximal mean discrepancy (MMD) (Gretton et al. (2012)), generalized maximal mean discrepancy (GMMD) (Balogoun et al. (2021(Balogoun et al. ( ,2022) and HSIC, this asymptotic distribution is an infinite sum of distributions and, therefore, can not be used for performing the test. That is why Pfister et al. (2018) proposed three approaches based on dHSIC: a permutation test, a bootstrap test and a gamma approximation of the limiting distribution. An alternative approach can be tackled as it was done in Makigusa and Naito (2020) and in Balogoun et al. (2021) for the MMD and the GMMD. It consists in constructing a test statistic from an appropriate modification of a naive estimator in order to yield asymptotic normality both under the null hypothesis and under the alternative. In this paper, such an approach is adopted for the dHSIC. It allows us to propose a new test for joint independence of several random variables valued into metric spaces, including functional variables. The rest of the paper is organized as follows. The dHSIC is recalled in Section 2, and Section 3 is devoted to its estimation by a modification of a naive estimator, and to the main results. A simulation study on functional data that allows to compare the proposed test to that of Pfister et al. (2018) is given in Section 4. All the proofs are postponed in Section 5. dHSIC AND JOINT INDEPENDENCE PROPERTY For an integer d ≥ 2, let X (1) , · · · , X (d) be random variables defined on a probability space (Ω, A , P ) and with values in compact metric spaces X 1 , · · · , X d respectively. Then, we consider the random variable X = (X (1) , · · · , X (d) ) with values in X := X 1 × X 2 × · · · × X d . For ℓ ∈ {1, · · · , d}, let us introduce a reproducing kernel Hilbert space (RKHS) H ℓ of functions from X ℓ to R with associated kernel K ℓ : X 2 ℓ → R. Each K ℓ is a symmetric function such that, for any f ∈ H ℓ and any t ∈ X ℓ , one has K ℓ (t, ·) ∈ X ℓ and f (t) = K ℓ (t, ·), f X ℓ (see Berlinet and Thomas-Agnan (2004)), where ·, · X ℓ denotes the inner product of X ℓ . Denoting by f 1 ⊗ · · · ⊗ f d the tensor products of functions defined by: (f 1 ⊗ · · · ⊗ f d ) (x 1 , · · · , x d ) = d ℓ=1 f ℓ (x ℓ ), we consider H := H 1 ⊗ · · · ⊗ H d the completed of the vector space spanned by the set {f 1 ⊗ · · · ⊗ f d / f ℓ ∈ H ℓ , ℓ = 1, · · · , d}. It is known that H is a RKHS associated with the kernel K satisfying: K (x, y) = d ℓ=1 K ℓ (x ℓ , y ℓ )(1) for any x = (x 1 , · · · , x d ) ∈ X and y = (y 1 , · · · , y d ) ∈ X . Throughout this paper, we make the following assumption: (A 1 ) : K ℓ ∞ := sup (t,s)∈X 2 ℓ K ℓ (t, s) < +∞, ℓ = 1, · · · , d; it implies that K ∞ < +∞ and ensures the existence of the kernel mean embeddings m ℓ = E K ℓ (X (ℓ) , ·) , ℓ = 1, · · · , d, and m = E (K(X, ·)). The random variables X (1) , · · · , X (d) are said to be jointly independent if P X = P X (1) ⊗ P X (2) ⊗ · · · ⊗ P X (d) , where P X (resp. P X (j) ) denotes the distribution of X (resp. X (j) ). For measuring this property, Pfister et al. (2018) introduced the d-variable Hilbert-Schimdt independence criterion (dHSIC) defined as dHSIC(X) = E (K(X, ·)) − d ℓ=1 m ℓ 2 H = E d ℓ=1 K ℓ (X (ℓ) , ·) − d ℓ=1 m ℓ 2 H ,(2) where d ℓ=1 m ℓ = m 1 ⊗ m 2 ⊗ · · · ⊗ m d . It is known that if the kernels K ℓ are characteristic ones, then dHSIC fully characterizes joint independence since the equality P X = P X (1) ⊗ P X (2) ⊗ · · · ⊗ P X (d) is equivalent to dHSIC(X) = 0 (see Pfister et al. (2018)). So, in order to test for joint independence, that is testing for the hypothesis H 0 : P X = P X (1) ⊗ P X (2) ⊗ · · · ⊗ P X (d) against H 1 : P X = P X (1) ⊗ P X (2) ⊗ · · · ⊗ P X (d) , a consistent estimator of dHSIC(X) can be used as test statistic. Pfister et al. (2018) introduced the statistic dHSIC n given by Let {X i } 1≤i≤n be an i.i.d. sample of X, with X i = (X (1) i , · · · , X (d) i ),dHSIC n = 1 n 2 n i=1 n j=1 d ℓ=1 K ℓ (X (ℓ) i , X (ℓ) j ) + 1 n 2d n i 1 =1 · · · n i 2d =1 d ℓ=1 K ℓ (X (ℓ) i 2ℓ−1 , X (ℓ) i 2ℓ ) − 2 n d+1 n i 1 =1 · · · n i d+1 =1 d ℓ=1 K ℓ (X (ℓ) i ℓ , X (ℓ) i ℓ+1 )(3) if n ≥ 2d, and dHSIC n = 0 if n ∈ {1, · · · , 2d − 1}. They proved that, under H 0 , the sequence n dHSIC n converges in distribution, as n → +∞, to 2d d +∞ m=1 λ m Z 2 m , where (λ m ) m≥1 is an appropriate sequence of positive real numbers and (Z m ) m≥1 is a sequence of independent standard normal random variables. This limiting distribution cannot be used for performing the test because it is an infinite sum of distributions. That is why Pfister et al. (2018) introduced three approach for this test based on the above estimator: a permutation test, a boostrap analogue and a procedure based on a gamma approximation. We propose in this paper another estimator for which we can obtain asymptotic normality under the null hypothesis. The main advantage is that it permits to avoid the use of permutation or bootstrap methods for achieving the test, so leading to a faster procedure. MODIFIED ESTIMATOR AND ASYMPTOTIC NORMALITY A natural approach for estimating dHSIC(X) consists in replacing in (2) each expectation by its empirical counterpart. This leads to the naive estimator D n = 1 n n i=1 K(X i , ·) − d ℓ=1 m ℓ 2 H = 1 n 2 n i=1 n j=1 d ℓ=1 K ℓ (X (ℓ) i , X (ℓ) j ) + 1 n 2d d ℓ=1 n i=1 n j=1 K ℓ (X (ℓ) i , X (ℓ) j ) − 2 n d+1 n i=1 d ℓ=1 n j=1 K ℓ (X (ℓ) i , X (ℓ) j ) ,(4)where m ℓ = 1 n n i=1 K ℓ (X (ℓ) i , ·) . Instead, we adopt an approach introduced in Makigusa and Naito (2020), and also used in Balogoun et al. (2021), consisting in introducing a modification in (4) in order to get a test statistic for which asymptotic normality can be obtained both under H 0 and under H 1 . For γ ∈ ]0, 1], let (w i,n (γ)) 1≤i≤n be a sequence of positive numbers satisfying : (A 2 ) : There exists a strictly positive real number τ and an integer n 0 such that for all n > n 0 : n 1 n n i=1 w i,n (γ) − 1 ≤ τ. (A 3 ) : There exists C > 0 such that max 1≤i≤n w i,n (γ) < C for all n ∈ N * and all γ ∈]0, 1]. (A 4 ) : For any γ ∈]0, 1], lim n→+∞ 1 n n i=1 w 2 i,n (γ) = w 2 (γ) > 1. An example of such sequence was given in Ahmad (1993) as w i,n (γ) = 1 + (−1) i γ. For this example, one has C = 2, w 2 (γ) = 1+γ 2 and τ is any positive real number. Another example is w i,n (γ) = 1 + sin(iπγ) which corresponds to τ = 1/| sin(πγ/2)|, C = 2 and w 2 (γ) = 3/2. We then propose to estimate dHSIC(X) by a modification of (4) given by D γ,n = 1 n 2 n i=1 n j=1 d ℓ=1 K ℓ (X (ℓ) i , X (ℓ) j ) + 1 n 2d d ℓ=1 n i=1 n j=1 K ℓ (X (ℓ) i , X (ℓ) j ) − 2 n d+1 n i=1 w i,n (γ) d ℓ=1 n j=1 K ℓ (X (ℓ) i , X (ℓ) j ) ,(5) and we take this estimator as test statistic for testing for H 0 against H 1 . For achieving this test, the asymptotic distribution under H 0 of the test statistic is needed. We will obtain an asymptotic normality result which is usable both under H 0 and under H 1 , so permitting to perform the test. For any (k, q) ∈ N * such that q ≥ k, we put ν q k = q ℓ=k m ℓ . Then, considering the functions U and V from X to R defined on any x = x (1) , · · · , x (d) ∈ X by U(x) = K 1 (x (1) , ·) ⊗ ν d 2 + d−1 ℓ=2 ν ℓ−1 1 ⊗ K ℓ (x (ℓ) , ·) ⊗ ν d ℓ+1 + ν d−1 1 ⊗ K d (x (d) , ·) − d ν d 1 , ν d 1 − m H + K(x, ·) − m, m H and V(x) = K(x, ·) − m, ν d 1 H , we have: Theorem 1 Assume that (A 1 ) to (A 4 ) hold. Then as n → +∞, one has √ n D γ,n − dHSIC(X) D −→ N (0, σ 2 γ ), where: σ 2 γ = 4V ar (U(X 1 )) + 4w 2 (γ)V ar (V(X 1 )) − 8Cov (U(X 1 ), V(X 1 )) . This theorem gives asymptotic normality in the general case. The particular case where H 0 holds can then be deduced. Indeed, in this case one has ν d 1 = d ℓ=1 m ℓ = m and, therefore, U = V. Since dHSIC(X) = 0 it follows that √ n D γ,n D −→ N (0, σ 2 γ ), where: σ 2 γ = 4(w 2 (γ) − 1)V ar (V(X 1 )) . This asymptotic variance in unknown since it depends on the distribution of X which is unknown. So, in order to perform the test, we have to seek a consistent estimator of it. The following proposition gives such an estimator: Proposition 1 Assume that (A 1 ) to (A 4 ) hold, and put σ 2 γ = 4(w(γ) 2 − 1) α 2 n , where α 2 n = 1 n n i=1 1 n n j=1 d ℓ=1 K ℓ (X (ℓ) i , X (ℓ) j ) 2 − 1 n 2 n i=1 n j=1 d ℓ=1 K ℓ (X (ℓ) i , X (ℓ) j ) 2 . Then, under H 0 , σ 2 γ converges in probability to σ 2 γ as n → +∞. We deduce from the previous theorem and proposition that, under the hypothesis H 0 , √ n σ −1 γ D γ,n D −→ N (0, 1) as n → +∞. This allows to perform the test: for a given significant level α ∈]0, 1[, one has to reject H 0 if D γ,n > n −1/2 σ γ Φ −1 (1−α/2) where Φ is the cumulative distribution function of the standard normal distribution. Remark 1. This test can easily be applied on functional data. Indeed, suppose that each X K ℓ (X (ℓ) i , X (ℓ) j ) = exp −η 2 ℓ X i − X j 2 = exp −η 2 ℓ 1 0 X (ℓ) i (t) − X (ℓ) j (t) 2 dt , where η ℓ > 0, and this term can be approximated by using trapezoidal rule: K ℓ (X (ℓ) i , X (ℓ) j ) ≃ exp − η 2 ℓ r ℓ −1 m=1 t (ℓ) m+1 − t (ℓ) m 2 X (ℓ) i (t (ℓ) m ) − X (ℓ) j (t (ℓ) m ) 2 + X (ℓ) i (t (ℓ) m+1 ) − X (ℓ) j (t (ℓ) m+1 ) 2 .(6) Then, D γ,n and α 2 n are to be computed by using (6). SIMULATIONS In this section, we present a simulation study made in order to evaluate the finite sample performance of the proposed test and compare it to the test of Pfister et al. (2018) based on a permutation method using dHSIC n . For convenience, we denote our test as T1, and the test of Pfister et al. (2018) as T2. We computed empirical sizes and powers through Monte Carlo simulations. We considered the case where d = 3, X 1 = X 2 = X 3 = L 2 ([0, 1]), and used the following models for generating functional data: X (1) (t) = √ 2 50 k=1 α k cos(kπt), X (2) (t) = √ 2 50 k=1 β k cos(kπt) + λf X (1) (t) , X (3) (t) = √ 2 50 k=1 ξ k cos(kπt) + λf X (1) (t) + λf X (2) (t) , where the α k s, the β k s and the ξ k s are independent and i.i.d. having the uniform distribution on [0, 1], f is a given function and λ is a real number giving the level of dependence between the functional variables. Indeed, H 0 holds in case λ = 0, and the dependence level increases with λ. Empirical sizes and powers were computed on the basis of 200 independent replicates. For each of them, we generated samples of size n = 100 of the above processes in discretized versions on equispaced values t 1 , · · · , t 51 in [0, 1], where t j = (j − 1)/50, j = 1, · · · , 51. For performing our method, we took γ = 0.32 and used the gaussian kernels K 1 (x, y) = K 2 (x, y) = K 3 (x, y) = exp −150 1 0 (x(t) − y(t)) 2 dt . The test statistics given in (3) and (5) were computed by approximating integrals involved in the above kernels by using the trapezoidal rule, as indicated in (6). The significance level was taken as α = 0.05. The method T2 was used with 100 permutations. Table 1 reports the obtained results. They are close to the nominal size for both methods when λ = 0. For λ = 0.25, T2 gives slightly larger powers than T1, but the differences are low. For m = 0.5, 1, the two methods give maximal power, except for the case where f (x) = sin(x) for which the powers are surprisingly low. This highlights the interest of the proposed test: it is powerful enough and is fast compared to the method of Pfister et al. (2018) based on permutations wich leads to very high computation times. PROOFS PRELIMINARY RESULTS We first give some technical lemmas which will be useful for proving the main theorems. f (x) method λ = 0 λ = 0.25 λ = 0. m ℓ − d ℓ=1 m ℓ D −→ W,(7) where W is a random variable having the normal distribution in H with mean 0 and covariance operator equal to that of K 1 (X (1) , ·)⊗ d ℓ=2 m ℓ + d−1 ℓ=2 ℓ−1 k=1 m k ⊗K ℓ (X (ℓ) , ·)⊗ d k=ℓ+1 m k + d−1 ℓ=1 m ℓ ⊗K d (X (d) , ·). Proof. Let us consider the random variables valued into H 1 × H 2 × · · · × H d defined as Z = K 1 (X (1) , ·), K 2 (X (2) , ·), . . . , K d (X (d) , ·) , Z i = K 1 (X (1) i , ·), K 2 (X (2) i , ·), . . . , K d (X (d) i , ·) , i = 1, · · · , n, with mean equal to µ = (m 1 , m 2 , . . . , m d ). For each ℓ ∈ {1, · · · , d}, we have √ n ( m ℓ − m ℓ ) = π ℓ ( H n ), where H n = √ n 1 n n i=1 Z i − µ and π ℓ is the canonical projection: π ℓ : (f 1 , f 2 , . . . , f d ) ∈ H 1 × H 2 × · · · × H d −→ f ℓ ∈ H ℓ . Then, from the decomposition √ n d ℓ=1 m ℓ − d ℓ=1 m ℓ = √ n( m 1 − m 1 ) ⊗ d ℓ=2 m ℓ + d−1 ℓ=2 ℓ−1 k=1 m k ⊗ √ n( m ℓ − m ℓ ) ⊗ d k=ℓ+1 m k + d−1 ℓ=1 m ℓ ⊗ √ n( m d − m d ) we get √ n d ℓ=1 m ℓ − d ℓ=1 m ℓ = Φ n H n ,(8) where Φ n is the random operator from H 1 × H 2 × · · · × H d to H such that Φ n (T ) = π 1 (T )⊗ d ℓ=2 m ℓ + d−1 ℓ=2 ℓ−1 k=1 m k ⊗π ℓ (T )⊗ d k=ℓ+1 m k + d−1 ℓ=1 m ℓ ⊗π d (T ). Considering the operator Φ from H 1 × H 2 × · · · × H d to H such that Φ(T ) = π 1 (T ) ⊗ d ℓ=2 m ℓ + d−1 ℓ=2 ℓ−1 k=1 m k ⊗ π ℓ (T ) ⊗ d k=ℓ+1 m k + d−1 ℓ=1 m ℓ ⊗ π d (T ),(9) we have Φ n H n − Φ H n H ≤ π 1 H n ⊗ d ℓ=2 m ℓ − d ℓ=2 m ℓ H + d−1 ℓ=2 ℓ−1 p=1 m ℓ ⊗ π ℓ H n ⊗ d p=ℓ+1 m p − d p=ℓ+1 m p H ≤ π 1 H n H d ℓ=2 m ℓ − d ℓ=2 m ℓ H + d−1 ℓ=2 ℓ−1 p=1 m ℓ H π ℓ H n H d p=ℓ+1 m p − d p=ℓ+1 m p H .(10) The cental limit theorem ensures that H n converges in distribution, as n → +∞, to a random variable H having a centered normal distribution in H with covariance operator equal to that of Z. Since π ℓ is continuous, we deduce that π ℓ H n converges in distribution, as n → +∞, to π ℓ (H). On the other hand, from the law of large numbers each m ℓ converges almost surely, as n → +∞, to m ℓ . Then, for any k ∈ {1, · · · , d − 1}, we deduce from the inequality d ℓ=k m ℓ − d ℓ=k m ℓ H = ( m k − m k ) ⊗ d ℓ=k+1 m ℓ + d−1 ℓ=k+1 ℓ−1 j=1 m j ⊗ ( m ℓ − m ℓ ) ⊗ d j=ℓ+1 m j + d−1 ℓ=1 m ℓ ⊗ ( m d − m d ) H ≤ m k − m k H k d ℓ=k+1 m ℓ H ℓ + d−1 ℓ=k+1 ℓ−1 j=1 m j H j m ℓ − m ℓ H ℓ d j=ℓ+1 m j H j + d−1 ℓ=1 m ℓ H ℓ m d − m d H d that d ℓ=k m ℓ − d ℓ=k m ℓ H converges almost surely, as n → +∞, to 0. Therefore, from (10) we deduce that Φ n H n − Φ H n converges in probability, as n → +∞, to 0; consequently, √ n d ℓ=1 m ℓ − d ℓ=1 m ℓ = Φ H n + o P (1) and, since Φ is contionuous, Slustky's theorrem implies the convergence in distribution , as n → +∞, to W = Φ(H). This random variable has a centered normal distribution in H with covariance operator equal to that of Φ(Z) = K 1 (X (1) , ·) ⊗ d ℓ=2 m ℓ + d−1 ℓ=2 ℓ−1 k=1 m k ⊗ K ℓ (X (ℓ) , ·) ⊗ d k=ℓ+1 m k + d−1 ℓ=1 m ℓ ⊗ K d (X (d) , ·). Lemma 2 Let (Z i ) 1≤i≤n be an i.i.d. sample of a random variable Z valued into a Hilbert space H and admitting a mean m Z . If there exists a real number M > 0 such that Z ≤ M, then the statistic s 2 n = 1 n n i=1 Z i , Z n 2 − 1 n n i=1 Z i , Z n 2 ,(11) where Z n = 1 n n i=1 Z i , is a consistent estimator of V ar Z, m Z . Proof. We have 1 n n i=1 Z i , Z n 2 − 1 n n i=1 Z i , m Z 2 = 1 n n i=1 Z i , Z n −m Z 2 + 2 n n i=1 Z i , m Z Z i , Z n −m Z ; and using the Cauchy-Shwarz inequality, we get the inequalities 1 n n i=1 Z i , Z n − m Z 2 ≤ M 2 Z n − m Z 2 and 2 n n i=1 Z i , m Z Z i , Z n − m Z ≤ 2M 2 m Z Z n − m Z from which we conclude that 1 n n i=1 Z i , Z n 2 − 1 n n i=1 Z i , m Z 2 = o P (1) since Z n − m Z = o P (1) . The law of large numbers and Slutsky's theorem allow to conclude that the sequence 1 n n i=1 Z i , Z n 2 converges in probability to E Z, m Z 2 as n → +∞. Similarly, from 1 n n i=1 Z i , Z n − 1 n n i=1 Z i , m Z = 1 n n i=1 Z i , Z n − m Z ≤ M Z n − m Z , we get 1 n n i=1 Z i , Z n − 1 n n i=1 Z i , m Z = o P (1), therefore, and 1 n n i=1 Z i , Z n converges in probability to E Z, m Z as n → +∞ just like 1 n n i=1 Z i , m Z . So, s 2 n is a consistent estimator of V ar Z, m Z . 5.2. PROOF OF THEOREM 1 2 m, d ℓ=1 m ℓ H and 1 n n i=1 w i,n (γ) K(X i , ·), d ℓ=1 m ℓ H = 1 n n i=1 (w i,n (γ) − 1) K(X i , ·), d ℓ=1 m ℓ − d ℓ=1 m ℓ H + 1 n n i=1 w i,n (γ)K(X i , ·), d ℓ=1 m ℓ H + m − m, d ℓ=1 m ℓ − d ℓ=1 m ℓ H + m, d ℓ=1 m ℓ H − 1 n n i=1 w i,n (γ) m, d ℓ=1 m ℓ H + 1 n n i=1 (w i,n (γ) − 1) m, d ℓ=1 m ℓ H , it follows √ n D γ,n − dHSIC(X) = A n + B n + C n , where A n = 1 √ n √ n ( m − m) 2 H + 1 √ n √ n d ℓ=1 m ℓ − d ℓ=1 m ℓ 2 H −2 m − m, √ n d ℓ=1 m ℓ − d ℓ=1 m ℓ H , B n = −2 1 n n i=1 (w i,n (γ) − 1) K(X i , ·), √ n d ℓ=1 m ℓ − d ℓ=1 m ℓ H −2 √ n 1 n n i=1 (w i,n (γ) − 1) m, d ℓ=1 m ℓ H and We will now decompose the term Φ H n , d ℓ=1 m ℓ − m H using (9). For ease of notation we set ν q k = q ℓ=k m ℓ , and we have π 1 H n ⊗ d ℓ=2 m ℓ = √ n(m 1 − m 1 ) ⊗ ν q 2 = 1 √ n n i=1 K 1 (X (1) i , ·) − m 1 ⊗ ν d 2 = 1 √ n n i=1 K 1 (X (1) i , ·) ⊗ ν d 2 − d ℓ=1 m ℓ , ℓ−1 k=1 m k ⊗ π ℓ H n ⊗ d k=ℓ+1 m k = ν ℓ−1 1 ⊗ √ n( m ℓ − m ℓ ) ⊗ ν d ℓ+1 = 1 √ n n i=1 ν ℓ−1 1 ⊗ K ℓ (X (ℓ) i , ·) − m l ⊗ ν d ℓ+1 = 1 √ n n i=1 ν ℓ−1 1 ⊗ K ℓ (X (ℓ) i , ·) ⊗ ν d ℓ+1 − d ℓ=1 m ℓ , and d−1 ℓ=1 m ℓ ⊗ π d H n = ν d−1 1 ⊗ √ n( m d − m d ) = 1 √ n n i=1 ν d−1 1 ⊗ K d (X (d) i , ·) − m d = 1 √ n n i=1 ν d−1 1 ⊗ K d (X (d) i , ·) − d ℓ=1 m ℓ . Hence Φ H n = 1 √ n n i=1 K 1 (X (1) i , ·)⊗ν d 2 + d−1 ℓ=2 ν ℓ−1 1 ⊗ K ℓ (X (ℓ) i , ·) ⊗ ν d ℓ+1 +ν d−1 1 ⊗K d (X (d) i , ·)−d d ℓ=1 m ℓ , and from (14) and (15) it follows that C n = D n + o P (1), where D n = 2 √ n n i=1 K(X i , ·) − m, m H − w i,n (γ) K(X i , ·) − m, d ℓ=1 m ℓ H + K 1 (X (1) i , ·) ⊗ ν d 2 + d−1 ℓ=2 ν ℓ−1 1 ⊗ K ℓ (X (ℓ) i , ·) ⊗ ν d ℓ+1 +ν d−1 1 ⊗ K d (X (d) i , ·) − d d ℓ=1 m ℓ , d ℓ=1 m ℓ − m H + o P (1) = 2 √ n n i=1 U(X i ) − w i,n (γ)V(X i ) + o P (1). Finally, √ n D γ,n −dHSIC(X) = E n +o P (1), where E n = 2 √ n n i=1 U(X i )− w i,n (γ)V(X i ) and, consequently, √ n D γ,n −dHSIC(X) has the same limiting distribution than E n ; it remains to derive this latter. Let us set s 2 n,γ = n i=1 V ar U(X i ) − w i,n (γ)V(X i ) ; by similar arguments as in the proof of Theorem 1 in Makigusa and Naito (2020) we obtain that, for any ε > 0, Thus lim n→+∞ n −1 s 2 n,γ = V ar (U(X 1 )) + w 2 (γ)V ar (V(X 1 )) − 2Cov (U(X 1 ), V(X 1 )) and, therefore, E n D → N 0, σ 2 γ . 5.3. PROOF OF PROPOSITION 1 Putting Z = K(X, ·), Z i = K(X i , ·) and Z n = 1 n n i=1 Z i , we have Z H = K(X, X) = d ℓ=1 K ℓ (X (ℓ) , X (ℓ) ) ≤ d ℓ=1 K ℓ 1/2 ∞ , Z i , Z n H = 1 n n j=1 Z i , Z j H = 1 n n j=1 K(X i , X j ) = 1 n n j=1 d ℓ=1 K ℓ (X (ℓ) i , X By applying Lemma 2 we obtain the convergence in probability of α 2 n to V ar ( K(X, ·), m H ) = V ar (V(X 1 )) as n → +∞ and, therefore, that of σ 2 γ to σ 2 γ . BIBLIOGRAPHY Aghoukeng Jiofack, J.G., and G.M. Nkiet. 2010. Testing for lack of dependence between functional variables.Statistics and Probability Letters 80, 1210-1217. = 1. If gaussian kernels are used, one has :|U (x)−w i,n (γ)V(x)|>εsn,γ } U(x) − w i,n (γ)V(x) 2 dP X (x)converges to 0 as n → +∞. Then, by Section 1.9.3 inSerfling (1980), Using (1) and the reproducing properties of K, it is easy to see thatFirst, using Cauchy-Schwartz inequality, we getThe law of large numbers ensures that m − m H → 0 as n → +∞. Then, from Lemma 1 we deduce that 1converge in probability to 0 as n → +∞.The central limit theorem gives the convergence in distrubution of m − m, as n → +∞, to a random variable having a normal distribution in H, hence 1 √ n √ n ( m − m) 2 H converges in probability to 0 as n → +∞. Then, from the preceding inequality we get A n = o P (1). Secondly, another use of Cauchy-Schwartz inequality gives for n large enoughsince, from (A 2 ), we have n 1 n n i=1 (w i,n (γ) − 1) ≤ τ . Lemma 1 in ManfoumbiDjonguet et al. (2022)H converges in distribution, as n → +∞, to an appropriate random variable (see Lemma 1), we deduce from (13) that B n = o P (1). Thirdly, reporting (8) in(12)givesFrom Modification of some goodness-of-fit statistics to yield asymptotic normal null distribution. I A Ahmad, Biometrika. 80Ahmad, I.A. 1993. Modification of some goodness-of-fit statistics to yield asymptotic normal null distribution. Biometrika 80, 466-472. Asymptotic normality of a generalized maximum mean discrepancy estimator. A K S Balogoun, G M Nkiet, C Ogouyandjou, Statistics and Probability Letters. 169108961Balogoun, A.K.S., G.M. Nkiet, and C. Ogouyandjou. 2021. Asymptotic normality of a generalized maximum mean discrepancy estimator. Statistics and Probability Letters 169, 108961. A k-sample test for functional data based on generalized maximum mean discrepancy. A K S Balogoun, G M Nkiet, C Ogouyandjou, 10.1007/10986-022-09572-xLithuanian Mathematical Journal. Balogoun, A.K.S., G.M. Nkiet, and C. Ogouyandjou. 2022. A k-sample test for functional data based on generalized maximum mean discrepancy. Lithuanian Mathematical Journal, DOI:10.1007/10986-022-09572-x. Reproducing kernel Hilbert spaces in probability and statistics. A Berlinet, C Thomas-Agnan, KluwerBerlinet, A., and C. Thomas-Agnan. 2004. Reproducing kernel Hilbert spaces in probability and statistics. Kluwer. Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data. T Górecki, M Krzysko, W Wo, Artificial Intelligence Review. 53Górecki, T., M. Krzysko, and W. Wo lynski. 2020. Independence test and canonical correlation analysis based on the alignment between kernel matrices for multivariate functional data. Artificial Intelligence Review 53, 475-499. A kernel two-sample test. A Gretton, K M Borgwardt, M J Rasch, B Schölkopf, A J Smola, Journal Machine Learning Research. 13Gretton, A., K.M. Borgwardt, M.J. Rasch, B. Schölkopf, and A.J. Smola. 2012. A kernel two-sample test. Journal Machine Learning Research 13, 723-776. Measuring statistical dependence with Hilbert-Schmidt norms. A Gretton, O Bousquet, A Smola, B Schölkopf, Algorithmic Learning Theory. S. Jain, H.U. Simon and E. TomitaBerlinSpringerGretton, A., O. Bousquet, A. Smola, and B. Schölkopf. 2005. Measuring statistical dependence with Hilbert-Schmidt norms. In Algorithmic Learning Theory (eds S. Jain, H.U. Simon and E. Tomita), 63-77. Berlin: Springer. . A Gretton, K Fukumizu, C H Teo, L Song, B Schölkopf, A J Smola, Gretton, A., K. Fukumizu, C.H. Teo, L. Song, B. Schölkopf, and A.J. Smola. A kernel statistical test of independence. Advances in Neural Information Processing Systems. J.C. Platt, D. Koller, Y. Singer and S.T. RoweisNew YorkCurran AssociatesA kernel statistical test of independence. In Advances in Neural Infor- mation Processing Systems 20 (eds J.C. Platt, D. Koller, Y. Singer and S.T. Roweis), 585-592. New York: Curran Associates. Testing for lack of dependence in the functional linear model. P Kokoszka, I Maslova, J Sojka, L Zhu, Canadian Journal of Statistics. 36Kokoszka, P., I. Maslova, J. Sojka, and L. Zhu. 2008. Testing for lack of dependence in the functional linear model. Canadian Journal of Statistics 36, 1-16. Testing independence of functional variables by angle covariance. T Lai, Z Zhang, Y Wang, L Kong, Journal of Multivariate Analysis. 182Lai, T., Z. Zhang, Y. Wang, and L. Kong. 2021. Testing independence of functional variables by angle covariance. Journal of Multivariate Analysis 182, 104711. Kernel two-sample and independence tests for non-stationary random processes. F Laumann, J Von Kügelgen, M Brahona, Engineering Proceedings. 531Laumann, F., J. Von Kügelgen, and M. Brahona. 2021. Kernel two-sample and independence tests for non-stationary random processes. Engineering Proceedings 5, 31. Asymptotic normality of a consistent estimator of maximum mean discrepancy in Hilbert space. N Makigusa, K Naito, Statistics and Probability Letters. 156108596Makigusa, N., and K. Naito. 2020. Asymptotic normality of a consistent estimator of maximum mean discrepancy in Hilbert space. Statistics and Probability Letters 156, 108596. Testing independence of functional variables by an Hilbert-Schmidt independence criterion estimator. Manfoumbi Djonguet, T K , A Mbina Mbina, G M Nkiet, arXiv:2206.11607Manfoumbi Djonguet, T.K., A. Mbina Mbina, and G.M. Nkiet. 2022. Testing independence of functional variables by an Hilbert-Schmidt independence criterion estimator. arXiv:2206.11607. Kernel-based tests for joint independence. N Pfister, P Bühlmann, B Schölkopf, J Peters, Journal of the Royal Statistical Society Series B Statistical Methodology. 80Pfister, N., P. Bühlmann, B. Schölkopf, and J. Peters. 2018. Kernel-based tests for joint independence. Journal of the Royal Statistical Society Series B Statistical Methodology 80, 5-31. Approximation Theorems of Mathematical Statistics. R J Serfling, WileyNew-YorkSerfling, R.J. 1980. Approximation Theorems of Mathematical Statistics. Wiley, New-York.

The toughness t(G) of a connected graph G is defined as t(G) = min{ |S| c(G−S) }, in which the minimum is taken over all proper subsets S ⊂ V (G) such that c(G − S) > 1, where c(G − S) denotes the number of components of G−S. Let λ denote the second largest absolute eigenvalue of the adjacency matrix of a graph. For any connected d-regular graph G, it has been shown by Alon that t(G) > 1 3 ( d 2 dλ+λ 2 − 1), through which, Alon was able to show that for every t and g there are t-tough graphs of girth strictly greater than g, and thus disproved in a strong sense a conjecture of Chvátal on pancyclicity. Brouwer independently discovered a better bound t(G) > d λ − 2 for any connected d-regular graph G, while he also conjectured that the lower bound can be improved to t(G) ≥ d λ − 1. We confirm this conjecture.MSC: 05C42 05C50

10.1137/20m1372652

2010.05065

fffdccc6e556af7a55b70924a624de6b4170eb1e

A proof of Brouwer's toughness conjecture 15 May 2021 Xiaofeng Gu A proof of Brouwer's toughness conjecture 15 May 2021arXiv:2010.05065v2 [math.CO]toughnesseigenvalueexpander mixing lemmapseudo-random graph The toughness t(G) of a connected graph G is defined as t(G) = min{ |S| c(G−S) }, in which the minimum is taken over all proper subsets S ⊂ V (G) such that c(G − S) > 1, where c(G − S) denotes the number of components of G−S. Let λ denote the second largest absolute eigenvalue of the adjacency matrix of a graph. For any connected d-regular graph G, it has been shown by Alon that t(G) > 1 3 ( d 2 dλ+λ 2 − 1), through which, Alon was able to show that for every t and g there are t-tough graphs of girth strictly greater than g, and thus disproved in a strong sense a conjecture of Chvátal on pancyclicity. Brouwer independently discovered a better bound t(G) > d λ − 2 for any connected d-regular graph G, while he also conjectured that the lower bound can be improved to t(G) ≥ d λ − 1. We confirm this conjecture.MSC: 05C42 05C50 The conjecture We use λ i (G) to denote the ith largest eigenvalue of the adjacency matrix of a simple graph G on n vertices, for i = 1, 2, · · · , n. By the Perron-Frobenius Theorem, λ 1 is always positive (unless G has no edges) and |λ i | ≤ λ 1 for all i ≥ 2. Let λ = max 2≤i≤n |λ i | = max{|λ 2 |, |λ n |}, that is, λ is the second largest absolute eigenvalue. For a d-regular graph, it is well known that λ 1 = d. Let c(G) denote the number of components of a graph G. The toughness t(G) of a connected graph G is defined as t(G) = min{ |S| c(G−S) }, where the minimum is taken over all proper subsets S ⊂ V (G) such that c(G − S) > 1. A graph G is k-tough if t(G) ≥ k. Graph toughness was introduced by Chvátal [10] in 1973 and is closely related to many graph properties, including connectivity, Hamiltonicity, pancyclicity, factors, spanning trees, etc. The study of toughness from eigenvalues was initiated by Alon [1] who showed that for any connected d-regular graph G, t(G) > 1 3 ( d 2 dλ+λ 2 − 1), through which, Alon was able to show that for every t and g there are t-tough graphs of girth strictly greater than g. This strengthened a result of Bauer, Van den Heuvel and Schmeichel [4] who showed the same for g = 3, and thus disproved in a strong sense a conjecture of Chvátal [10] that there exists a constant t 0 such that every t 0 -tough graph is pancyclic. Brouwer [5] independently discovered a better bound and showed that t(G) > d λ − 2 for a connected d-regular graph G. He mentioned in [5] that the bound might be able to be improved a little to d λ − 1, and then posed the exact conjecture as an open problem in [6]. Some partial results have been provided in [8]. However, no substantial progress has been made for more than two decades. Most recently, the author improved the bound to t(G) > d λ − √ 2 in [11]. Conjecture 1 (Brouwer [5,6]). For any connected d-regular graph G, t(G) ≥ d λ − 1. It is mentioned by Brouwer [5] that there are infinitely many graphs G such that t(G) ≤ d/λ with equality in many cases (for example, strongly regular graphs constructed in [5] and [9], and many Kneser graphs [14]). Brouwer [5] also pointed out that Conjecture 1, if true, is tight. To see sharpness, one may notice that the toughness can be arbitrarily close to 0, while d/λ ≥ 1 holds for all d-regular graphs. Cioabȃ and Wong [9] briefly described an explicit construction of such extremal graphs. In this paper, we confirm Conjecture 1. Theorem 1. Let G be a connected d-regular graph. Then t(G) ≥ d λ − 1. The proof Our main tool is the expander mixing lemma. A d-regular graph on n vertices with the second largest absolute eigenvalue at most λ is called an (n, d, λ)-graph. It is well known that an (n, d, λ)graph for which λ = Θ( √ d) is a very good pseudo-random graph behaving, in many aspects, like a truly random graph G(n, p). The quantitative definition of pseudo-random graphs was introduced by Thomason [15,16] who defined jumbled graphs. The celebrated expander mixing lemma is usually attributed to Alon and Chung [2]. This idea appeared earlier with a different form in the PhD thesis [12] of Haemers. We present this lemma in Theorem 2 (see [13] for a complete proof). A variation can be found in [3, Chapter 9] by Alon and Spencer. We refer readers to the informative survey [13] by Krivelevich and Sudakov for more about the expander mixing lemma and pseudo-random graphs. As mentioned in [13], the expander mixing lemma is a truly remarkable result, connecting edge distribution and graph spectrum, and providing a very good quantitative handle for the uniformity of edge distribution based on graph eigenvalues. Theorem 2 (Expander Mixing Lemma). Let G be an (n, d, λ)-graph. Then for every two subsets A and B of V (G), e(A, B) − d n |A||B| ≤ λ |A||B| 1 − |A| n 1 − |B| n .(1) In particular, e(A) − d 2n |A| 2 ≤ λ 2 |A| 1 − |A| n .(2) The following lemma has been proved and used by Brouwer and Haemers [7]. For the sake of completeness, we include a short proof here. Lemma 3 ([7]) . Let x 1 , · · · , x c be positive integers such that c i=1 x i ≤ 2c − 1. Then for every integer ℓ with 0 ≤ ℓ ≤ c i=1 x i , there exists an I ⊂ {1, · · · , c} such that i∈I x i = ℓ. Proof. The proof provided by Brouwer and Haemers [7] goes by induction on c. The case c = 1 is trivial. Let c ≥ 2 and assume that First we show that c ≤ λn d + λ . x 1 ≤ · · · ≤ x c . Suppose it is true for c − 1 integers. Let ℓ ′ = ℓ if ℓ ≤ c − 1 and ℓ ′ = ℓ − x c otherwise. Since c − 1 ≤ c−1 i=1 x i = c i=1 x i − x c ≤ 2(c − 1) − 1, In fact, (4) can be obtained from the well-known Hoffman ratio bound. Here we give a direct proof. To see this, let U be a set of vertices that consists of exactly one vertex from each component of G − S. Then |U | = c and e(U ) = 0. By (2), e(U ) − d 2n |U | 2 ≤ λ 2 |U | 1 − |U | n , and thus d n |U | ≤ λ 1 − |U | n , which implies that c = |U | ≤ λn d+λ . If |B| ≤ 2λn d+λ , then tc = n − |B| ≥ (d−λ)n d+λ . Together with (4), we have t = (n − |B|)/c ≥ (d − λ)n d + λ · d + λ λn = d − λ λ = d λ − 1, contrary to (3). Thus, we may assume that |B| > 2λn d + λ ,(5) By (4) and (5), we have |B| > 2c, that is |B| ≥ 2c + 1.(6) Let V 1 , · · · , V c denote the vertex sets of the c components of G − S. Without loss of generality, we may assume that |V 1 | ≤ · · · ≤ |V c |. Claim 1: c−1 i=1 |V i | ≥ c. Proof of Claim 1: Otherwise, c−1 i=1 |V i | = c− 1, that is, each V i is a single vertex for i = 1, · · · , c− 1. Let A = ∪ c−1 i=1 V id n |A||B| ≤ λ |A||B| 1 − |A| n 1 − |B| n < λ |A||B| 1 − |B| n , and so d n |A||B| 2 < λ 2 |A||B| 1 − |B| n , which implies that d 2 |A||B| λ 2 n < n − |B| = tc. Thus t > |A| c · d 2 λ 2 · |B| n = c − 1 c · d 2 λ 2 · |B| n = 1 − 1 c · d 2 λ 2 · |B| n ≥ d 2 2λ 2 · |B| n . By (5), |B| n > 2λ d+λ and we have t > d 2 2λ 2 · 2λ d + λ = d 2 λ(d + λ) = d 2 − λ 2 + λ 2 λ(d + λ) = d 2 − λ 2 λ(d + λ) + λ 2 λ(d + λ) > d λ − 1, contrary to (3). This completes the proof of Claim 1. Claim 2: V 1 , · · · , V c can be partitioned into two disjoint sets X and Y such that e(X, Y ) = 0, |X| ≥ c and |Y | ≥ c. Proof of Claim 2: If |V c | ≥ c, then let X = ∪ c−1 i=1 V i and Y = B − X = V c . Then |Y | ≥ c and by Claim 1, |X| ≥ c. Thus we may assume that |V c | ≤ c − 1. We also notice that |V c | ≥ 3, for otherwise, |B| = c i=1 |V i | ≤ 2c, contradicting (6). Thus 3 ≤ |V c | ≤ c − 1. Let ℓ = c − |V c |. Then 1 ≤ ℓ ≤ c − 3. If c−1 i=1 |V i | ≤ 2(c − 1) − 1 = 2c − 3 , then by Lemma 3, there exists an I ⊂ {1, · · · , c − 1} such that i∈I |V i | = ℓ. Let X = ∪ i∈I V i ∪ V c and Y = B − X. Clearly |X| = ℓ + |V c | = c and |Y | = |B| − |X| ≥ c + 1. If c−1 i=1 |V i | > 2(c−1)−1 = 2c−3, then let V ′ i be a nonempty subset of V i for each i = 1, 2, · · · c−1 such that c−1 i=1 |V ′ i | = 2c − 3. We can easily do it by removing vertices from some V i 's one by one until the total number of remaining vertices is exactly 2c − 3 (keep at least one vertex in each subset). By Lemma 3, there exists an I ⊂ {1, · · · , c − 1} such that i∈I |V ′ i | = ℓ. Then i ∈I,i<c |V ′ i | = (2c−3)−ℓ ≥ (2c−3)−(c−3) = c. Let X = ∪ i∈I V i ∪V c and Y = B −X = ∪ i ∈I,i<c V i . Clearly |X| ≥ ℓ + |V c | = c and |Y | = i ∈I,i<c |V i | ≥ i ∈I,i<c |V ′ i | ≥ c. This completes the proof of Claim 2. For every two subsets A and B of V (G), let e(A, B) denote the number of edges with one end in A and the other one in B (edges with both ends in A ∩ B are counted twice). We use e(A) to denote the number of edges with both ends in A, and thus e(A, A) = 2e(A). by inductive hypothesis, there exists an I ′ ⊂ {1, · · · , c − 1} such that i∈I ′ x i = ℓ ′ . Let I = I ′ or I = I ′ ∪ {c}, as desired. Now we are ready to present the proof of the main theorem. Proof of Theorem 1. Suppose to the contrary that , suppose S is a subset of V (G) such that |S| c(G−S) = t(G). Let B = V (G − S). Denote |V (G)| = n, c(G − S) = c and t(G) = t. Then |S| = tc and so |B| = n − tc. and thus |A| = c − 1 and e(A, B) = 0. By (1), By Claim 2, V 1 , · · · , V c can be partitioned into two disjoint sets X and Y such that e(X, Y ) = 0, |X| ≥ c and |Y | ≥ c.Since e(X, Y ) = 0, by (1), d n |X||Y | ≤ λ |X||Y | 1 − |X| n 1 − |Y | n , and sowhich implies thatWithout loss of generality, we may assume that |X| ≤ |Y |. By(7), we haveAlso, since |Y | = n − |S| − |X|, by(7)By (8), we haveand we have |X| ≤ λ d (|S| + |X|) .completing the proof of the theorem. Tough Ramsey graphs without short cycles. N Alon, J. Algebraic Combin. 4N. Alon, Tough Ramsey graphs without short cycles, J. Algebraic Combin. 4 (1995), 189-195. Explicit construction of linear sized tolerant networks. N Alon, F R K Chung, Discrete Math. 72N. Alon and F. R. K. Chung, Explicit construction of linear sized tolerant networks, Discrete Math. 72 (1988), 15-19. . N Alon, J H Spencer, The Probabilistic Method. WileyN. Alon and J. H. Spencer, The Probabilistic Method, Wiley, 2016. Toughness and triangle-free graphs. D Bauer, J Van Den Heuvel, E Schmeichel, J. Combin. Theory Ser. B. 65D. Bauer, J. van den Heuvel and E. Schmeichel, Toughness and triangle-free graphs, J. Combin. Theory Ser. B 65 (1995), 208-221. Toughness and spectrum of a graph. A E Brouwer, Linear Algebra Appl. 226A. E. Brouwer, Toughness and spectrum of a graph, Linear Algebra Appl. 226/228 (1995), 267-271. Spectrum and connectivity of graphs. A E Brouwer, CWI Quarterly. 9A. E. Brouwer, Spectrum and connectivity of graphs, CWI Quarterly 9 (1996), 37-40. Eigenvalues and perfect matchings. A E Brouwer, W H Haemers, Linear Algebra Appl. 395A. E. Brouwer and W. H. Haemers, Eigenvalues and perfect matchings, Linear Algebra Appl. 395 (2005), 155-162. Connectivity, toughness, spanning trees of bounded degrees, and spectrum of regular graphs. S M Cioabȃ, X Gu, Czechoslovak Math. J. 66S. M. Cioabȃ and X. Gu, Connectivity, toughness, spanning trees of bounded degrees, and spectrum of regular graphs, Czechoslovak Math. J. 66 (2016), 913-924. The spectrum and toughness of regular graphs. S M Cioabȃ, W Wong, Discrete Appl. Math. 176S. M. Cioabȃ and W. Wong, The spectrum and toughness of regular graphs. Discrete Appl. Math. 176 (2014), 43-52. Tough graphs and hamiltonian circuits. V , Discrete Math. 5V. Chvátal, Tough graphs and hamiltonian circuits, Discrete Math. 5 (1973), 215-228. Toughness in pseudo-random graphs. X Gu, European J. Combin. 92103255X. Gu, Toughness in pseudo-random graphs, European J. Combin., 92 (2021), 103255. W H Haemers, Eigenvalue techniques in design and graph theory. PhD thesisW. H. Haemers, Eigenvalue techniques in design and graph theory, PhD thesis, 1979. Pseudo-random graphs, More sets, graphs and numbers. M Krivelevich, B Sudakov, Springer15BerlinM. Krivelevich and B. Sudakov, Pseudo-random graphs, More sets, graphs and numbers, Bolyai Soc. Math. Stud., 15, Springer, Berlin, 2006, 199-262. D Park, A Ostuni, N Hayes, A Banerjee, T Wakhare, W Wong, S Cioabȃ, arXiv:2008.08183The toughness of Kneser graphs. math.COD. Park, A. Ostuni, N. Hayes, A. Banerjee, T. Wakhare, W. Wong and S. Cioabȃ, The tough- ness of Kneser graphs, arXiv:2008.08183 [math.CO]. Pseudo-random graphs. A Thomason, Proceedings of Random Graphs. M. KarońskiRandom GraphsPoznań33A. Thomason, Pseudo-random graphs, in: Proceedings of Random Graphs, Poznań 1985, M. Karoński, ed., Annals of Discrete Math. 33 (1987), 307-331. Random graphs, strongly regular graphs and pseudo-random graphs. A Thomason, LMS Lecture Note Series. C. Whitehead123Surveys in CombinatoricsA. Thomason, Random graphs, strongly regular graphs and pseudo-random graphs, Surveys in Combinatorics, 1987, C. Whitehead, ed., LMS Lecture Note Series 123 (1987), 173-195.

In this paper, we modify the adaptive cubic regularization method for large-scale unconstrained optimization problem by using a real positive definite scalar matrix to approximate the exact Hessian. Combining with the nonmonotone technique, we also give a variant of the modified algorithm. Under some reasonable conditions, we analyze the global convergence of the proposed methods. Numerical experiments are performed and the obtained results show satisfactory performance when compared to the standard trust region method, adaptive regularization algorithm with cubics and the simple model trust-region method.

1904.07440

2caa6da5b74b1f7bde9c1361c90b31f99705d260

A modified adaptive cubic regularization method for large-scale unconstrained optimization problem * 16 Apr 2019 Yutao Zheng School of Mathematics and Statistics Lanzhou University 730000LanzhouP.R. China College of Mathematics and Information Science Henan Normal University 453007XinxiangChina Bing Zheng School of Mathematics and Statistics Lanzhou University 730000LanzhouP.R. China A modified adaptive cubic regularization method for large-scale unconstrained optimization problem * 16 Apr 2019Unconstrained optimizationBarzilai-Borwein step lengthAdaptive cubic regularization methodNonmonotone line search AMS Subject Classifications(2010): 65K0590C3049M37 In this paper, we modify the adaptive cubic regularization method for large-scale unconstrained optimization problem by using a real positive definite scalar matrix to approximate the exact Hessian. Combining with the nonmonotone technique, we also give a variant of the modified algorithm. Under some reasonable conditions, we analyze the global convergence of the proposed methods. Numerical experiments are performed and the obtained results show satisfactory performance when compared to the standard trust region method, adaptive regularization algorithm with cubics and the simple model trust-region method. Introduction Consider the following unconstrained optimization problem min f (x), x ∈ R n(1) where f : R n → R is a continuously differentiable function and its gradient g(x) is available. In addition to the trust-region (TR) [10] and line-search [24] methods, Cartis et al. [8] suggested a third alternative for solving (1) -the use of a cubic overestimator of the objective function as a regularization technique. Let x k be the current iterate point, g k = g(x k ) and B k denotes the exact Hessian of f at x k or its symmetric approximation. According to the adaptive regularization algorithm using cubics (ARC) proposed in [8], at each iteration k, the cubic model is m k (s) = f (x k ) + s T g k + 1 2 s T B k s + 1 3 σ k s 3 ,(2) where σ k > 0 is an adaptive parameter which can be viewed as the reciprocal of the trust-region radius. The trial step s k is computed as an approximate global minimizer of m k (s). The next iterate x k + s k is accepted if the value of the metric function ρ k = f (x k ) − f (x k + s k ) f (x k ) − m k (s k ) is greater than some positive constant of η 1 . The adaptive parameter σ k is updated by a similar mechanism in the trust region method. Compared to the standard trust region approach, the ARC method has a better numerical performance for small-and medium-scale test problems, see [8] for more details. Another good point for ARC method is its worst-case complexity when the exact second order information is provided, which means that we can find an ǫ-approximate first-order critical point for an unconstrained problem with Lipschitz continuous Hessians in at most O(ǫ −2/3 ) evaluations of the objective function (and its derivatives). Since then, more and more researchers take their attention to this new topic. Bianconcini et al. [5] supplied a viable alternative to the implementation of ARC method using the GLRT routine of GALAHAD library [16]. Bergou et al. [3] considered the energy norm of s M = √ s T Ms instead of the general Euclidean norm s , where M is a symmetric positive definite matrix. Bianconcini and Sciandrone [4] introduced the line search and nonmonotone techniques to the ARC algorithm. Some worst-case evaluation complexity results can be found in [6,7,9]. We refer the readers to [2,15,18,21,22] and references therein for more related work. However, from the view of numerical simulation, the TR or ARC method will be not suitable for large-scale problems, it may be very expensive to evaluate the exact Hessians. Moreover, choosing the true or a quasi-Newton approximate Hessian as B k will make finding the minimizer of the cubic model (2) more difficult and more complex, see more details in [8]. Such of these drawbacks can restrict the applications of the TR and ARC methods in practice. Recently, Zhou et al. [30] introduced a nonmonotone adaptive trust region method with line search based on diagonal updating the Hessians, then Zhou et al. [31] embedded the Barzilai-Borwein step length [1] in the framework of simple model trust-region method. The resulted algorithm simplifies the computation of solving the trust-region subproblems and shows better numerical performance when compared with the global Barzilai-Borwein method [25]. In this paper, we will take B k as a real positive definite scalar matrix γ k I, where γ k > 0 is the Barzilai-Borwein step length or some of its variants, which will inherit some certain quasi-Newton property. The minimizer of the resulted cubic model can be easily determined, and at the same time, the convergence of the algorithm is also maintained. The rest of this paper is organized as follows. In Sect. 2, we propose a modified adaptive regularization algorithm by introducing the Barzilai-Borwein parameter. Under some certain conditions, the global and strong convergence of the modified method is studied in Sect. 3. With nonmonotone technique, we also present a variant of the MARC algorithm and analyze the corresponding convergence in Sect. 4. Numerical experiments are performed in Sect. 5. Finally, in Sect. 6, we give some conclusions to end this paper. The MARC Method We first review the Barzilai-Borwein gradient method [1] which is given by x k+1 = x k − D k g k , where D k = 1 γ k I. Ask D k to have the certain quasi-Newton property, that is, γ k = arg min γ s k−1 − 1 γ y k−1 = s T k−1 y k−1 s T k−1 s k−1 ,(3) where s k−1 = x k − x k−1 and y k = g k − g k−1 . Due to its very satisfactory performance compared to the steepest descent method and better convergence property, Barzilai-Borwein gradient method now has been developed into a competitive method for large scale problems and got broad attention of numerous experts. Moreover, the Barzilai-Borwein step length is also widely applied in other optimization algorithms, such as the trust region method [31] and the conjugate gradient method [12], and was used to solve constrained optimization problem [11,20], multiobjective optimization problem [23] and so on. By using the real positive definite scalar matrix γ k I to approximate the Hessian, where γ k is defined by (3) as before, the subproblem (2) in the ARC algorithm can be written as min m k (s) = f (x k ) + g T k s + 1 2 γ k s T s + 1 3 σ k s 3 , s ∈ R n ,(4) which can be easily solved. Since ∇m k (s) = g k + γ k s + σ k s s, the solution s k of problem (4) is parallel to −g k . Let s k = −α k g k , (4) is equivalent to min α>0 φ(α) = f (x k ) − g k 2 α + 1 2 γ k g k 2 α 2 + 1 3 σ k g k 3 α 3 . Asking φ ′ (α) = g k 2 (σ k g k · α 2 + γ k · α − 1) = 0 yields a positive solution of α k = −γ k + γ 2 k + 4σ k g k 2σ k g k = 2 γ k + γ 2 k + 4σ k g k . Thus the exact solution of (4) is s k = − 2 γ k + γ 2 k + 4σ k g k g k ,(5) which implies that s k ≤ g k σ k .(6) Since the Barzilai-Borwein steplength γ k would be very large or negative, the truncation technique will be used, we restrict γ k to the interval [γ min , γ max ]. Now, recall the framework of the adaptive cubic regularization algorithm and the trust region method, we state our algorithm as below: Algorithm 1 Modified Adaptive Regularization algorithm using Cubics (MARC) Require: x 0 , η 2 ≥ η 1 > 0, c 1 ≥ 1 ≥ c 2 > 0, γ max > γ 0 > γ min > 0 and σ 0 > 0. For k = 0, 1, · · · until convergence, 1: Compute s k by (5). 2: Compute f (x k + s k ) and ρ k = f (x k ) − f (x k + s k ) f (x k ) − m k (s k ) ;(7) 3: Set x k+1 = x k + s k if ρ k ≥ η 1 , x k otherwise. 4: Set σ k+1 =    c 2 σ k , if ρ k > η 2 , [very successful iteration], σ k , if η 1 ≤ ρ k ≤ η 2 , [successful iteration], c 1 σ k , otherwise, [unsuccessful iteration].(8) 5: Compute γ k+1 by (3), (9) or (10). Set γ k+1 = max{γ min , min{γ k+1 , γ max }}. Note that the scalar γ k plays an important role in Algorithm 1, hence, another two formulas developed by Yabe et al. [27] and Zheng et al. [29] to compute γ k are considered, they are defined as γ k+1 = s T k y k + θ k [2(f k − f k+1 )] + (g k + g k+1 ) T s k s T k s k ,(9) where θ k ∈ [0, 3], and γ k+1 = r T k w k r T k r k ,(10) where r k = s k − ψ k s k−1 and w k = y k − ψ k y k−1 , ψ k ≥ 0 is a scalar. The γ k+1 in (10) reduces to (3) if ψ k = 0 for all k. Convergence Analysis Our convergence analysis is similar to that of the ARC method in [8]. However, the real positive scalar matrix, a special approximation of Hessian can simplify the analyses of some conclusions in some sense. Throughout this and the next section, we denote the index set of all successful iterations of the MARC algorithm by S {k ≥ 0 : k is successful or very successful}. We first give a lower bound on the decrease in f predicted from the cubic model. Lemma 3.1. Let f ∈ C 1 . Suppose that the step s k is computed by (5). Then for all k ≥ 0, we have that f (x k ) − m k (s k ) ≥ g k 12 min g k γ k , 1 2 g k σ k .(11) Proof. By a direct computation, we have f (x k ) − m k (s k ) =f (x k ) − m k (−α k g k ) =α k g k 2 1 − 1 2 γ k α k − 1 3 σ k g k α 2 k ≥ g k 2 6(γ k + 2 σ k g k ) ≥ g k 12 min g k γ k , 1 2 g k σ k , since 1 γ k +2 √ σ g k ≤ 1 √ γ 2 k +4σ g k ≤ α k ≤ 1 γ k and α k ≤ 1 √ σ k g k . In the above proof, we remark that α k is also less than 2 1 2 γ k + √ 1 4 γ 2 k + 4 3 σ k g k , which is the positive root of h(α) = 1 − 1 2 γ k α − 1 3 σ k g k α 2 = 0. This fact gives that h(α k ) > 0. Furthermore, h(α k ) > 1 6 since α k ≤ 1 γ k gives γ k α k ≤ 1 and α k ≤ 1 √ σ k g k gives σ k g k α 2 k ≤ 1. An auxiliary lemma is given next. Lemma 3.2. Let f ∈ C 1 . Suppose that ǫ > 0 and I is an infinite index set such that g k ≥ ǫ, k ∈ I,(12) and g k σ k → 0, k → ∞, k ∈ I.(13) Furthermore, if for some x * ∈ R n , x k → x * , k → ∞, k ∈ I,(14) then each iteration k ∈ I that is sufficiently large is very successful, and σ k+1 ≤ σ k holds for all k ∈ I sufficiently large. Proof. We first estimate the difference between the function and the model at x k + s k . A Taylor expansion of f (x k + s k ) around x k gives f (x k + s k ) − m k (s k ) = g(ξ k ) − g k T s k − 1 2 γ k s k 2 − 1 3 σ k s k 3 for some ξ k on the line segment (x k , x k + s k ), which, together with (6), further gives f (x k + s k ) − m k (s k ) ≤ g(ξ k ) − g k + γ k 2 g k σ k · g k σ k .(15) On the other hand, it follows from (11) and the boundedness of γ k that f (x k ) − m k (s k ) ≥ g k 12 min g k γ max , 1 2 g k σ k holds for all k. By employing the limit in (13), for all k ∈ I sufficiently large, we have f (x k ) − m k (s k ) ≥ g k 24 g k σ k . Thus, by (15), we obtain that r k :=f (x k + s k ) − f (x k ) − η 2 [m k (s k ) − f (x k )] =f (x k + s k ) − m k (s k ) + (1 − η 2 )[m k (s k ) − f (x k )] ≤ g k σ k g(ξ k ) − g k + γ max 2 g k σ k − (1 − η 2 )ǫ 24 ,(16) for all k ∈ I sufficiently large. (6) and (13) imply that s k → 0 as k → ∞, k ∈ I. Therefore ξ k → x * as k → ∞, k ∈ I due to (14). Since g is continuous, we conclude that g(ξ k ) − g k → 0 as k → ∞, k ∈ I. Thus, the limits in (13) and (16) imply that r k < 0, for all k ∈ I sufficiently large. Recalling that Since ξ k ∈ (x k , x k + s k ), we have ξ k − x * ≤ x k − x * + s k . Also,r k < 0 ⇔ ρ k = f (x k ) − f (x k + s k ) f (x k ) − m k (s k ) > η 2 , which means that the k-th iteration is very successful, and hence σ k+1 ≤ σ k . The following lemma shows that if there are only finitely many successful iterations, then all later iterates are first-order critical points. Lemma 3.3. Let f ∈ C 1 . Suppose that there are only finitely many successful iterations. Then x k = x * for all sufficiently large k and g(x * ) = 0. Proof. See the proof of Lemma 2.4 in [8]. Now, we are ready to show the global convergence of the MARC algorithm. We conclude that if f is bounded from below, either g k = 0 for some finite k, or there is a subsequence of {g k } converging to zero. Theorem 3.1. Let f ∈ C 1 , {x k } is the sequence generated by Algorithm 1. If {f (x k )} is bounded below, then lim inf k→∞ g k = 0.(17) Proof. Lemma 3.3 shows that (17) is true when there are only finitely many successful iterations. Now we assume that infinitely many successful iterations occur, and recall the notation S. We prove (17) by contradiction. Assume that there is some ǫ > 0 such that g k ≥ ǫ(18) holds for all k ≥ 0. We first prove that (18) implies that k∈S g k σ k < +∞,(19) which further gives that g k /σ k → 0, and hence σ k → ∞ as k → ∞. It follows from (11), (18) and the construction of algorithm that f (x k ) − f (x k+1 ) ≥η 1 [f (x k ) − m k (s k )] ≥ η 1 ǫ 12 min ǫ γ max , 1 2 g k σ k , k ∈ S.(20) Since the sequence {f (x k )} is monotonically decreasing and bounded below, it is convergent. As k → ∞, the minimum on the right-hand side of (20) will be attained at g k /(2 √ σ k ) and the left-hand side of (20) converges to zero. Thus we obtain f (x k ) − f (x k+1 ) ≥ η 1 ǫ 24 g k σ k ,(21) for all k ∈ S sufficiently large. Summing up (21) over all sufficiently large iterations provides that f (x k 0 ) − f (x j+1 ) = j k=k 0 ,k∈S [f (x k ) − f (x k+1 )] ≥ η 1 ǫ 24 j k=k 0 ,k∈S g k σ k(22) holds for some index k 0 sufficiently large and for any j ∈ S, j ≥ k 0 . Since {f (x j+1 )} is convergent, letting j → ∞ in (22) yields (19). Now we show that the sequence of iterates {x k } is a Cauchy sequence. It follows from (6) and the construction of the algorithm that x l+r − x l ≤ l+r−1 k=l x k+1 − x k = l+r−1 k=l,k∈S s k ≤ l+r−1 k=l,k∈S g k σ k , for l ≥ 0 sufficiently large and any r ≥ 0, whose right-hand term tends to zero as l → ∞ due to (19). Thus {x k } is a Cauchy sequence, and there is some x * ∈ R n such that x k → x * , k → ∞.(23) Then all the conditions of Lemma 3.3 hold with I := S, and all k ∈ S sufficiently large are very successful. Now, if all k sufficiently large belong to S, then by (8), σ k+1 ≤ σ k for all k sufficiently large, and so {σ k } is bounded above. This, however, contradicts σ k → ∞. Thus (18) cannot hold. It remains to show that all sufficiently large iterations belong to S. Conversely, let {k i } denote an infinite subsequence of very successful iterations such that k i − 1 is unsuccessful for all i ≥ 0. Then, σ k i = c 1 σ k i −1 and g k i = g k i −1 for all i. Thus, we deduce that g k i −1 /σ k i −1 → 0, i → ∞.(24) It follows from (18), (23) and (24) that (12)- (14) are satisfied with I := {k i − 1 : i ≥ 0}, and so Lemma 3.2 provides that k i − 1 is very successful for all i sufficiently large. This contradicts our assumption that k i − 1 is unsuccessful for all i. Furthermore, if we assume that g is uniformly continuous, the whole sequence of gradients {g k } converges to zero. Theorem 3.2. Let f ∈ C 1 and its gradient g is uniformly continuous, {x k } is the sequence generated by Algorithm 1. If {f (x k )} is bounded below, then lim k→∞ g k = 0.(25) Proof. Lemma 3.3 implies that (25) holds if there are finitely many successful iterations. Now assume that there is an infinite subsequence {t i } ⊆ S such that g t i ≥ 2ǫ, for some ǫ > 0 and for all i. By Theorem 3.1, for each t i , there exists a first successful iteration l i > t i such that g l i < ǫ. Thus {l i } ⊆ S, and we have g t i ≥ ǫ,(26) for all i and for all k with t i ≤ k < l i . Let K {k ∈ S : t i ≤ k < l i }, where {t i } and {l i } were defined above. Since K ⊆ S, it follows from (11), (18) and the construction of algorithm that f (x k ) − f (x k+1 ) ≥ η 1 ǫ 12 min ǫ γ max , 1 2 g k σ k , k ∈ K.(27) Since {f (x k )} is monotonically decreasing and bounded from below, it is convergent, i.e., f (x k ) − f (x k+1 ) → 0 as k → ∞, thus g k /σ k → 0, k → ∞, k ∈ K, which, together with (6) and (27), implies for all t i ≤ k < l i , k ∈ S, i sufficiently large, f (x k ) − f (x k+1 ) ≥ η 1 ǫ 24 s k . Summing up the above inequation over k with t i ≤ k < l i , we have x t i − x l i ≤ l i −1 k=t i x k+1 − x k = l i −1 k=t i s k ≤ 24 η 1 ǫ [f (x t i ) − f (x l i )], for all i sufficiently large. Since {f (x k )} is convergent, {f (x t i ) − f (x l i )} converges to zero as i → ∞. Thus x t i − x l i → 0 as i → ∞, and hence g t i − g l i → 0 since g is uniformly continuous. Then we have reached a contradiction, since g t i − g l i ≥ g t i − g l i ≥ ǫ for all i > 0. Remark: We could set γ min = 0 in Algorithm 1, which means that we will ignore the second-order term when γ k < 0. In this case, the model m k (s) = f (x k ) + g T k s + σ k 3 s 3 is just the linear model plus a term of cubic regularization, the minimizer of m k (s) is s = − g k √ σ k g k , which can be reduced from (2) with γ k = 0, and s k = g k /σ k . Furthermore, f (x k ) − m k (s k ) = 2 3 g k σ k g k . Then the argument of Lemma 3.1 still holds. Thus this change will not affect our above analysis. A Variant of MARC Algorithm Recently, nonmonotone technique is always applied to develop more efficient optimization algorithms. Combining with the nonmonotone line search of Grippo et al. [19], Raydan [25] proposed a globally convergent Barzilai and Borwein gradient method for large scale unconstrained optimization problem, Deng et al. [13] and Sun [26] generalized the above nonmonotone rule into the trust region framework. Bianconcini and Sciandrone [4] introduced the line search and nonmonotone techniques to the ARC algorithm. In this section, we will present a variant of MARC algorithm based on Hager and Zhang's nonmonotone technique [28]. To determine the step-size in the line search methods, Hager and Zhang [28] considered the following type of nonmonotone line search rule f (x k + α k d k ) ≤ C k + σα k g T k d k , where C k is defined as (7), then we state a variant of MARC algorithm in Algorithm 2. C k = η k−1 Q k−1 C k−1 + f (x k ) Q k , in which Q k = η k−1 Q k−1 + 1. The initial value C 0 = f 0 and Q 0 = 1. Replacing f (x k ) with C k in Algorithm 2 A Variant of MARC Algorithm (VMARC) Require: x 0 , η 2 ≥ η 1 > 0, c 1 ≥ 1 ≥ c 2 , γ max > γ 0 > γ min > 0, σ 0 > 0, C 0 , Q 0 For k = 0, 1, · · · until convergence, 1: Compute s k by (5). 2: Compute f (x k + s k ) and ρ k = C k − f (x k + s k ) f (x k ) − m k (s k ) ; 3: Set x k+1 = x k + s k if ρ k ≥ η 1 , x k otherwise. 4: Set σ k+1 =    c 2 σ k , if ρ k > η 2 [very successful iteration], σ k , if η 1 ≤ ρ k ≤ η 2 [successful iteration], c 1 σ k , otherwise [unsuccessful iteration].(28) 5: Compute γ k+1 by (3), (9) or (10). Set γ k+1 = max{γ min , min{γ k+1 , γ max }}. 6: Update C k and Q k when k is a very successful or successful iteration. Now we analyze the convergence of this variant of the MARC algorithm. Some of the following analysis is similar to that of MARC method in Sect. 3. For the completeness of the paper, we give a sketch of the proof. k ≥ 0, we have f k+1 ≤ C k+1 ≤ C k .(29) Proof. Obviously, there is f (x k+1 ) ≤ C k . Since C k+1 is a convex combination of C k and f (x k+1 ), we have f k+1 ≤ C k+1 ≤ C k for each k. We note that the conclusions of Lemma 3.2 and Lemma 3.3 in Section 2 also hold in this case. Lemma 4.2. Let f ∈ C 1 . Suppose that ǫ > 0 and I is an infinite index set such that g k ≥ ǫ, k ∈ I, and g k σ k → 0, k → ∞, k ∈ I. Furthermore, if for some x * ∈ R n , x k → x * , k → ∞, k ∈ I, then each iteration k ∈ I that is sufficiently large is very successful, and σ k+1 ≤ σ k holds for all k ∈ I sufficiently large. Proof. Firstly, we have that f (x k + s k ) − m k (s k ) ≤ g(ξ k ) − g k + γ k 2 g k σ k · g k σ k , and f (x k ) − m k (s k ) ≥ g k 24 g k σ k , for all k ∈ I sufficiently large. Thus, by (15), we obtain that r k :=f (x k + s k ) − C k − η 2 [m k (s k ) − f (x k )] =f (x k + s k ) − m k (s k ) + (1 − η 2 )[m k (s k ) − f (x k )] + (f k − C k ) ≤ g k σ k g(ξ k ) − g k + γ max 2 g k σ k − (1 − η 2 )ǫ 24 + (f k − C k ),(30) for all k ∈ I sufficiently large. Since ξ k belongs to the line segment (x k , x k + s k ), we have ξ k − x * ≤ x k − x * + s k . In addition, s k → 0 and ξ k → x * , as k → ∞, k ∈ I and g is continuous, we conclude that g(ξ k ) − g k → 0 as k → ∞, k ∈ I. Thus, the limit of g k /σ k , (30) and (29) imply that r k < 0, for all k ∈ I sufficiently large. Recalling that r k < 0 ⇔ ρ k = f (x k ) − f (x k + s k ) f (x k ) − m k (s k ) > η 2 , we conclude that the k-th iteration is very successful, and hence σ k+1 ≤ σ k . Theorem 4.1. Let f ∈ C 1 . Suppose that {x k } is the sequence generated by Algorithm 2. If {f (x k )} is bounded below, then lim inf k→∞ g k = 0.(31) Proof. By way of contradiction, we assume that there is some ǫ > 0 such that g k ≥ ǫ(32) holds for all k ≥ 0. It follows from ρ k ≥ η 1 and Lemma 2 that f (x k+1 ) ≤ C k − η 1 ǫ 6 √ 2 min ǫ γ max , 1 2 g k σ k , k ∈ S. which, together with (32) and the definition of C k , implies C k − C k+1 =C k − η k Q k C k + f k+1 Q k+1 ≥C k − η k Q k C k + C k − η 1 ǫ 6 √ 2 min ǫ γmax , 1 2 g k σ k Q k+1 = η 1 ǫ 12Q k+1 min ǫ γ max , 1 2 g k σ k .(33) Since {f (x k )} is bounded below and f k ≤ C k , so {C k } is bounded below. Also, (33) implies that {C k } is monotonically decreasing, and hence it is convergent. Therefore the left-hand side of (33) converges to zero as k → ∞, thus, g k /σ k → 0, k → ∞. The rest analysis is similar to that of Theorem 3.1, so we omit it. Furthermore, if we assume that g is uniformly continuous, we have the strong convergence of the VMARC method. Proof. The proof is similar to that of Theorem 3.2, so we omit it. Numerical Experiments In this section, through testing a set of 54 unconstrained optimization problems from CUTEst library [17], we analysis the effectiveness of Algorithm VMARC and compare it with the TRMSM algorithm of Zhou et al. [31], the standard trust region method and adaptive regularization algorithm with cubics. All the codes were written in Fortran 77 with double precision and run on a personal desktop computer with 3.60 GHz CPU, 4.0 GB memory and 64-bit Ubuntu operation system. In all our tests, the parameters in Algorithm VMARC are as follows: σ 0 = 1, η 1 = 0.1, η 2 = 0.75, c 1 = 5, c 2 = 0.2, γ max = 10 6 , ψ k = 0.2 and η k = 0.7. We call the VMARC type method employing (3), (9), (10) as the γ k parameter the MARC1 method, MARC2 method and MARC3 method, respectively. The corresponding TRMSM type methods are the TRMSM1 method, TRMSM2 method and TRMSM3 method. Our experiments contains two parts. In the first part, we give some numerical comparisons between TRMSM3, MARC3, standard TR and ARC methods. During this work, the TRU and ARC packages (Fortran implementation of the standard TR and ARC methods) of GALAHAD [16] are used and we change the stopping criteria of MARC3 and TEMSM3 methods as TRU and ARC packages, that is, we stop the iteration when g k ≤ max{10 −5 , 10 −9 g 0 }. The detailed numerical results 1 are included in Table 1 with the the number of iterations (iter), the number of function-gradient evaluations (nf) and the cost time (cpu). The sign '-' denotes a failure of the algorithm. For each problem, some more information are also given in Table 1, r is the ratio of the cost time of computing the exact Hessians once and cost time of one function-gradient calculation, the 4th column Time gives the cost time of 20 functiongradient calculations. For different r, when the exact Hessian is used, the TRU and ARC method always need less iterations but more cost time than the TRMSM3 and MARC3 methods. Firstly, when r is very large, such as 3641 for problem VARDIM, calculating the exact Hessians even once could cost much computing resource. On the other hand, even if r is small, another inner iterative algorithm for solving trust region or cubic model subproblems is still needed and this step may cost much time. But for the TRMSM3 or MARC3 method, we only use the function value and the gradient information, and at each step, O(n) operations are needed to form γ k . Next, we mainly compare MARC with TRMSM methods when employing different γ k . We terminate the iteration when g k ∞ ≤ 10 −6 (1 + |f (x k )|). In addition, we also stop the algorithm if the number of iterations exceeds 5000. The detailed numerical results are given in Tables 2-3. Dolan and Moré's [14] performance profiles are used to display the behaviors of these methods. Figure 1 shows comparison of different methods based on cpu time. By using the same γ k parameter, the VMARC algorithm obviously outperform the TRMSM algorithm. Among these six methods, the MARC3 method has the best numerical performance, which can solve about 45% of the tested problems in the shortest time. Similar conclusions can be draw from Figure 2, which show the performance profiles based on iterations and function evaluations. Conclusion In this paper, combing with the Barzilai-Borwein step size, we present a modified adaptive cubic regularization method for large-scale unconstrained optimization problem. With nonmonotone technique, a variant of the modified method is also proposed. Convergence of the method is analyzed under some reasonable assumptions. Numerical experiments are performed to show the effectiveness of the new methods. Lemma 4 . 1 . 41Let {x k } be the sequence generated by Algorithm 2. Then for any Theorem 4 . 2 . 42Let f ∈ C 1 and its gradient g is uniformly continuous, {x k } is the sequence generated by Algorithm 2. If {f (x k )} is bounded below, then lim k→∞ g k = 0. Figure 1 : 1Performance profiles based on CPU time Figure 2 : 2Performance profiles based on iterations and function evaluations Table 1 : 1Numerical comparisons between ARC/TR and TRMSM/MARC methodsProblem Dim r Time ARC TRU TRMSM3 MARC3 iter cpu iter cpu iter ng cpu iter ng cpu POWER 5000 8328.6 2.67-03 26 27.497 33 32.747 55 125 0.023 92 170 0.032 VARDIM 2000 3641.1 1.59-03 27 6.873 719 182.221 52 157 0.015 5 36 0.003 BROWNAL 200 586.47 1.49-02 7 0.349 7 0.315 6 23 0.004 4 11 0.002 PENALTY1 1000 456.2 1.02-03 28 0.759 37 0.933 12 46 0.003 6 14 0.001 FMINSURF 1024 246.9 2.41-03 373 32.209 117 8.416 1172 1799 0.242 821 1548 0.202 SPARSQUR 5000 218.2 1.07-02 22 0.205 20 0.153 22 41 0.024 17 22 0.013 DQRTIC 2000 180.7 3.11-03 28 0.009 36 0.012 39 82 0.007 31 47 0.004 SROSENBR 5000 140.9 4.05-03 12 0.016 10 0.012 27 59 0.015 29 49 0.013 NONDIA 5000 102.1 5.61-03 29 0.033 24 0.03 4 25 0.008 5 14 0.004 WOODS 4000 87.8 4.23-03 56 0.062 67 0.077 195 329 0.089 453 860 0.224 DQDRTIC 5000 73.1 7.72-03 9 0.013 2 0.004 20 29 0.014 35 36 0.017 ARWHEAD 5000 70.5 8.26-03 10 0.015 6 0.009 13 29 0.013 14 21 0.01 ENGVAL1 5000 65.8 8.70-03 12 0.035 9 0.026 25 34 0.018 22 23 0.012 SINQUAD 5000 63.1 9.38-03 34 0.089 45 0.118 - - -305 591 0.428 CRAGGLVY 5000 40.8 1.44-02 17 0.102 15 0.082 144 246 0.234 138 263 0.252 DIXMAANJ 3000 36.7 5.79-03 50 0.273 76 0.28 683 1075 0.355 444 839 0.277 DIXMAANJ 9000 275.1 1.72-02 68 1.086 80 0.873 531 840 0.826 410 776 0.752 DIXMAAND 3000 35.7 5.91-03 16 0.082 17 0.056 10 17 0.005 11 12 0.004 DIXMAAND 9000 108.3 1.74-02 17 0.326 43 0.275 9 16 0.016 12 13 0.014 MOREBV 5000 31.2 1.90-02 2 0.286 1 0.312 134 213 0.083 129 247 0.091 COSINE 2000 24.2 3.69-03 42 0.053 - - 11 13 0.004 15 17 0.005 EDENSCH 2000 21.3 4.42-03 17 0.027 16 0.024 27 36 0.009 29 30 0.008 BROYDN7D 5000 20.2 2.92-02 767 13.791 679 6.553 2325 3563 6.938 2039 3688 7.159 EXTROSNB 5000 13.3 1.01-03 685 16.134 703 1.788 35 45 0.015 76 143 0.045 EG2 1000 6.9 1.96-03 7 0.003 3 0.001 5 16 0.002 6 11 0.001 SENSORS 1000 1.7 5.24-00 683 316.162 33 36.078 42 66 22.958 29 33 9.99 Table 2 : 2Numerical results of VMARC and TRMSM methodsProblem n TRMSM1 MARC1 TRMSM2 MARC2 TRMSM3 MARC3 iter nf cpu iter nf cpu iter nf cpu iter nf cpu iter nf cpu iter nf cpu ARWHEAD 10000 10 27 0.0250 10 18 0.0170 10 27 0.0264 9 17 0.0161 13 30 0.0285 12 20 0.0197 BDQRTIC 2000 3832 5964 1.3863 2469 4801 1.0979 3366 5242 1.1989 2188 4213 0.9635 1007 1584 0.3808 884 1691 0.4010 BOX 10000 3015 4689 6.1013 2009 3891 4.9861 2770 4305 5.5816 1793 3462 4.4329 698 1085 1.4385 668 1278 1.6854 BROWNAL 400 8 27 0.0200 10 17 0.0129 8 27 0.0198 10 17 0.0129 10 29 0.0212 12 19 0.0146 BROYDN7D 5000 2448 3752 7.1970 2178 3961 7.5734 2152 3272 6.2765 2195 3949 7.5585 2252 3455 6.6699 2007 3656 7.0330 BRYBND The complete data can be downloaded from https://pan.baidu.com/s/1CWF2WoQSdSksIbwi-4AXsg. 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In this paper we provide a lower bound for the long time on-diagonal heat kernel of minimal submanifolds in a Cartan-hadamard ambient manifold assuming that the submanifold is of polynomial volume growth. In particular cases, that lower bound is related with the number of ends of the submanifold.

1310.4643

151f1b9b0977778b196052194b8b612cea399d9f

LARGE TIME BEHAVIOR OF THE ON-DIAGONAL HEAT KERNEL FOR MINIMAL SUBMANIFOLDS WITH POLYNOMIAL VOLUME GROWTH Vicent Gimeno LARGE TIME BEHAVIOR OF THE ON-DIAGONAL HEAT KERNEL FOR MINIMAL SUBMANIFOLDS WITH POLYNOMIAL VOLUME GROWTH In this paper we provide a lower bound for the long time on-diagonal heat kernel of minimal submanifolds in a Cartan-hadamard ambient manifold assuming that the submanifold is of polynomial volume growth. In particular cases, that lower bound is related with the number of ends of the submanifold. INTRODUCTION Let M m be a m-dimensional minimally immersed submanifold into a simply connected ambient manifold N n with sectional curvatures K N bounded from above by K N ≤ 0. S. Markvorsen proved in [Mar86] -following [CLY84]-that the heat kernel H of M m is bounded from above by the heat kernel H m,0 of the Euclidean space R m , namely: (1.1) H(t, x, y) ≤ H m,0 (t, r x (y)) = 1 (4πt) This paper deals with lower bounds to the on-diagonal heat kernel assuming certain restriction on the volume growth. In order to define that appropriate behavior on the growth of the extrinsic volume, recall that given a minimal submanifold M m properly immersed in a Cartan-Hadamard manifold N with sectional curvatures K N bounded from above by K N ≤ 0 and denoting by ω m the volume of a radius one geodesic ball in R m and by B N R (p) the geodesic ball in N of radius R centered at p, by the monotonicity formula (see for instance [MP12,theorem 2.6.9] and [Pal99]) for any point p ∈ M m the function (1.3) Q(R) = Vol(M m ∩ B N R (p)) ω m R m , is a non decreasing function. Throughout this paper a complete minimal submanifold properly immersed in a Cartan-hadamard ambient manifold is called a minimal submanifold of polynomial volume growth if there exists a constant E depending on M m such that: (1.4) lim R→∞ Q(R) ≤ E < ∞. Under such volume growth behavior we can state the behavior of the long time asymptotic for the on-diagonal heat kernel by the main theorem of this paper. The main theorem makes use of the following constant C m depending only on the dimension m of the submanifold (1.5) C m := Γ m 2 , 2 m 2 Γ m 2 2 m Γ( m 2 ) , Work partially supported by DGI grant MTM2010-21206-C02-02. where Γ(z) and Γ(z 1 , z 2 ) in the above expression denote the gamma function and the incomplete gamma function respectively, i.e, Γ(z) := ∞ 0 t z−1 e −t dt. Γ(z 1 , z 2 ) := ∞ z2 t z1−1 e −t dt. For minimal submanifolds with an extrinsic volume growth controlled by the above constant C m we can state the main result of this paper: Main Theorem. Let M m be a complete m-dimensional submanifold properly immersed in a simply connected ambient manifold N with sectional curvatures K N bounded from above by K N ≤ 0. Suppose that M m is of polynomial volume growth, and that (1.6) E < 1 C m , Then, the heat kernel H of M m satisfies (1.7) (1 − EC m ) 2 E ≤ lim sup t→∞ (4πt) m 2 H(t, x, x) ≤ 1. It is not hard to find examples of complete minimal submanifolds properly and minimally immersed in a Cartan-Hadamard ambient manifold with polynomial volume growth. Indeed, for a complete minimal surface embedded in R 3 , by a well known result (see [Oss86,JM83] and introduction in [GP13]), if the surface has finite total curvature then the surface has polynomial volume growth (quadratic area growth) and the constant E given in equation (1.4) is equal to the number of ends of the surface. This is the case of the catenoid or the Costa surface (with E = 2 for the catenoid and E = 3 for the Costa surface). It is also known that there exist other surfaces with quadratic area growth but without finite total curvature and even without finite topological type. An example of that kind of surface is the Scherk singly periodic surface (see introduction in [MW07]) which has E = 2. Since C 2 ∼ 0.14 1 C 2 ∼ 7.39, we can apply the main theorem to the catenoid, the Costa and the Scherk surface, obtaining (1 − 0.28) 2 2 ≤ lim sup t→∞ (4πt) H(t, x, x) ≤ 1, for the catenoid and the Scherk singly periodic surface, and (1 − 0.41) 2 3 ≤ lim sup t→∞ (4πt) H(t, x, x) ≤ 1, for the Costa surface. As we have shown, there are several examples where the volume growth is related with the number of ends of the submanifold. In fact, the following theorem allow us to achieve inequality (1.4) under certain decay of the norm of the second fundamental form and to give a topological meaning to lim R→∞ Q(R) Theorem 1.1 (see theorem 2.2 of [Qin95] and [GP12]). Let M m be an m−dimensional complete immersed minimal submanifold in R n which satisfies (1.8) lim R→∞ sup x∈M m r(x)≥R r(x) A (x) = 0, where A denotes the second fundamental form. Then, the number of ends E (M m ) of M m is given by: (1.9) lim R→∞ Q(R) = E(M m ) provided either of the following two conditions is satisfied: (1) m = 2, n = 3 and each end of M m is embedded. (2) m ≥ 3. Hence, we can state the following corollary showing the relation between the number of ends and the lower bound for the heat kernel under the assumptions of the above theorem (see introduction of [GSC09] for a complete overview on the two sides estimates for the heat kernel on manifolds with ends): Corollary 1.2. Let M m be an m−dimensional complete immersed minimal submanifold in R n which satisfies(1.11) E(M m ) < 1 C m , the heat kernel H of M m satisfies (1.12) (1 − E(M m )C m ) 2 E(M m ) ≤ lim sup t→∞ (4πt) m 2 H(t, x, x) ≤ 1. If M 2 is a minimal surface in R 3 , by the Gauss formula the second fundamental form is related with the Gaussian curvature K G of M 2 by (1.13) K G = − 1 2 |A| 2 , in view of [MPR13, theorem 1.2] it seems that in the particular case of complete embedded minimal surfaces in R 3 if there exists a constant C such that |K G |R 2 ≤ C, then: |K G |R 2 ≤ C → M 2 |K G | < ∞ → lim R→∞ sup x∈M m r(x)≥R r(x)|A|(x) = 0. Hence, the condition given in equation (1.10) in the above corollary can be replaced in the particular case of complete embedded minimal surfaces in R 3 by |K G |R 2 ≤ C. Recall also that a particular case when equality (1.10) holds is (see [Qin95]) when M m |A| m dV < ∞ i.e,. when the submanifold has finite scalar curvature (see also [And84]). Let us finally remark that Remark a. Given a manifold M n with non-negative Ricci curvature Rc > 0, Bishop-Gromov volume comparison theorem asserts that for any o ∈ M n the relative volume quotient Vol(B M n R (o)) ω n R n is non-increasing in the radius R (being B M n R (o) the geodesic ball of radius R centered at o). The relative volume quotient converges to a non-negative number Θ: lim R→∞ Vol(B M n R (o)) ω n R n = Θ ≥ 0. If Θ > 0, one says that the manifold M n has maximal volume growth. P. Li proved in [Li86] (see also [Xu13]) that if M n has Rc > 0 and maximal volume growth, then (1.14) lim t→∞ Vol B M n √ t (y) H (t, x, y) = ω n (4π) − n 2 . Therefore (1.15) lim t→∞ (4πt) n 2 H(t, x, y) = 1 Θ . In some sense, our main theorem can be understood (partially) as a reverse of the Li's theorem because at least on dimension 2, by the Gauss formula (equation (1.13)), a submanifold properly and minimally immersed in a Cartan-Hadamard ambient manifold has non-positive sectional curvature (instead of Rc > 0) and because, by the monotonicity formula, the extrinsic quotient given in equation (1.3) is non-decreasing (instead of nonincreasing like the relative volume quotient). Despite of the weakness of the inequalities (1.7) in comparison to equality (1.15) observe, however, that a non-negatively Ricci-curved manifold with maximal volume growth must have finite fundamental group (see [Li86]) but that is not true for minimal submanifolds of a Cartan-Hadamard with polynomial volume growth (see for instance the singly periodic Scherk surface (figure 1)). The most well known examples of heat kernels of minimal submanifolds M m in the Euclidean space R n are when M m is a totally geodesic submanifold R m in R n . Observe that in that case E = 1 if C m were 0 the inequality (1.7) would be an exact equality. Therefore, it is a natural question to ask the following open question Open question. Is it possible to improve the main theorem changing C m by 0? The structure of the paper is as follows In §2 we recall the definition and several properties of the heat kernel on a Riemannian manifold and provide proposition 2.1 which states that every complete minimal submanifold with polynomial volume growth is stochastically complete. With those previous requirements we can, in §3, to prove the main theorem. In other words, the Cauchy problem with Dirichlet boundary conditions (2.2)    ∂v ∂t = ∆v , v| t=0 = v 0 (x) , has a solution (2.3) v(x, t) = M H(t, x, y)v 0 (y)dµ y , provided that v 0 is a bounded continuous positive function. Moreover the heat kernel has the following properties: (1) Symmetry in x, y that is H(t, x, y) = H(t, y, x). (2) The semigroup identity: for any s ∈ (0, t) (2.4) H(t, x, y) = M H(s, x, z)H(t − s, z, y)dV(z). (3) For all t > 0 and x ∈ M , (2.5) M H(t, x, y)dV(y) ≤ 1. If M is the Euclidean space R n then, due to the homogeneity and isotropy of the Euclidean space, the heat kernel H n,0 (t, x, y) depends only on t and ρ(x, y) = dist(x, y), and is given by the classical formula (2.6) H n,0 (t, ρ(x, y)) = 1 (4πt) n 2 e − ρ 2 (x,y) 4t . A manifold M satisfying for all x ∈ M and all t > 0 (2.7) M H(t, x, y)dV(y) = 1, is said to be stochastically complete. In the following proposition is proved that a complete submanifold of polynomial volume growth is stochastically complete Hence, by [Gri99,theorem 9.1] M m is stochastically complete. Finally in order to conclude this preliminary section let us recall here the coarea formula Theorem 2.2 (Coarea formula, see [Sak96,Cha84]). Let f be a proper C ∞ function defined on a Riemannian manifold (M n , g). Now we set (2.10) Ω t := {p ∈ M ; f (p) < t} , V t := Vol(Ω t ), Γ t := {p ∈ M ; f (p) = t} , A t := Vol n−1 (Γ t ). Then for an integrable function u on M n the following hold: (1) Let g t be the induced metric on Γ t from g. Then (2.11) M n u|∇f |dν g = ∞ −∞ dt Γt udν gt . (2) t → V t is of class C ∞ at a regular value t of f such that V t < +∞, and (2.12) d dt V t = Γt 1 |∇f | dν gt . PROOF OF THE MAIN THEOREM First of all, let us denote by D R (x) the extrinsic ball of radius R cantered at x, i.e., D R (x) := M m ∩ B N R (x), therefore Q(R) is given by Q(R) = Vol(D R (x)) ω m R m . Note that D R (x) is the sublevel set of the extrinsic distance function r x : H(t/2, x, y) 2 dV(y), for any extrinsic ball D R (x). Applying now the Cauchy-Schwarz inequality (3.1) D R (x) = {p ∈ M m ; r x (p) < R} .(3.3) 1 ≥ (4πt) m 2 H(t, x, x) ≥ (4πt) m 2 D R (x) H(t/2, x, y)dV(y) 2 Vol(D R (x)) , Since by proposition 2.1 M m is stochastically complete (3.4) 1 ≥ (4πt) m 2 H(t, x, x) ≥ (4πt) m 2 1 − M m \D R (x) H(t/2, x, y)dV(y) 2 Vol(D R (x)) , Applying the polynomial volume growth property (3.5) 1 ≥ (4πt) m 2 H(t, x, x) ≥ (4πt) m 2 1 − M m \D R (x) H(t/2, x, y)dV(y) 2 Eω m R m , for all R > 0. If we choose (3.6) R = R t := (4π) 1 2 ω 1 m m t 1 2 = 2 m 2 Γ m 2 1 m t 1 2 , we obtain (3.7) 1 ≥ (4πt) m 2 H(t, x, x) ≥ 1 − M m \D R t (x) H(t/2, x, y)dV(y) 2 E , We need now the following proposition Proposition 3.1. Suppose that lim R→∞ Q(R) = E then (3.8) M m \D R t (x) H(t/2, x, y)dV(y) ≤ E (C m + δ(t)) , being δ a smooth function with δ → 0 when t → ∞. Proof. By inequality (1.1) (3.9) M m \D R t (x) H(t/2, x, y)dV(y) ≤ M m \D R t (x) H m,0 (t/2, r x (y))dV(y) by coarea formula (theorem 2.2) (3.10) M m \D R t (x) H(t/2, x, y)dV(y) ≤ ∞ Rt ∂D S (x) H m,0 (t/2, r x (y)) |∇r x | dV s (y)ds . Taking into account the definition of R t (equation (3.6)) and that ω m = 2π Making use that Q(s) = E when s → ∞ the proposition is proven. Hence for t large enough we can apply the above proposition in equation (3.7) (3.14) 1 ≥ (4πt) Therefore, taking limits the theorem follows. r x (y) the distance in N from x to y. In particular for the on-diagonal heat kernel H(t, x, x) of M m one can state that FIGURE 1 . 1The catenoid, the Costa surface and the Scherk singly periodic surface are examples of minimal surfaces immersed in R 3 with polynomial volume growth which is equivalent to quadratic area growth when the submanifold is a surface. if m = 2 and n = 3, each end of M m is embedded. Or, (2) m ≥ 3. Then, if the number of ends E(M m ) of M m is bounded from above by be a Riemannian manifold with (possibly empty) smooth boundary ∂M , and denote by ∆ the Laplacian on M . The heat kernel on M is a function H(t, x, y) on (0, ∞) × M × M which is the minimal positive fundamental solution to the heat equation (2.1) ∂v ∂t = ∆v . Proposition 2 . 1 . 21Let M m be a m-dimensional complete minimal submanifold properly immersed in a Cartan-Hadamard ambient manifold. Suppose that M m is of polynomial volume growth, then M m is stochastically complete Proof. Since M m has polynomial volume growth by equation (1.4), for any o ∈ M and any R ∈ R + we have (2.8) Vol(M m ∩ B N R (o)) ≤ Eω m R m . But since the geodesic ball B M m R (o) of radius R in M m is a subset of the extrinsic ball M m ∩ B N R (o), Making use of the upper bounds for the heat kernel (equation 1.2) and the semigroup property of the heat kernel (equation 2. s) (Vol(D s (x)) ds. The derivative d dR Vol(D R (o)) = (Vol(D R )) satisfies (3.11) (Vol(D R )) = mω m Q(R)R m−1 + ω m R m Q(R) (log(Q(R)) . m 2 H 2(t, x, x) ≥ [1 − E (C m + δ(t))] 2 E. The compactification of a minimal submanifold in euclidean space by the gauss map. Michael T Anderson, unpublished preprintMichael T. Anderson, The compactification of a minimal submanifold in euclidean space by the gauss map, unpublished preprint, 1984. Eigenvalues in Riemannian geometry. Isaac Chavel, Pure and Applied Mathematics. Burton Randol115Academic Press IncWith an appendix by Jozef Dodziuk. MR 768584 (86g:58140Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Aca- demic Press Inc., Orlando, FL, 1984, Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk. MR 768584 (86g:58140) Heat equations on minimal submanifolds and their applications. Peter Shiu Yuen Cheng, Shing-Tung Li, Yau, 1033-1065. MR 761578Amer. J. Math. 106558171Shiu Yuen Cheng, Peter Li, and Shing-Tung Yau, Heat equations on minimal submanifolds and their applications, Amer. J. Math. 106 (1984), no. 5, 1033-1065. MR 761578 (85m:58171) Volume growth, number of ends, and the topology of a complete submanifold. Vicent Gimeno, Vicente Palmer, Journal of Geometric Analysis. EnglishVicent Gimeno and Vicente Palmer, Volume growth, number of ends, and the topology of a complete submanifold, Journal of Geometric Analysis (2012), 1-22 (English). Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces. MR 3047082Israel J. Math. 1942, Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces, Israel J. Math. 194 (2013), no. 2, 539-553. MR 3047082 Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Alexander Grigor Yan, Bull. Amer. Math. Soc. (N.S.). 362MR 1659871 (99k:58195Alexander Grigor yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135-249. MR 1659871 (99k:58195) Alexander Grigor, &apos; Yan, Laurent Saloff-Coste, Heat kernel on manifolds with ends. 5935499Alexander Grigor'yan and Laurent Saloff-Coste, Heat kernel on manifolds with ends, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 5, 1917-1997. MR 2573194 (2011d:35499) The topology of complete minimal surfaces of finite total Gaussian curvature. P Luquésio, William H Jorge, Meeks, Iii, Topology. 22253006MRLuquésio P. Jorge and William H. Meeks, III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology 22 (1983), no. 2, 203-221. MR 683761 (84d:53006) Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Peter Li, Ann. of Math. 847950258259MRPeter Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2) 124 (1986), no. 1, 1-21. MR 847950 (87k:58259) On the heat kernel comparison theorems for minimal submanifolds. Steen Markvorsen, Proc. Amer. Math. Soc. 97358160MRSteen Markvorsen, On the heat kernel comparison theorems for minimal submanifolds, Proc. Amer. Math. Soc. 97 (1986), no. 3, 479-482. MR 840633 (87i:58160) A survey on classical minimal surface theory. H William, Joaquín Meeks, Pérez, MR 3012474University Lecture Series. 60American Mathematical SocietyWilliam H. Meeks, III and Joaquín Pérez, A survey on classical minimal surface theory, University Lecture Series, vol. 60, American Mathematical Society, Providence, RI, 2012. MR 3012474 Local removable singularity theorems for minimal laminations. H William, Meeks, Joaquín Iii, Antonio Pérez, Ros, arXiv:1308.6439William H. Meeks, III, Joaquín Pérez, and Antonio Ros, Local removable singularity theorems for minimal laminations, 2013, arXiv:1308.6439. Minimal surfaces with the area growth of two planes: the case of infinite symmetry. H William, Michael Meeks, Wolf, 441-465. MR 2276776J. Amer. Math. Soc. 20253008William H. Meeks, III and Michael Wolf, Minimal surfaces with the area growth of two planes: the case of infinite symmetry, J. Amer. Math. Soc. 20 (2007), no. 2, 441-465. MR 2276776 (2007m:53008) A survey of minimal surfaces. Robert Osserman, MR 852409. New YorkDover Publications Inc8753012second ed.Robert Osserman, A survey of minimal surfaces, second ed., Dover Publications Inc., New York, 1986. MR 852409 (87j:53012) Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications. Vicente Palmer, MR 1724821J. London Math. Soc. 253050Vicente Palmer, Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their appli- cations, J. London Math. Soc. (2) 60 (1999), no. 2, 607-616. MR 1724821 (2000j:53050) On the volume growth and the topology of complete minimal submanifolds of a euclidean space. Chen Qing, J. Math. Sci. Univ. Tokyo. 2Chen Qing, On the volume growth and the topology of complete minimal submanifolds of a euclidean space, J. Math. Sci. Univ. Tokyo 2 (1995), 657-669. . Takashi Sakai, Riemannian Geometry, Translations of Mathematical Monographs. 149American Mathematical SocietyTranslated from the 1992 Japanese original by the author. MR 1390760 (97f:53001Takashi Sakai, Riemannian geometry, Translations of Mathematical Monographs, vol. 149, American Mathematical Society, Providence, RI, 1996, Translated from the 1992 Japanese original by the author. MR 1390760 (97f:53001) Guoyi Xu, ; Department, Universitat Mathematics-Init, Castelló Jaume I, De La, Plana, arXiv:1310.2382SPAIN E-mail address: [email protected] time behavior of the heat kernel. Guoyi Xu, Large time behavior of the heat kernel, 2013, arXiv:1310.2382. DEPARTMENT OF MATHEMATICS-INIT, UNIVERSITAT JAUME I, CASTELLÓ DE LA PLANA, SPAIN E-mail address: [email protected]

In the era of big data, logistic planning can be made data-driven to take advantage of accumulated knowledge in the past. While in the movie industry, movie planning can also exploit the existing online movie knowledge library to achieve better results. However, it is ineffective to solely rely on conventional heuristics for movie planning, due to a large number of existing movies and various real-world factors that contribute to the success of each movie, such as the movie genre, available budget, production team (involving actor, actress, director, and writer), etc. In this paper, we study a "Blockbuster Planning" (BP) problem to learn from previous movies and plan for low budget yet high return new movies in a totally data-driven fashion. After a thorough investigation of an online movie knowledge library, a novel movie planning framework "Blockbuster Planning with Maximized Movie Configuration Acquaintance" (BigMovie) is introduced in this paper. From the investment perspective, BigMovie maximizes the estimated gross of the planned movies with a given budget. It is able to accurately estimate the movie gross with a 0.26 mean absolute percentage error (and 0.16 for budget). Meanwhile, from the production team's perspective, BigMovie is able to formulate an optimized team with people/movie genres that team members are acquainted with. Historical collaboration records are utilized to estimate acquaintance scores of movie configuration factors via an acquaintance tensor. We formulate the BP problem as a non-linear binary programming problem and prove its NPhardness. To solve it in polynomial time, BigMovie relaxes the hard binary constraints and addresses the BP problem as a cubic programming problem. Extensive experiments conducted on IMDB movie database demonstrate the capability of BigMovie for an effective data-driven blockbuster planning.

10.1109/bigdata.2018.8622316

1810.10175

cb9f3eefdf5038a3058ceba597c4b5bd74840fdd

Data-driven Blockbuster Planning on Online Movie Knowledge Library Ye Liu Jiawei Zhang [email protected] Department of Computer Science Florida State University FLUSA Chenwei Zhang [email protected] Philip S Yu [email protected] Institute for Data Science Tsinghua University BeijingChina Department of Computer Science University of Illinois at Chicago ILUSA Data-driven Blockbuster Planning on Online Movie Knowledge Library Index Terms-Knowledge Base DiscoveryBlockbuster Con- figuration PlanningData-driven Application In the era of big data, logistic planning can be made data-driven to take advantage of accumulated knowledge in the past. While in the movie industry, movie planning can also exploit the existing online movie knowledge library to achieve better results. However, it is ineffective to solely rely on conventional heuristics for movie planning, due to a large number of existing movies and various real-world factors that contribute to the success of each movie, such as the movie genre, available budget, production team (involving actor, actress, director, and writer), etc. In this paper, we study a "Blockbuster Planning" (BP) problem to learn from previous movies and plan for low budget yet high return new movies in a totally data-driven fashion. After a thorough investigation of an online movie knowledge library, a novel movie planning framework "Blockbuster Planning with Maximized Movie Configuration Acquaintance" (BigMovie) is introduced in this paper. From the investment perspective, BigMovie maximizes the estimated gross of the planned movies with a given budget. It is able to accurately estimate the movie gross with a 0.26 mean absolute percentage error (and 0.16 for budget). Meanwhile, from the production team's perspective, BigMovie is able to formulate an optimized team with people/movie genres that team members are acquainted with. Historical collaboration records are utilized to estimate acquaintance scores of movie configuration factors via an acquaintance tensor. We formulate the BP problem as a non-linear binary programming problem and prove its NPhardness. To solve it in polynomial time, BigMovie relaxes the hard binary constraints and addresses the BP problem as a cubic programming problem. Extensive experiments conducted on IMDB movie database demonstrate the capability of BigMovie for an effective data-driven blockbuster planning. I. INTRODUCTION The movie industry attracts great interests from both movie investors and the public audience because of its high profits and entertainment nature. Attracted by the huge market, lots of investors are inquiring about identifying high-gross movies to invest in. Besides recognizing profitable movies, it is rewarding to provide a reasonable and promising planning for a new movie at its developmental stage, which has been greatly ignored in previous works due to the complexity of various factors, including the movie genre and production team (actor, actress, writer and director). The booming movie industry has accumulated thousands of previous movies as well as their gross statistics, which may serve as a movie knowledge library to help achieve better results for future movie planning. Therefore it is no longer efficient to rely on conventional heuristics for comprehensive movie planning [1]. Data-driven movie planning methods are in great need to exploit the accumulated knowledge to support the decision-making process when planning for a new movie. The data-driven planning has shown a huge success on the well-known TV series "House of Cards", produced by Netflix, using the data collected from viewer 1 . Generally, popular movie genres and renowned movie stars are the favorable choices during the planning so as to maximize the gross. But remuneration of the movie stars and movie's available budget also need to be considered in the movie planning. Meanwhile, a seamless collaboration among team members is the premise of high gross. For example, it will always be easier for directors to continue working on a new movie with the movie genre and production team members that they are acquainted with. And the old acquaintances can always have a tacit understanding and easy to arouse spark when they cooperate in their new movies. Problem Studied: In this paper, a research problem, namely the "Blockbuster Planning" (BP) problem, is introduced. Given an online movie knowledge library which consists of the existing movie information, we plan the movie configuration including genre and production team for a new movie under a pre-specified budget. We note that although there are occasions where a low budget production with unknown stars becomes a hit, we focus on the common cases involving known persons with available data. The objective of an optimal planning is to achieve: (1) the maximized gross, and (2) the optimized acquaintance among the movie configuration factors. The BP problem studied in this paper is a novel research problem, and few existing methods can be applied to solve it. The BP problem significantly differs from related works, such as (1) movie gross prediction [2], (2) viral marketing [3], [4], (3) team formation [5], [6]. (1) The movie gross prediction problem [7] studied in existing works merely focuses on inferring the movie gross while the BP problem aims at providing the optimal planning of various movie factors which can lead to the optimal gross for investors. (2) BP and the viral marketing problems [4] are both planning problems that aimed at maximizing certain target objectives, but they are solving totally different problems in distinct scenarios: a) viral marketing problems are usually studied in online social networks based on certain information diffusion models, while the BP problem is studied in the online movie knowledge libraries instead; b) viral marketing problems aim at maximizing the infected users, while BP's objective is to maximize the movie gross; c) instead of selecting the optimal users in viral marketing problems, the BP problem aims at planning for an optimal movie factor configurations. Recently, a variation of the LT model named PNP [8] is proposed for the movie design problem. The objective of PNP is very similar to our work except that PNP aims to attract most of the target users but our model aims to achieve the maximum gross under the given budget. (3) Different from conventional team formation problems [5], where team members are planned for the entrepreneurial team project base on satisfying skill qualification and minimizing the communication cost of the team members, our method also aims to maximize movie gross. The BP problem is challenging to solve due to: • Unknown Movie Success Factors: What are the contributing factors in the success of a movie? Few research works have ever been studied this problem, and relevant movie factors are still unknown. • Movie Gross/Budget Function: How much gross (budget) can a movie make (require), given a configuration of the movie success factors? A proper estimation of the movie gross and budget will be required for studying the BP problem. • Movie Configuration Acquaintance Function: How to compute the acquaintance scores among the movie configuration factors? A function that can measure acquaintance properly is needed in defining the BP problem. • NP Hardness: Based on our analysis, we demonstrate that the BP problem is actually an NP-hard problem, and no solution exists that can solve it in polynomial time if P = NP. To solve the aforementioned challenges, a new movie planning framework "Blockbuster Planning with Maximized Movie Configuration Acquaintance" (BigMovie) is proposed in this paper. With a thorough analysis of an online movie knowledge library dataset, IMDB, a set of factors affecting movie success are identified. The effectiveness of these extracted factors are validated in Section IV. The acquaintance scores of the movie configuration factors can be calculated based on an acquaintance tensor constructed with the historical collaboration records which is discussed in great detail in Section V. The BP problem is formulated as a constrained optimization problem with hard binary constraints, which aims at maximizing the inferred gross function as well as the acquaintance measure. We further demonstrate that BP is at least as difficult as the Knapsack problem and the Maximal Clique problem, which renders the BP problem to be NP-hard as well. By relaxing the hard constraints, we introduce an approximation solution to resolve the problem in polynomial time. For the experimental result, we can see BigMovie outperforms the competitors. In addition, at the end of the paper, the case study is provided, which demonstrate that by using BigMovie, a lucrative movie planning can be achieved. II. PROBLEM FORMULATION In this section, we will first define several important concepts used in this paper, and then provide the formulation of the BP problem. A. Notation At the beginning of this section, we will first define some notations used in this paper. Throughout this paper, we will use lower case letters (e.g., x) to denote scalars, lower case bold letters (e.g., x) to denote column vectors, upper case letters (e.g., X) to denote elements of matrices, upper case calligraphic letters (e.g., X ) to denote sets, and bold-face upper case letters (e.g., X) to denote matrix and high-order tensors. T is used to represent the transpose of a vector (e.g., x T ). || · || 1 is the 1 -norm of vector (e.g., ||x|| 1 ). .., c l } is the set of l production team members. The node set C can be divided into C t ∪ C s ∪ C w ∪ C d , which denote the set of actors, actresses, writers and directors, respectively. Link E represents the relationship between movie production team and movies. For instance, link ((c i , m j ) ∈ E) indicates participation of a production team member c i in a movie m j . And set A denotes the attribute of node set M. For the movie m i , the relative attribute is B. Terminology Definition A (mi) = A g (mi) ∪ {a b (mi) , a g (mi) } , where A g (mi) is the genre list of movie m i , a b (mi) is the budget of movie m i and a g (mi) is the gross of movie m i . Definition 2. Movie Configuration: Each movie m i ∈ M in the online knowledge library will have an unique configuration , covering movie production team (involving actor, actress, writer and director), movie genre, etc, which can be represented as vector x (mi) =[x t (mi) , x s (mi) , x d (mi) , x w (mi) , x g (mi) ] ∈ R 1×N , where x t (mi) represents the list of all actor, x s (mi) is the list of all actress, x w (mi) represents the list of all writer, x d (mi) represents the list of all director and x g (mi) represents the list of all genre of a movie m i . N is the sum of length of those lists. We will provide detailed representations in Section 4.1. Besides those factors covered in the above movie configuration definition, various other relevant factors (e.g., movie language, production country, etc.) can also be effectively incorporated with a simple extension to the definition, which will not be studied in this paper. Definition 3. Movie Configuration Acquaintance: Given two movie team members c i , c j ∈ C, c i = c j and a movie genre g k ∈ A g (mi) , their acquaintance can be represented as Acquaintance(c i , c j , g k ), denoting their historical collaboration frequency. For instance, if crews c i and c j participate t times in g k genre movie, Acquaintance( c i , c j , g k ) = t. C. Problem Formulation Definition 4. Blockbuster Planning Problem: Given a fixed budget B, the objective of BP is to plan a movie configuration x that achieves maximum movie gross and maximum movie configuration acquaintance simultaneously, subject to the budget B. Let Budget(x) denotes the cost by using movie configuration x, Gross(x) estimates the gross earned by using x and Acquaintance(x) measures the acquaintance of movie configuration. Formally, the BP problem aims at inferring the optimal movie configuration x * which can maximize the following objective function x * = arg max x α · Gross(x) + β · Acquaintance(x), (1) s.t. Budget(x) ≤ B In above equation, the concrete representation of function Budget(x) and Gross(x) will be provided in Section IV and function Acquaintance(x) will be provided in Section V-A. α is the coefficient of Gross(x) and β is the coefficient of function Acquaintance(x). Analysis of parameters α and β will be provided in the Section V-E. III. ONLINE MOVIE KNOWLEDGE LIBRARY STATISTICAL ANALYSIS Before introducing the method to solve the blockbuster planning problem, in this section, we first study the IMDB 2 datasets to provide some statistical analysis about the factors affecting movie gross. The analysis of the IMDB movies focuses on several important aspects like the gross, budget, genres and production team information (Actor, Actress, Director, Writer), which provides fundamental insights for the blockbuster planning framework. Among the crawled IMDB movies, only 3, 156 movies contain the gross and budget information, and they belong to 24 genres and cover 72, 786 actors, 38, 951 actresses, 4, 576 writers and 1, 682 directors. A. General Movie Information Statistics In this section, we study general information, like budget and genre regarding the movie gross. The results are shown in Figure 1. In this figure, we provide the information distribution of IMDB datasets in terms of their production years. In the plot, each circle denotes a movie, whose x axis and y axis denote the movie gross and the production year, respectively. Meanwhile, the circle diameter represents the budget of the movies (larger circle corresponding to movies with bigger budgets). Additionally, we use different colors, shown in the color bar below the figure, to represent the corresponding genre of each movie. According to Figure 1, we observe that the number of movies produced in recent years are increasing. For instance, according to our dataset, the number of movies produced in years 1980,1990,2000,2010, and 2015 are 12, 58, 77, 137 and 158 respectively. Besides the movie numbers, we also Recent Movies Have Higher Budget And Gross: According to the movie budget data, a majority of the movies produced before 2000 have budgets under $200 million while recent movies have relatively higher budgets (i.e., the circles in recent years are much larger). Among the top ten movies receiving the highest budget, six of them were produced within the past five years. Simultaneously, the movie gross of the past ten years is much higher than before (i.e., dot in recent years are much higher). Few of movies produced before 2000 had a gross of more than $250 million while some movies produced after 2000 reached more than $750 million on gross, which shows the growth of the movie industry. Among top ten highest gross movies, five of them were produced within the past five years, for example movie "Jurassic World" ($652 million), "The Avengers" ($623 million) and "The Hunger Games: Catching Fire" ($424 million). The movie "Avatar" (produced in 2009) achieves the highest gross in our dataset, which is $760 million. Additionally, the growth of the movie industry brings a big gross discrepancy from the recent movies, because the gross variance between movies produced before 2000 is smaller than movies produced after 2000. Movie Genre Distribution And Performance: Generally, each movie can belong to more than three movie genres. For all movies, the top three movie genres on most movies include "Drama", "Adventure" and "Fantasy". Movies belonging to any of those three genres are more than 91% of the total movies. In order to further analyze the overall genre preference of audiences, we show the violin plot on gross of all movies in Figure 2. In this figure, the horizontal bar in each box denotes the median gross of each genre, vertical bar denotes the range of gross in each genre, and the width of the violin shows the quantity of the movie in the same gross. By comparing the positions of the horizontal bar of each movie genre, we observe that the median movie gross fluctuates widely on different genres. For instance, the median gross of the "Animation" and "Adventure" genres are $85 million and $68 million respectively, but those of the "Film-Noir" and "News" genres only have $89k and $95k. Additionally, the box height of some movie genres, like "News" and "Short", are relatively short compared to the remaining movie genres. By studying the data, we observe that these movie genres are of a relatively small minority, and less than ten movies in total belong to these genres according to our IMDB dataset. B. Movie Production Team Statistics After analyzing the common movie information, we believe that production team information which influence movie gross are more important than those common movie information. For example, it's more likely that an audience watch a movie due to his/her favorite actress or actor participation. Therefore, in this section, we will analyze some latent movie information such as the movie production team information, which are actor, actress, writer and director. Moreover, we will discuss the movie configuration acquaintance and why it's important to consider it when planning the blockbuster. 1) Production Team and Movie Gross: We show the stacked bar plot of the top ten movie production team members whose movies have the highest accumulative gross. In each stacked bar, the different color represents different movies. The height of the bar represents gross of the given movie, and the higher the bar is, the higher the gross is. Movie Gross vs. Actor: Frank Welker's movies have the highest gross according to Figure 3(a); He participated in 66 movies based on our dataset and most of them, he acts as voice actor. His movies earned a total of $6, 579.99 million with an average of $99.7 million. Among the top ten gross maker actors, Stan Lee is the actor who has the highest average gross of $217.8 million and he is second highest grossing actor. From Figure 3(a), we can know that most of the those actors act in an average of more than 30 movies, which means that the famous actors are very popular. Movie Gross vs. Actress: Compared to actors, actresses relatively act in less movies as shown in Figure 3(b). Cate Blanchett is the actress who acts in the highest number of movies; She was involved in 35 movies, and she ranks fifth in the top ten actresses. The number of movies she has acted in is much less than Frank Welker. That shows that actors usually act in more movies than actresses. Generally speaking, we can make a conclusion that actresses act in fewer movies than actors. Movie Gross vs. Writer: From Figure 3(c), we observe that the total gross difference between the last seven writers is not obvious. Their movies' gross only have a few million difference. But compared to the first writer and the last writer, their difference is observable. The total gross of first writer is twice that of the last writer, which shows that famous writers are in great demand. Movie Gross vs. Director: Directors are responsible for the whole production process of the movies, and their crucial roles may determine the movie quality. The Figure 3(d) shows that top ten directors participated in relatively few movies. Steven Spielberg participated in 23 movies which is the highest of all the directors in our dataset. The average gross of each director is around $165.7 million dollars, which is higher than actor, actress and writer. We can see that best directors relatively act less time of making movie, but each movie they making have a high gross. We show the different character between actor, actress, writer and director regarding to movie gross. But they all have a strong relation to the movie gross and are the necessary factors for planning the blockbuster. 2) Movie Configuration Acquaintance: We show that the production team has a strong connection with the movie gross. Some of them are the guarantee to a high gross movie. To ensure a high gross movie, effective collaboration among production team also needs to be analyzed. We will see that production members have a strong collaboration with each other. There are six different types of collaborations, which are shown in the Table I. Actor and Actress Collaboration: Bernard Lee and Lois Maxwell participated in seven movies, like "Dr. No" which is the first James Bond film, and the genre of those movies are "Action", "Adventure" and "Thriller". The second frequent collaborating partners are Johnny Depp and Helena B. Carter. They participated in recent well-known movies, like "Sweeney Todd: The Demon Barber of Fleet Street" and "Alice in Wonderland". Actor and Writer Collaboration: Daniel Radcliffe and J.K. Rowling collaborated in the series of "Harry Potter". For the third frequent partners, Bernard Lee and Ian Fleming collaborated in many "Action", "Adventure" or "Thriller" genre movies and actress Lois Maxwell also participated in most of those movies, which shows strong collaboration among the three of them. Actor and Director Collaboration: Johnny Depp and Tim Burton collaborated in eight movies which makes them the most frequent partners. Among those eight movies, Helena B. Carter also participated in five of them, like "Corpse Bride" and "Dark Shadows". Adam Sandler and Dennis Dugan collaborated seven times, and in those movies, Tim Herlihy also participated. Actress and Writer Collaboration: Lois Maxwell was a famous actress during the 1960s and 1970s. She collaborated fourteen times with Ian Fleming and eleven times with Richard Maibaum. The most common genres they participated in are "Action", "Thriller" and "Adventure". Actress and Director Collaboration: Mia Farrow and Woody Allen collaborated in the "Comedy" or "Drama" genre movies eleven times and those movies have "Comedy" or "Drama" genre. Director and Writer Collaboration: Directors and writers collaborate more often than the other relationships. Ethan Coen and Joel Coen even collaborated sixteen times, with most of the movie genres being "Comedy" or "Crime". All of these collaborations show a strong relationship between the production team members and movie genre. The movies with high collaborations have a high gross and are well-known by viewers. Besides, the binary relationship cannot fully represent those collaborations. For example, Bernard Lee as actor, Ian Fleming as actress and Ian Fleming as writer, participated in many movies together. Moreover, those movies have the same genre, "Action", "Adventure" and "Thriller", which shows that only considering the collaboration between team members is not enough. Instead, considering the collaboration between team members based on movie genre is necessary. In our dataset, there are a lot of same or more complex collaborations like this. Therefore, the movie configuration acquaintance must need be studied when we make the blockbuster planning. The more details of movie configuration acquaintance term will be discussed in the Section V-A. IV. MOVIE CONFIGURATION VERIFICATION To verify the effectiveness of these factors aforementioned on estimating the movie gross and budget and to learn the weight of movie configuration, in this section, we will build a prediction model to learn their correlations. A set of features (i.e., the configuration) will be extracted for the movies based on each of the factors first. After that, a regression model will be built to project the movie configurations to their budget and gross. A. Feature Extraction and Movie Budget/Gross Estimation Features like actor, actress, writer and director are a bagof-words. Moreover, the relationship between a movie and its feature is one-to-many. For example, each movie belongs to more than one genre. And each movie has more than one actor or actress. We use e to represent an element in the movie m i configuration x (mi) =[x t (mi) , x s (mi) , x d (mi) , x w (mi) , x g (mi) ]. We use the binary value to set the element. Namely, for example, if actor t j participates in movie m i , then x tj (mi) equals to 1, otherwise, it equals to 0. In the same way, we can get the vector representation of x s (mi) , x d (mi) , x w (mi) and x g (mi) . After extracting all the features of a movie, we can train an approximation model of budget function and gross function. Since the cost and income of each production team member can not be negative, we use Lasso linear regression [9] and force the coefficients to be non-negative. Formally, they can be represented as: Budget(x) = min w b ,b b ||B − (w T b x + b b )|| 2 2 + λ||w b || 1 s.t.w b ≥ 0(2)||G − (w T g [B, x] + b g )|| 2 2 + λ||w g || 1 s.t.w g ≥ 0 (3) where x is the movie configuration and B is the budget and G is the gross of movie in the IMDB knowledge base. w b and w g are the weights and b b and b g are the intercepts of function Budget(x) and function Gross(x) function respectively. λ is the coefficient of the 1 -norm regularization, which we set to 0.1 in the experiment. B. Movie Budget/Gross Estimation Experiment Results In the experiments, 80% of 3156 movies are used as training data and 20% are used as testing data. The 5-fold cross validations are performed on training data. We analyze the effectiveness of those features on function Budget(x) and function Gross(x) separately. To measure the performance, we use the mean absolute percentage error (MAPE) as the evaluation metrics which represent as follow: M AP E = 100 n * n t=1 (A t − E t ) A t (4) where A t is the actual value and E t is the estimated value. The following is our compared methods: Models using all information • ALL: Method ALL builds the gross approximation with all features, which are genre, actor, actress, writer and director. Models using partial information Using all features together gives us the lowest MAPE, 16.243% as shown in the left part of Figure 4. When looking at the production team information, we find that writer and director have relative low MAPE, 16.679% and 16.280%, which implies writer and director positively correlated to the movie budget. While in reality, the salary of writer and director can determine how much the movie producer needs to invest in the movie. Production team information achieves a relatively lower MAPE than genre, due to wide difference of movie budgets in the same genre; therefore genre is not a good factor for the budget. By comparing all features in the gross approximation, we can observe that Director achieves the lowest MAPE (23.417%). Actor gets the second-lowest MAPE (23.859%). Compared to those two models, the performance of ALL is not good. It's probably because feature actress, writer and genre have a large MAPE, meaning that those features have no (or weak) correlation to the movie gross. Therefore, combining all features together will achieve 26.515% on MAPE. Such result is reasonable because a director with great reputation is more likely to produce a good movie. Moreover, similarly as the conclusion in Section 4, production team information can reach a relatively lower MAPE than the genre. Even if the performance of ALL is not as good as same only feature such as director or actor, the performance is still good. And our goal is to learn a function which can map the movie configuration with gross and budget. V. BLOCKBUSTER PLANNING: BIGMOVIE Since we have already demonstrated that good collaboration between production team members is the safeguard for the profit of movie. In this section, we first formulate the movie configuration acquaintance. Based on the movie gross, budget estimation and movie configuration acquaintance function, we provide the joint objective function of BigMovie and a cubic programming algorithm to effectively solve the objective. A. Movie Configuration Acquaintance As we discussed previously, it is advantageous to have production team members that have a strong collaboration to the other specific members and furthermore have strong acquaintance to the certain movie genre. If team members have already participated together before, they will have chemistry when they participate in the next movie, which may stimulate the increase of movie gross. Besides, production team members that have joined in a certain movie genre previously can more easily work together when making the same genre type movies. Those two types of acquaintances have a great effect on the movie gross and movie budget, so finding the mathematical representation of the movie configuration acquaintance is important. Movie Configuration Acquaintance Function: We discuss in section III-B2 the binary relationship between two members cannot well represent their collaboration and movie genre need to be considered as well. To solve these problems, we use a three dimensional tensor W a ∈ R C×C×G to represent movie configuration acquaintance, where the C is the dimension for the size of all cast, G is the dimension for the size of all movie genres. We propose to define the movie configuration acquaintance as follow: Acquaintance(x) = C−1 n=0 C−1 m=0 G−1 l=0 W a [n][m][l]·x[n]·x[m]·x[l],(5) where n and m are the production team member ∈ C. And l is the movie genre ∈ A g (mi) B. Joint Objective Function Maximize movie gross can be mathematically represented as: max x N −1 i=0 w g [i + 1] · x[i] + b g + w g [0] · B, where w g is the weight and b g is the intercept of function Gross(x) that we learned from section IV-A. The movie budget bound can be mathematically represented as: N −1 i=0 w b [i] · x[i] + b b ≤ B, where w b and b b are weights and intercepts of function Budget(x) we learned from section IV-A. The objective of the BP problem is to find the optimal movie configuration that can maximize the movie gross and movie configuration acquaintance while not exceeding the movie budget bound. So the joint objective function represents as: max x α( N −1 i=0 w g [i + 1] · x[i] + b g + w g [0] · B) (6) + β C−1 n=0 C−1 m=0 G−1 l=0 W a [n][m][l] · x[n] · x[m] · x[l] s.t. N −1 i=0 w b [i] · x[i] + b b ≤ B, ∀i : x i ∈ {0, 1} where α and β are parameters to adjust movie gross estimation and movie configuration acquaintance which are studied in Section V-E. C. Prove NP-Hardness In this section, we prove that the Blockbuster Planning with maximize movie configuration acquaintance problem is a NPhard problem. In Equation 6 of the BP problem, two objectives equations are involved: the gross equation weighted by α, and the acquaintance equation weighted by β. By assigning α = 1 and β = 0, we will show that the Knapsack problem can be reduced to the BP problem, which is a classic NP-hard problem. Meanwhile, by assigning α = 0 and β = 1, we will show that the Maximal Clique problem can be reduced to the BP problem. Given a set of items, each with a weight and a value, the Knapsack problem aims at picking the items to be included in a bag so that the total weight is less than a given limit while maximizing the total value. By treating items as features in the movie configuration vector x with corresponding values in vector w g and weights in vector w b , Knapsack problem can be exactly reduced to the BP problem (with α = 1 and β = 0), where the bag limit is denoted as the provided budget B. If we can identify an optimal movie configuration vector x, the items corresponding to the features with value 1 can be selected, which will be the optimal solution to the Knapsack problem. Given an undirected graph formed by a finite set of nodes and a set of undirected edges, the k-Clique problem aims at determining whether there exist a clique involving k nodes in graph or not. Let a tensor W p denote whether nodes can form a triangle in the input graph or not. If nodes n i , n j , n k can form a triangle, then W p [i][j][k] = 1, and 0 if not. By treating each node in the graph as a feature to be determined in the movie movie configuration vector x and assign vectors w g , w b to be 1, the problem of identifying a clique of size k in the input graph (i.e., k-Clique problem) can be reduced to the problem of obtaining the optimal value of k(k − 1)(k − 1) in the BP problem, where the budget B takes value k. If we can identify an optimal value k(k − 1)(k − 1), then the nodes corresponding to features with value 1 will be selected to for a clique of size k in the input graph. Therefore, the BP problem containing these two objectives makes itself at least as difficult as the Knapsack problem [10] and the Maximal Clique problem [11], which renders the BP problem to be NP-hard. D. Solving the Objective Since this problem is NP-complete [12] and no polynomialtime solutions can solve the problem efficiently, we propose to solve the problem with two steps: (1) integer constraint relaxation, and (2) result post-processing. We relax the integer constraint on variables, and allow them to take real values in the range of [0, 1] to help address the problem in polynomial time [13]. Based on the obtained real-valued solution x denoting the score of the feature-gross links, we post-process the variable to binary values by pruning with a confidence threshold θ in [0, 1]. For the variables, e.g., if x i > θ, we will map x i to value 1; otherwise, x i will be mapped to value 0. After post-processing, we can obtain the final optimal movie plan by finding corresponding genre g i and team member c i for the movie m i having values x i = 1. E. Experiments In this section, we give the experimental analysis of BigMovie, and evaluate its performance for designing new movies. We seek to answer two main questions: Q1. How well does BigMovie quantitative performance? Q2. Is the movie planned by BigMovie reasonable? 1) A1: Quantitative Evaluation: In order to quantitatively measure the performance, we use our method to design the movie production team setting or movie genre to see whether our model can match with the ground truth, the movie setting in the database. The BP problem is a new problem, and no existing methods can be applied to address it directly. To show the advantages of the framework BigMovie, we compare some other methods with BigMovie measured by accuracy and F1 score on production team and genre. The comparison methods used in the experiments are listed as follows: • BigMovie: framework in the paper that achieves the maximum gross and maximizes the movie configuration acquaintance. By setting α = 1, then the performance of different β will be studied in the experiments. • MaxG: aims to maximize the movie gross, which sets the objective function Eq. (6) with α = 1 and β = 0. • MaxA: only considers the movie configuration acquaintance by setting the objective function Eq. (6) with α = 0 and β = 1. • Greedy [14]: iteratively picks the greedy choice to maximize movie gross, by always choosing the maximum ratio of wg w b . We verify the effectiveness of production team and genre planning on 3, 156 IMDB movies separately. When studying the production team feature, we set the production team as unknown while the movie genre is given, and vice versa. Observation 1. As shown in Figures 5 and 6, Big-Movie outperforms the competing baselines MaxG, MaxA and Greedy and obtains higher accuracy and F1 score. From the experiment results, we can see that BigMovie can get more than 90% accuracy on both genre and production team, which is consistently better than other methods on different β. When β = 0.0001, the best performance is achieved on production team and genre, as shown in Figures 5(a), 5(b) and 6(a), 6(b). With β = 0.0001, we study different ratios of positive and negative samples that are randomly selected, as shown in Figure 5(c), 5(d), 6(c) and 6(d). Observation 2. Maximizing both the movie gross and movie configuration acquaintance simultaneously can achieve best performance. For methods only depending on maximizing the gross, MaxG and Greedy, because they do not depend on β, their performances do not change with different β. Since we use Lasso with non-negative constraint to learn the weight of budget and gross, most of weights we learned are zeros. Therefore, most candidates are not selected in the production team study, when using MaxG. Such planning is less ideal because not choosing any candidate is not the goal of an optimal planning. The Greedy method achieves higher accuracy than MaxG, but still has the same problem as MaxG. In the genre study, MaxG select all the candidate genre, so when the planning scale gets larger, its performance gets worse. For the MaxA method which only uses movie configuration acquaintance part of objective function, in both production team and genre planning, MaxA doesn't have the problem of not selecting any samples. This shows the importance of considering movie configuration acquaintance. But when the negative ratio gets bigger, MaxA performs worse. Since BigMovie outperforms MaxA, the results show the importance of considering simultaneously maximizing the movie gross and movie configuration acquaintance. 2) A2: Case Study: We show a case study to demonstrate the reasonable and effectiveness of the proposed method. We choose "The Avengers" to plan, which has the fourth-highest movie gross on our dataset, $623.27 million. For fairness, we delete the other sequel movies of "Avengers". We provide movie genre "Action", "Adventure" and "Sci-Fi", movie budget and 250 random candidates to build an about 20 casts production team. The detail of planned movie configuration and actual movie configuration is shown in Table II. Observation 3. BigMovie is rational and interpretive on planning the blockbuster movie. In the planned movie configuration, we get a $654.52 million on gross which is higher than the gross in original movie, $623.27 million. All the members in the original movie are selected by BigMovie. Besides the actual members, we plan one more actor, actress and writer and two more directors which are shown with underscores. For actors, we selected Sebastian Stan. We find that the reason why Sebastian Stan was selected is that he has a strong collaboration with Chris Evans as they collaborated twice and he participated in Marvel's "Captain America" series of movies, which share the same genre with "The Avengers". For actress, Elizabeth Olsen was selected. She has a strong collaboration with others, like Scarlett Johansson, Cobie Smulders and Chris Evans, because they participated in the movie "Captain America: The Winter Soldier". Additionally, she has participated in many the "Action", "Adventure" and "Sci-Fi" type movies, as she participated in the same genre movie "Godzilla". For writer, Joe Simon was selected. He is the writer in "Captain America" series of movies. And he collaborated with Chris Evans twice as well. For director, Jon Favreau was planned, as he is the director for the "Iron Man" series of movie. He collaborated with Robert Downey Jr. three times. Michael Bay, who is the director for the "Transformers" movie series, has collaborated with Scarlett Johansson in the movie "The Island". Besides, "Iron Man" series and "Transformers" series have the same genre as the "The Avengers" series. This planned configuration shows the effectiveness of our model, because it has a higher gross than the actual configuration and even if some casts was mistaken by us, all the planned casts are reasonable. Definition 1 . 1Online Movie Knowledge Library: An online movie knowledge library can be represented as an undirected graph G = (M, C, E, A), where node set M = {m 1 , m 2 , ..., m n } denotes the set of n movies in the library and C = {c 1 , c 2 , . Fig. 1 . 1Movie General information of IMDB Movie Fig. 2 . 2Movie Gross Distribution of IMDB Movie Genres discovered several important observations regarding the movie budget, gross and genre information. Fig. 3 . 3Accumulative Movie Gross vs. Production Team Member. From the left to the right which are Actor, Actress, Writer and Director Fig. 4 . 4The MAPE of different approximation models Gross(x) = min wg,bg • Genre: Method Genre builds the gross and budget approximation model with the genre information alone. • Actor: Method Actor just uses the actor feature to build the gross and budget approximation model. • Actress: Method Actress builds the gross and budget approximation model with the actress feature. • Writer: Method Writer builds the gross and budget approximation model with writer feature. • Director: Method Director builds the gross and budget approximation model with only director feature. The experiment results are available inFigure 4. The left part shows the MAPE on predicting the movie budget and the right part shows the MAPE of movie gross estimated by different methods. Fig. 5 .Fig. 6 . 56Planning production team with different β and different predicted ratio measured by Accuracy and F 1 (a) Accuracy with different β (b) F 1 with different β (c) Accuracy with different predicted ratio (d) F 1 with different predicted ratio Planning movie genre with different β and different predicted ratio measured by Accuracy and F 1 TABLE I THE ICOLLABORATION AMONG PRODUCTION TEAM http://www.imdb.com VIII. ACKNOWLEDGEMENTSThis work is supported in part by NSF through grants IIS-1526499, IIS-1763325, and CNS-1626432, and NSFC 61672313. This work is also partially supported by NSF through grant IIS-1763365 and by FSU through the startup package and FYAP award.ActressElizabethTABLE II THE PLANNED AND ACTUAL MOVIE CONFIGURATION OF MOVIE "THE AVENGERS"VI. RELATED WORKWe have clearly illustrated the significant differences of the BP problem from the existing works in Section I. In this section, we provide a brief review of recent developments on related works. Movie Gross Prediction People use different resource of information to predict the movie gross. Mestyán and Yasseri[2]used the knowledge base, Wikipedia, to predict the movie box office. Joshi et al.[15]use the sentiment analysis on movie reviews to predict the movie gross. The recent analysis of the movie gross was done through social media, like Twitter and YouTube[16][17]. Viral Marketing This problem focuses on finding a small set of seed nodes in a social network that maximizes the spread of influence. Kempe et al.[4],[18]first proposed two basic diffusion models, namely independent cascade model(IC) and linear threshold model(LT). These two models set the foundation of almost all existing algorithms finding seed in social networks[19]. The major drawback of their algorithm is that its inefficiency and ineffectiveness to the large networks. Later, Chen[20]proposed a greedy algorithm to approximate the influence regions of nodes. However, when the scales beyond million-sized graphs, greedy algorithm becomes unfeasible. Chen et al. proposed to use local directed acyclic graphs to explore a large-scale influence maximization algorithm[21]. Team Formation Lappas et al. first proposed this problem[5]. They described an approach that defined the total communication cost among the social relationships to select a subset of experts to form a qualified team for certain projects. Recently, Nikolaev et al. proposed the EngTFP problem to find the subset of users that was the most important for keeping the whole user base together[22]. Different from those two works that find a subset of users to form a qualified team for certain projects, several recent works focus on training the team members[23][24]. Their motivation is to build a team so that teammates can benefit from interaction to improve their skills.VII. CONCLUSIONIn this work, we studied the Blockbuster Planning (BP) problem where professional movie planning are made by exploring the accumulated knowledge in the online movie knowledge library. A novel movie planning framework named BigMovie is introduced, where we first build the gross estimation function by analyzing and investigating the real-world online movie library dataset. The weights of the movie factors learned by the gross estimation are easily interpretable, and can be directly applied to the objective function for blockbuster planning. The BigMovie framework is optimized to maximize the movie gross as well as the production team preference simultaneously. In addition, the limited budget is used as a hard constraint for the objective function to guarantee the plan achievement. Extensive experiments have been done on the real-world dataset to demonstrate the effective and advantages of the proposed framework in addressing the BP problem. . 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Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple.

10.1063/1.4975410

1605.07035

10ca667457a2ce8a01ed39cbc65f217ca71dd8fb

The graded product of real spectral triples 23 May 2016 Shane Farnsworth Max Planck Institute for Gravitational Physics Am Mühlenberg 114476Golm, GermanyEU The graded product of real spectral triples 23 May 2016 Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple. Introduction Forming the product between two geometric spaces is a basic operation in geometry. In noncommutative geometry (NCG) not only do product geometries provide a rich set of example spaces for mathematicians to explore, but they are also of great physical interest, because they arise in the description of gauge theories (including the standard model of particle physics, and its extensions) coupled to Einstein gravity. Unfortunately, in the spectral triple formulation of NCG, the traditional prescription for taking the product of geometric spaces has problems. In particular, although it should be expected that the product operation be commutative and associative, and to transform naturally under unitaries, it does not; and given two geometries T 1 and T 2 with well-defined 'KO-dimensions' d 1 and d 2 respectively, it should be expected that their product T 1,2 = T 1 × T 2 also has a well defined KO dimension d i,j = d 1 + d 2 (mod 8), but in general it does not. In this paper, we point out that these difficulties (and others) ultimately result from the fact that the traditional prescription incorrectly uses the ungraded tensor product to describe the product between graded spaces. We show that by switching to the graded tensor product, all of these issues are neatly resolved. The paper is organised as follows: In Sections 2 and 3 we cover review material. In particular we start in Subsection 2.1 by briefly reviewing the idea of KO-dimension, after which in Subsection 2.2 we review the traditional prescription for taking the product between two or more real NCGs and describe what goes wrong in general. In Section 3 we review graded tensor products as they are defined for star differential graded algebras ( * -DGAs). The material from Section 4 onwards is new. In Sections (4.1), (4.2), and (4.3) we use the graded tensor product developed for * -DGAs to redefine the product between NCGs. Then in Subsection 4.4 we provide a useful mnemonic for constructing the full KO-dimension table. In Subsection 4.5 we briefly detail how our new prescription builds upon the previous approaches introduced in [1][2][3][4]. 2 The traditional product prescription (and its shortcomings) The goal of this section is two fold: (i) We begin in Subsection 2.1 by reviewing the idea of KO-dimension, which is the notion of dimension we will be making use of in this paper. (ii) In Subsection 2.2 we review the traditional prescription for taking the product between two real spectral triples in NCG. We show what goes wrong, and briefly discuss some of the previous proposed solutions. KO-dimension There are several equally good ways of defining the dimension of an ordinary Riemannian geometry. The most familiar definition is given in terms of the number of linearly independent basis elements in the tangent space of a smooth manifold. Alternatively one could instead make use of Weyl's law, which relates the asymptotic growth of the eigenvalues of the Laplace operator on a manifold to the metric dimension of the underlying space [5]. A less familiar notion is so called 'KO-dimension', which exists for Riemannian spin geometries and more generally for 'real' NCGs. KO-dimension can be understood in the following functional sense: Consider a familiar four dimensional Riemannian spin geometry equipped with the flat Dirac operator D = −iγ µ ∂ µ , the Dirac gamma five matrix γ = γ 0 γ 1 γ 2 γ 3 , and the charge conjugation operator J = γ 0 γ 2 • cc, where we are using the basis of hermitian gamma matrices γ a given in [6, §3.4]. If one checks, then what one finds is that the operators {D, J, γ} satisfy the following conditions: J 2 = I, JD = DJ, Jγ = γJ, (2.1) where { , , } = {−1, +1, +1}. An analogous treatment can be performed in any metric dimension [7, §B], however in general the signs { , , } will depend on the dimension mod 8 of the underlying manifold 1 . Said another way, the signs { , , } define the 'KOdimension' of a spin geometry, and this idea continues to make sense for real NCGs. The notion of KO-dimension has many deep connections with Clifford algebras, Bott-periodicity, homology, etc (see e.g. [1,[7][8][9][10][11][12]), but the functional definition outlined here is all that will be necessary for understanding the rest of the paper. In table 1 we collect the various signs corresponding to each KO-dimension as they are usually presented in the NCG literature. 0 1 2 3 4 5 6 7 +1 +1 −1 −1 −1 −1 +1 +1 +1 −1 +1 +1 +1 −1 +1 +1 +1 −1 +1 −1 Product non-commutative geometries NCG is a generalization of Riemannian geometry which (amongst other applications) provides an elegant framework for describing gauge theories coupled to gravity. In this capacity, it's main physical interest is in constraining the allowed extensions of the standard model of particle physics [10,[16][17][18][19][20][21][22][23][24][25][26][27]. The basic idea of NCG is to replace the familiar manifold and metric data {M, g} of Riemannian geometry with a 'spectral triple' of data {A, H, D}, which consist of a 'coordinate' algebra A that provides topological information, a Dirac operator D which provides metric information, and a Hilbert space H that provides a place for A and D to interact. A spectral triple is said to be 'real' and 'even' if it is also equipped with an anti-unitary real structure operator J [28] and a Z 2 grading operator γ on H respectively. We will call a spectral triple which is not equipped with a non-trivial grading operator 'odd'. The benefit of this 'spectral' approach to geometry is that it continues to make sense even when the input algebra A is non-commutative, hence the name 'non-commutative geometry'. For a review see for example [6,10,11,15,[29][30][31]. To build a sensible NCG, the data {A, H, D, J, γ} should not be selected arbitrarily, but instead must satisfy a number of geometric conditions and axioms (which generalize the conditions satisfied by commutative Riemannian geometries, see e.g. [10,11,16] for details). In particular, a 'real' NCG must have a well defined KO-dimension, which in practice means that the operators {D, J, γ} must satisfy the 'real structure' conditions outlined in Eqs. (2.1) for an appropriate set of signs { , , }. A useful trick for finding new and interesting geometric spaces which satisfy the NCG axioms is to build product geometries from spaces which are already known to satisfy the NCG axioms. Unfortunately, as we will discuss now, the traditional prescription for taking the product between two or more real spectral triples does not always result in a product space with a well defined KO-dimension. In the traditional prescription, a product NCG is defined as follows: Given two real spectral triples [1,2,10,11,28]: T i = {A i , H i , D i , J i , γ i } and T j = {A j , H j , D j , J j (, γ j )}, the first of which is necessarily even, their product T i × T j is defined by T i,j = {A i,j , H i,j , D i,j , J i,j (, γ i,j )} whereA i,j = A i ⊗A j , H i,j = H i ⊗H j , D i,j = D i ⊗I j + γ i ⊗D j , (2.2a) γ i,j = γ i ⊗γ j , J i,j = J i ⊗J j , and where ⊗ is the usual tensor product (see e.g. [1, §4]), and the Z 2 grading operator γ i,j is only defined if both T i and T j are even. The product given in Eqs. (2.2a) does not always form a well defined spectral triple satisfying the real structure conditions of Eqs. (2.1). In fact for the signs given in Table 1, this product only makes sense if the first spectral triple T i is of KO-dimension 0 or 4 (mod 8): When T i is of KO-dimension 2 or 6 (i.e. when i = −1) then the product geometry T i,j fails to satisfy the real structure condition J i,j D i,j = i,j D i,j J i,j of Eqs. (2.1), and when the first spectral triple T i is of odd KO-dimension the product Dirac operator D i,j is not defined at all (because an odd T i will not be equipped with a non-trivial grading operator γ i ). Worse still, the definitions given in Eqs. (2.2) are inherently non-symmetric in the sense that even when a product geometry T i,j is well defined, T j,i is not necessarily. A partial solution to the above mentioned problems is obtained if one makes two important observations: (i) The first observation, which was emphasised in [1,4], is that for even spectral triples there is a second equally good choice for the product Dirac operator: D i,j = D i ⊗γ j + I i ⊗D j . (2.2b) The two choices of Dirac operator given in Eqs. (2.2) are unitarily equivalent, D i,j = U D i,j U * , with the unitary operator U given by [4]: U = 1 2 (I i ⊗I j + γ i ⊗I j + I i ⊗γ j − γ i ⊗γ j ). (2.3) ( ii) The second observation which was emphasised in [1] is that Table 1 should be extened to include 12 instead of 8 possible KO-dimension signs. This is because in each even KOdimension there are two equally good ways of defining the real structure operator: If J U is a real structure operator with KO-dimension signs { U , U , U }, then the composition J L = γJ U is also an anti-unitary operator satisfying the real structure conditions given in (2.1) with signs { L L , − L , L }. The 'U' and 'L' subscripts stand for 'upper' and 'lower' respectively -the reason for our naming convention will become apparent in Section 4. For odd spectral triples the grading operator is trivial γ ∝ I, and the upper and lower sign choices { , } are degenerate. When taken together, these two observations extend the applicability of the product defined in Eqs. (2.2) significantly [1,4]. For example, if the product between a certain pair of even spectral triples T i and T j is not well defined, then one can always find a well defined product triple T i,j by first replacing either the real structure operator J i with γ i J i , or by replacing J j with γ j J j (i.e. if the product between triples T i and T j is not well defined, then 0 2 4 6 0 2 4 6 1 3 5 7 +1 +1 −1 −1 +1 −1 −1 +1 +1 −1 −1 +1 −1 −1 −1 −1 +1 +1 +1 +1 −1 +1 −1 +1 +1 −1 +1 −1 +1 −1 +1 −1 L U L U U L U Lreplacing T i = {A i , H i .D i , J i , γ i } with T i = {A i , H i .D i , γ i J i , γ i } in the product will always result in a well defined geometry T i,j ). Similarly, products which are poorly defined when using the Dirac operator D i,j may make sense if instead the unitariliy equivalent choice of Dirac operator D i,j is used. What is more, the definitions given in Eqs. (2.2) have been extended to include the odd-odd cases in [1][2][3]. Despite these improvements, the product as defined in Eqs. (2.2) remains problematic: • Undefined products: For even spectral triples there are two equally good choices for the real structure operator {J, γJ}. Therefore when forming the product of any two real, even spectral triples there are four possible combinations for the product real structure operator (i.e. J i ⊗J j , γ i J i ⊗J j , J i ⊗γ j J j , or γ i J i ⊗γ j J j ) , while only two of these four possibilities may correspond to a well defined product geometry. To understand what goes wrong for two of the four choices it is useful to examine the KO-dimension signs { i,j , i,j , i,j } corresponding to a product space T i,j = T i × T j . For the definitions given in Eqs. (2.2) these are given by: i,j = i j , i,j = i = i j , i,j = i j , (2.4a) or i,j = i j , i,j = i j = j , i,j = i j . (2.4b) where the product signs with 'tildes' • Transformation under unitaries: Despite the two Dirac operators D i,j and D i,j being unitarily equivalent, it does matter which one is used when taking the product of two even spaces [1]. While some products are always well defined regardless of which Dirac operator is selected, others depend on the choice between D i,j and D i,j , while other products are never well defined. In addition, product triples as defined in Eqs. (2.2) are not stable under the unitary transformation of the Dirac operator given in Eq. (2.3), in the sense that while the product algebra A i,j and grading γ i,j are invariant under conjugation by U , the real structure operator J i,j is not. It transforms along with the Dirac operator. { i,j , i,j , i,j } correspond to the choice of Dirac operator D i,j , while those without tildes { i,j , i,j , i,j } correspond to the choice D i, • Commutativity and Associativity: The product defined in Eqs. (2.2) is noncommutative in the sense that while T i,j may be well defined, T j,i is not necessarily. Perhaps more troubling however is that the product is not associative, in the sense that while a product ( T i × D T j ) × D T k may be well defined, the product T i × D (T j × D T k ) is not necessarily (where the D and D subscripts indicate which choice of Dirac operator is being used for the product). • Obscure grading factors: The two product Dirac operators defined in Eqs. (2.2) include grading factors. These factors are introduced to ensure that the total Dirac operator squares to D 2 i,j = D 2 i ⊗I j + I i ⊗D 2 j , which implies that the dimensions add d i,j = d i + d j [4]. Grading factors also appear when translating between 'upper' and 'lower' real structure operators J L = γJ U . The distinction between 'upper' and 'lower' spectral triples and between D i,j and D i,j does seem to matter, and so it would be good to understand what is it that governs the appearance of the various grading factors in well defined product geometries. • Obscure KO-dimension signs: The product as defined in Eqs. (2.2) together with the KO-dimension table as presented in Table 2, provides little hint as to why certain products work, and why others fail. There is no obvious pattern behind the various KO-dimension signs, and no good reason for distinguishing those even signs for which = +1 from those satisfying = −1 as is done in the literature (see e.g. e.g. [10][11][12][13][14][15][16]). A number of solutions to the above mentioned problems have already been proposed. In particular the authors in [2][3][4] provide new definitions for the product real structure operator J i,j , each of which includes various clever insertions of grading factors γ i and γ j , which depend explicitly on the KO-dimensions of the two spectral triples being multiplied. While it is always possible to form well defined products in this way, the definitions already proposed offer no real explanation for the various obscure grading factors which are forced to appear. They also either depend on lookup tables, or unnaturally distinguish those KO-dimension signs for which is positive. Stability of the various definitions under the unitary transformation given in Eq. (2.3) has also not been discussed. In Section 4 we will show that a much more natural definition for the product between spectral triples is given in terms of the graded tensor product. The new definitions we provide are simple, and neatly resolve all of the various problems and questions which arise for the product defined in Eqs. (2.2). Graded tensor products The purpose of this section is to provide a brief review of * -DGAs, as well as to review graded tensor products as they are defined for * -DGAs. For a more complete account see the second section and the appendix of [19]. Differential graded star algebras A Z graded vector space H (over a field F), is a vector space which decomposes into the direct sum of vector spaces H i (each defined over the field F): H = i∈Z H i . (3.1) Any element h ∈ H i is said to be of 'degree' or 'order' |h| = i ∈ Z. A graded algebra A over the field F, is defined to be a graded vector space over F which is equipped with a bi-linear product over F, A × A → A, which respects the grading on A in the sense: |aa | = |a| + |a | for a, a ∈ A. A graded algebra A is said to be involutive if it is equipped with an anti-linear operator * : A → A which satisfies: (a * ) * = a, (3.2a) (aa ) * = (−1) |a|.|a | a * a * , (3.2b) for a, a ∈ A. 2 A graded algebra is said to be differential if it is equipped with a linear first order differential operator d : A → A, which satisfies: d 2 = 0, (3.3a) d[aa ] = d[a]a + (−1) |a| ad[a ] (3.3b) for a, a ∈ A. An algebra A is said to be a * -DGA if it is equipped with an involution * and a differential d satisfying Eqs. for a ∈ A. 3 Graded tensor products The action of linear operators on graded vector spaces can be defined in the same way as is done for spaces which are ungraded. In particular, a linear operator O on a graded vector space H is a map from H to itself satisfying: O(α 1 h 1 + α 2 h 2 ) = α 1 Oh 1 + α 2 Oh 2 ,(3.(O ⊗ O )(h ⊗ h ) ≡ (−1) |O ||h | (O h ⊗ O h ), (3.6a) or alternatively: (O ⊗ O )(h ⊗ h ) ≡ (−1) |O ||h | (O h ⊗ O h ), (3.6b) for h ∈ H, h ∈ H . The choice between the 'Kozul' signs given in Eqs. (3.6) is purely conventional, but will be of consequence when we later define the graded product between NCGs 4 . It is easy to show that the graded tensor product is associative. The definitions given in (3.6) are all that is needed to construct the graded tensor product of two * -DGAs. Given two graded algebras A and A , the order of an element a ⊗ a ∈ A ⊗ A is defined to be |a ⊗ a | = |a | + |a |. Multiplication between any two elements a 1 ⊗ a 1 and a 2 ⊗ a 2 in A ⊗ A is defined following (3.6) to be: a 1 a 2 ⊗ a 1 a 2 ), (3.7a) or alternatively: (a 1 ⊗ a 1 )(a 2 ⊗ a 2 ) ≡ (−1) |a 1 ||a 2 | ((a 1 ⊗ a 1 )(a 2 ⊗ a 2 ) ≡ (−1) |a 1 ||a 2 | (a 1 a 2 ⊗ a 1 a 2 ), (3.7b) depending on the 'Kozul' sign convention chosen. If A , and A are equipped with star operations * , and * respectively, then the star operation on the product algebra A = A ⊗ A is defined to be: * = * ⊗ * . (3.7c) If A , and A are equipped with differential operators d and d respectively, then the differential on the product algebra A = A ⊗ A is defined to be: d = d ⊗ I + I ⊗ d . (3.7d) The graded tensor product as given in Eqs. This is the graded product which we will employ in Section 4. A new product prescription (and its advantages) In this section we apply the graded tensor product reviewed in Subsection 3.2 to redefine the tensor product of two real, spectral triples. We consider the even-even, even-odd, and odd-odd cases separately. Our goal will be to ensure that the product geometries we define always have a well defined KO-dimension. Before we begin it should be noted that in addition to this dimensional requirement, product geometries must also satisfy a number of other geometric conditions in order to qualify as NCGs [10,11,16]. We will not discuss these extra conditions here, but instead refer the reader to the relevant sections of [1][2][3][4] to see that this will indeed always be the case. The even-even case The graded tensor product which we reviewed in Subsection 3.2 is directly applicable when constructing a product geometry from two real even spectral triples. For even spectral triples {A, H, D, J, γ} the Hilbert space H is Z 2 graded, with the degree of its elements distinguished by the grading operator γ. The degree of the algebra representation π with respect to the grading on H, and also that of the operators {D, J, γ} is determined by the NCG axioms, a review of which can be found for example in [6,10,11,16,29]. The grading operator is both hermitian and unitary γ = γ * = γ −1 , which means that it is equipped with eigenvalues ±1. We say that elements h ∈ H which satisfy γh = h are of 'even' degree, while elements satisfying γh = −h are of 'odd' degree. The representation π of the input algebra A on H is even with respect to the grading on H, which means that it satisfies [π(a), γ] = 0 for all a ∈ A. Meanwhile the Dirac operator is of odd degree with respect to the grading on H, which means that it satisfies {D, γ} = 0. The degree of the real structure operator depends on the KO-dimension of the geometry: Jγ = γJ. For a more complete discussion of the Z 2 grading on H see also [19]. Following the prescription outlined in Subsection 3.2 we define the graded product between two real, even spectral triples T i = {A i , H i , D i , J i , γ i } and T j = {A j , H j , D j , J j , γ j } as T i,j = {A i,j , H i,j , D i,j , J i,j , γ i,j }, where: A i,j = A i ⊗ A j , H i,j = H i ⊗ H j , D i,j = D i ⊗ I j + I i ⊗ D j , (4.1) J i,j = J i ⊗ J j , γ i,j = γ i ⊗ γ j , and where the lack of 'hats' indicates that we are using the graded tensor product of Subsection 3.2. We note that the real structure operator in a spectral triple may be viewed as a star operation on the input Hilbert space (as described in [17][18][19]), and so the form of the product real structure operator J i,j in (4.1) follows directly from Eq. (3.7c). Similarly, the Dirac operator of a spectral triple may be understood as deriving from the differential operator of a * -DGA (as for example in [19]), and so the form of D i,j in (4.1) follows directly from Eq. (3.7d). To compare our new definitions with the traditional definitions given in Eqs. (2.2), as well as to compare with the product triples defined in [1][2][3][4], we have only to re-express our graded tensor product in terms of the un-graded tensor product, which we do now: Because the representations of the algebras A i , A j and grading operators γ i , γ j are of even order, the action of the product algebra A i,j and product grading operator γ i,j given in eq (4.1) may be expressed on H i,j exactly as in eq (2.2): H i,j = H i ⊗H j , A i,j = A i ⊗A j , γ i,j = γ i ⊗γ j . (4.2a) The Dirac operators {D i , D j } however are of odd order, while the order of the real structure operators {J i , J j } depends on their KO-dimension signs { i , j }. Re-expressing the operators J i,j and D i,j of Eqs. (4.1) using the un-graded tensor product results in the appearance of grading operators: J i,j = J i γ (1− j )/2 i ⊗J j , D i,j = D i ⊗I j + γ i ⊗D j , (4.2b) or J i,j = J i ⊗J j γ (1− i )/2 j , D i,j = D i ⊗γ j + I i ⊗D j , (4.2c) where the two choices depend on the Kozul sign convention chosen (see Subsection 3.2). These two choices are unitarily equivalent, with the unitary transformation given as in eq (2.3). As would be expected given the unitary equivalence of {D i,j , J i,j } and { D i,j , J i,j }, the signs { i,j , i,j , i,j } corresponding to a product triple T i,j do not depend on which 'Kozul' sign convention is chosen: i,j = (−1) (1− i )(1− j )/4 i j , i,j = i j = i j , i,j = i j . (4.3) Comparing with the KO-dimension signs of the traditional product prescription in Eq. (2.4), the signs in Eq. (4.3) are completely symmetric and do not depend on what order the tensor product is taken in (i.e. both T i,j and T j,i are always well defined). Our naming convention for the KO-dimension table now also becomes apparent: The product between two even 'upper' ('lower') spectral triples is always well defined and results in an 'upper' ('lower') product triple of the correct KO-dimension. One can also check that the product between three 'upper' ('lower') spectral triples always remains well defined and is associative. It should be stressed that the graded product automatically organizes the KO-dimension table into a closed set of 'upper' and 'lower' signs in this way, and this is not something we have introduced by hand (i.e. we have not made an arbitrary choice such as L = +1 for all even dimensions as is regularly done in the NCG literature). We re-arrange the KO-dimension signs according to our 'upper' and 'lower' classification in Table 3, with the 'upper' signs for a given KO-dimension placed above the corresponding 'lower' signs. The 'upper' signs are those for which U = U , while the 'lower' signs satisfy L = − L . With this presentation a clear pattern between the signs emerges: { n+1,U , n+1,U } = { n,L , n,L } (where we remind the reader that for odd KO-dimensions the 'upper' and 'lower' signs { , } are degenerate). Every real, even spectral triple is equipped with both an 'upper' and a 'lower' real structure, and eqs (4.1) and eqs (4.2) consistently define how to take their product. The even-odd cases Our next goal is do define the product between odd and even spectral triples. The Hilbert space H i,j , and algebra A i,j will be the same as in Eq. (4.2a), but now only the even dimensional space will be equipped with a non-trivial grading operator. We therefore choose {D i,j , J i,j } or { D i,j , J i,j } from Eqs. (4.2), according to whether the even triple is the first one or the second one in the product respectively (a similar choice was made in [1]). Making use of Eqs. (4.2) in this way however presents us with a puzzle: how do we define the signs in odd dimensions? We take inspiration from Clifford algebras 5 , and define: n+1,L = n,U ,(4.4) for all n ∈ Z 8 6 . We have included these additional signs for the odd cases in table 3. With these definitions in place, the product between an upper (lower) 2n dimensional geometry and an upper (lower) 2m + 1 dimensional geometry, according to Eqs. (4.2), yields a geometry with upper (lower) KO-dimension 2(m + n) + 1. 7 . In practice however we will never be making any practical use of the identification γ = {I, iI} when constructing product geometries (i.e. we will never build a product grading operator γ i,j where for example γ i = iI). 0 1 2 3 4 5 6 7 +1 +1 +1 +1 −1 −1 −1 −1 −1 −1 +1 +1 +1 −1 −1 −1 +1 +1 +1 −1 −1 −1 +1 +1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 +1 −1 The odd-odd case For the odd-odd cases there is no non-trivial grading operator to work with and so we can no longer make use of the product given in Eq. (4.2). Taking inspiration from [1, 2] however we define: A i,j = A i ⊗A j , H i,j = H i ⊗H j ⊗C 2 , γ i,j = I i ⊗I j ⊗σ 3 , D i,j = D i ⊗I ⊗σ 1 + I i ⊗D j ⊗σ 2 , J i,j = J i ⊗J j ⊗σ (1− i )/2 1 (iσ 2 ) (1+ j )/2 • cc, (4.5) where the σ i are Pauli matrices, and once again the signs are determined for odd KOdimensions using Eq. (4.4). The representation of the algebra is understood to be trivial 6 Note that we could have also chosen n+1,U = n,L , which would have resulted in a more aesthetically pleasing presentation of the KO-dimension table, but at the same time would have also propagated various signs through the definition for the tensor product between odd-even and odd-odd spectral triples. 7 When γj = iI the product Dirac operator Di,j = Di ⊗Ij + γi ⊗Dj transforms as U Di,jU * = Diγi ⊗i + γ ⊗D, while the real structure operator Ji,j = Jγ on the C 2 factor, i.e. π(a i ⊗ a j ) = π i (a i ) ⊗ π j (a j ) ⊗ I C 2 [1]. With these definitions in place, the product between an upper (lower) 2n + 1 dimensional geometry and an upper (lower) 2m + 1 dimensional geometry yields a geometry with upper (lower) KO-dimension 2(m+n+1) without the need for the lookup tables that were required in [1,2]. Finally, just as in the even-even and even-odd cases, the odd-odd product KO-dimension signs depend symmetrically on their constituent KO-dimension signs: i,j = (−1) (1+ i )(1+ j )/4 i j , i,j = − i j = − i j , i,j = − i j . (4.6) A useful Mnemonic Having defined the product between real spectral triples, we are now able to introduce a useful mnemonic for 'deriving' the full KO-dimension table. We proceed in three steps: Step 1. There are 2 3 = 8 possible sign combinations { , , } corresponding to the even KO-dimension cases, and 2 2 = 4 sign combinations { , } corresponding to the odd KO-dimension cases. Begin by matching the 8 even cases into 4 pairs according to the relation J U = γJ L . Note that it is not yet important to know which set of signs in each pair should be labelled 'upper', and which should be labelled 'lower', only which pairs belong together. Step 2. It is now possible to determine which of the even sign cases corresponds to KO-dimension 0 mod 8, and which of the even sign cases corresponds to KO-dimension 4 mod 8. The product of two KO-dimeinsion 0 mod 8 spectral triples is again a KO-dimension 0 mod 8 spectral triple. This is the only KO-dimension which has this property, and so we can use equations (4.3) directly to distinguish which signs correspond to KO-dimension 0 mod 8. Similarly the product of two KO-dimension 4 mod 8 spectral triples gives a spectral triple of KO-dimension 0 mod 8, and so the signs corresponding to KO-dimension 4 mod 8 are also readily distinguishable. Step 3. Usually modular arithmetic would prevent us from going any further, however the 'upper' signs { n+1,U , n+1,U } corresponding to a spectral triple of KO-dimension n mod 8 match the 'lower' signs { n,L , n,L } corresponding to a spectral triple of KO-dimension n − 1 mod 8, while these 'upper and 'lower' signs in the odd cases are degenerate. We therefore have: { 2n,L , 2n,L } = { 2n+1,U , 2n+1,U } = { 2n+1,L , 2n+1,L } = { 2n+2,U , 2n+2,U }. (4.7) Equation (4.7) is restrictive enough that it allows the 'upper' and 'lower' signs of KOdimension 0 mod 8 to be distinguished. Alternatively we could have distinguished 'upper' signs from 'lower' by noting that for 'upper' signs U = U , while for 'lower' signs L = − L . This is enough information to fill out the remainder of table 3. Concluding remarks We conclude this section with a brief recap of the advantages of our graded product of spectral triples, which we introduced in Eqs. (4.1), (4.2) and (4.5). • Well defined products: The first point to note is that our product is always well defined for any pair of real spectral triples of any KO-dimensions including the oddodd cases. In particular our product does not rely on the various look-up tables which were required for the odd-odd cases in [1,2]. Furthermore, our product is associative, and as is clear from Eqs. (4.3) and (4.6) it is also symmetric in the sense that if a product space T i,j is well defined, then so is T j,i . • Meaning behind the grading factors: The authors in [2][3][4] all found ways of cleverly inserting grading factors into their definitions for the product of real structure operators in order to construct well defined product spectral triples. In our formulation the appearance of grading factors in both the Dirac and real structure operators is natural, and is no longer a mystery. They result automatically when translating between the graded tensor product and the ungraded tensor product. • KO-Dimension patterns: Our product naturally distinguishes the 'upper' from the 'lower' KO-dimension signs. What is more, once this naming convention is adopted a number of patters emerge in the KO-dimension table which were previously obscured by the arbitrary distinction between KO-dimension signs for which = +1 and those for which = −1. • KO-dimension table extension: Our product suggests a natural extension of the KO-dimension table, in which there are 8 rather than 4 possible odd KO-dimension combinations. We close with the product table corresponding to our prescription in Table 4. In order to appreciate just how simple our product is, comparison should be made for example with Tables 2-5 of [1] and tables 2-5 of [4]. Our product for the odd-odd cases also avoids the need for lookup tables which can be seen for example in Table 6 of [1], and Table 2.3 of [2]. Note: During the write-up of this work we learned that the authors C. Brouder, N. Bizi and F. Besnard have also constructed a product of spectral triples similar to that of [3,4] for Lorentzian spectral triples, which they will likely publish along with future work. We make note of this as their work has some similarities to our own which were obtained independently. 0 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 0 L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 0 U 0 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 1 U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 0 U 2 U 2 U 3 U 4 U 5 U 6 U 7 U 0 U 1 U 3 U 3 U 4 U 5 U 6 U 7 U 0 U 1 U 2 U 4 U 4 U 5 U 6 U 7 U 0 U 1 U 2 U 3 U 5 U 5 U 6 U 7 U 0 U 1 U 2 U 3 U 4 U 6 U 6 U 7 U 0 U 1 U 2 U 3 U 4 U 5 U 7 U 7 U 0 U 1 U 2 U 3 U 4 U 5 U 6 U 0 L 0 L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 1 L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 0 L 2 L 2 L 3 L 4 L 5 L 6 L 7 L 0 L 1 L 3 L 3 L 4 L 5 L 6 L 7 L 0 L 1 L 2 L 4 L 4 L 5 L 6 L 7 L 0 L 1 L 2 L 3 L 5 L 5 L 6 L 7 L 0 L 1 L 2 L 3 L 4 L 6 L 6 L 7 L 0 L 1 L 2 L 3 L 4 L 5 L 7 L 7 L 0 L 1 L 2 L 3 L 4 L 5 L 6 L j . It is clear from Eqs. (2.4) what must go wrong: For certain real structure combitations it is not possible to satisfy i,j = i = i j and/or i,j = i j = j . In tables 2-5 of [1], and 2-5 of [4] the authors give a full listing of which product geometries have a well defined KO-dimension, along with those which do not. (3.2) and (3.3) respectively, along with the condition d[a * ] = ±d[a] * , (3.4) (3.6) is defined such that the product of two * -DGAs as given in Eqs.(3.7) is itself a * -DGA which satisfies Eqs. (3.2), (3.3), and (3.4). the construction of odd-even and odd-odd product geometries. In this presentation a clear pattern emerges: { n+1,U , n+1,U , n+1,L } = { n,L , n,L , n,U }. Note that the reader may wish to view these new odd signs as corresponding to the two choices γ = {I, iI}, which leave the upper and lower signs { , } degenerate and which satisfy [D, γ] = [γ, π(a)] = 0. While γ = iI no longer satisfies the usual defining condition γ 2 = I [11], both choices γ = {I, iI} are unitary, which means that we can still make use of the unitary transformation given in Eq. (2.3) i )/2 Jj, where U is the unitary given in Eq. (2.3). When γj = I the product Dirac and real structure operators are invariant under the unitary transformation given in Eq. (2.3): Di,j = U Di,jU * , and Ji,j = U Ji,jU * . • Transformation under unitaries: Our product always remains well defined under the unitary transformation given in Eq. (2.3). Unlike in previous work, we stress that the Dirac operator and the real structure operator of an even spectral triple transform non-trivially under the action of the unitary operator given in Eq. (2.3). In addition the unitary equivalence of the two choices {D i,j , J i,j } and { D i,j , J i,j } is linked to the choice of Kozul sign in the graded tensor product. Table 1 . 1Mod 8 KO-dimension table as it is traditionally presented in the NCG literature (see e.g.[10][11][12][13][14][15][16]). With this presentation no obvious patterns emerge in the signs. Table 2 . 2ExtendedMod 8 KO-dimension table as presented in [1], with even KO-dimension signs grouped according to their sign. 'Even' KO-dimension signs corresponding to our 'upper' ('lower') naming convention are marked with a 'U' ('L'). 5a ) 5awhere h 1 , h 2 ∈ H, and α 1 , α 2 ∈ F. An operator O is said to be of 'degree' or 'order' |O| = j ∈ Z if it maps elements of H i into elements of H i+j , i.e. O : H i → H i+j . Notice that any element a ∈ A j of a graded algebra A (as defined above in Subsection 3.1) can be thought of as an operator of degree j on A, i.e. a :A i → A i+j .Given two graded vector spaces H and H over the field F and graded linear operators O : H → H and O : H → H respectively, their graded tensor product is defined as follows: the product vector space H is the tensor product of the vector spaces H and H , where the degree of an element h ⊗ h ∈ H ⊗ H is defined to be |h ⊗ h | ≡ |h | + |h |. The product operator O ⊗ O : H ⊗ H → H ⊗ H is defined to be of order |O ⊗ O | = |O | + |O |, while its action on H is defined such that: Table 3 . 3CompleteMod 8 KO-dimension table: Black entries correspond to the KO-dimension signs { , , } of Eq. (2.1). We introduce the red entries for odd KO-dimensions to facilitate Table 4 . 4The graded product table for real spectral triples. As explained in[13,14], 'KO-dimension' is a misnomer, and really only corresponds to metric dimension for commutative Riemannian geometries. If for example we had instead considered the familiar 4D Lorentzian spin geometry with Dirac operator D = −iγ µ ∂µ, gamma five matrix γ = iγ 0 γ 1 γ 2 γ 3 , and a charge conjugation operator JU = γ 2 • cc, then we would have found signs { , , } corresponding to the 'KO-signature' 3 − 1 = 2 case. Note that our choice of sign convention here corresponds to 'convention 2' as outlined in[19].3 For a natural generalization of condition (3.4) see[19]. The ungraded tensor product is defined with no 'Kozul' sign, i.e. (O ⊗O )(h ⊗h ) = (O h ⊗O h ), see e.g. [1, §4]. An irreducible representation of the d = 2n + 1 dimensional Clifford algebra can be constructed by extending the irreducible representation of the d = 2n dimensional Clifford algebra by γ, or alternatively a reducible representation can be constructed as a sub-algebra of a representation of the d = 2n + 2 dimensional Clifford algebra. See[7, §B] for details. Acknowledgements: We would like to thank John Barrett, Nadir Bizi, Latham Boyle, Christian Brouder, and Matilde Marcolli for useful discussions during the writing of this work. This work was supported by the Max Planck Society, and in part by the European Cooperation in Science and Technology association. Product of real spectral triples. L Dabrowski, G Dossena, arXiv:1011.4456International Journal of Geometric Methods in Modern Physics. 81833math-phL. Dabrowski and G. 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Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by 4 × 4 Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scattering and spin-orbit interaction of Dirac fermions in graphene. We use the reducible Darboux transformation to construct exactly solvable models of these systems where backscattering is absent, i.e. the models are reflectionless. arXiv:2110.05816v1 [quant-ph]

10.1140/epjp/s13360-022-02611-z

2110.05816

2ad89fbffd22775caa6486e033412c012161a321

Form-preserving Darboux transformations for 4 × 4 Dirac equations M Castillo-Celeita Nuclear Physics Institute Czech Academy of Science Prague/ŘežCzech Republic V Jakubský Nuclear Physics Institute Czech Academy of Science Prague/ŘežCzech Republic K Zelaya Nuclear Physics Institute Czech Academy of Science Prague/ŘežCzech Republic Form-preserving Darboux transformations for 4 × 4 Dirac equations Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by 4 × 4 Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scattering and spin-orbit interaction of Dirac fermions in graphene. We use the reducible Darboux transformation to construct exactly solvable models of these systems where backscattering is absent, i.e. the models are reflectionless. arXiv:2110.05816v1 [quant-ph] Introduction The Dirac equation rules relativistic dynamics of spin-1 2 quantum particles, which predicts the existence of antiparticles [1], hyperfine structure in the hydrogen atom and other muonic atoms [2,3], among other phenomena. Notably, the one-and two-dimensional Dirac equations emerge naturally in other physical scenarios such as optics [4][5][6][7][8], geometrical and topological phases [9][10][11], graphene layers [12][13][14][15], and Bose-Einstein condensates [16]. In these scenarios, predictions obtained from the Dirac equation can be tested in non-relativistic setups, which are usually more accessible to implement, such as in microwave cavities [17,18]. From the mathematical-physics perspective, solvable models play an essential role by providing further insight into physical phenomena. Exact solutions grant us global information of the system and are by construction free of error. This is in contrast to numerical approaches, where numerical solutions bring only local information and have intrinsic error. On the other hand, solvable models provide a test field for computational methods, and they can serve in perturbation analysis as the initial unperturbed system. In this regard, supersymmetric quantum mechanics offers us effective tools for construction of exactly solvable models of both nonrelativistic and relativistic systems [19][20][21]. Darboux transformation forms the backbone of supersymmetric quantum mechanics. In the context of the Dirac equation, there are two possible implementations followed in the literature. In the first approach, Darboux transformation is not applied directly on the Dirac Hamiltonian, but rather on the Schrödinger-like operator that corresponds to the square of Dirac Hamiltonian, see e.g. [22][23][24][25][26][27][28]. In the second case, Darboux transformation is used directly to modify Dirac energy operators. For 2×2 Dirac operator 1 , it was discussed in [29,30]. For general one-dimensional Dirac operator given in terms of n × n matrices, it was discussed in [31]. This approach was utilized e.g. for construction of the new exactly solvable models based on Heun equation [32], describing mechanical deformations of nanotubes [33][34][35] or in the description of optical systems [8,36]. Darboux transformation can serve well in constructing a new Dirac Hamiltonian that describes an exactly solvable system. Yet, it can be challenging to obtain it in the required and physically relevant form. In the current article, we shall discuss some specific Darboux transformations for 4 × 4 Dirac operators such that, after introducing some appropriate unitary transformations, we still keep control over the structure of the new Hamiltonian. This allows us to discuss some examples of interest in physics of graphene. The work is organized as follows. In section 2, we review the Darboux transformation for a generic m × m Dirac operator, where we show that the general construction makes it challenging to keep control over the resulting model. In section 3, we discuss a class of 4 × 4 Dirac operators that are reducible. We show there that these operators allow us to use reduced or partial Darboux transformation in order to obtain energy operators for specific physical systems. In section 4, we discuss in detail reflectionless systems described by 2 × 2 Dirac Hamiltonian. These results are used directly in the section 5, where solvable, reflectionless systems with distortion scattering or spin-orbit interaction are constructed. We discuss possible caveats that emerge in case of non-reducible Darboux transformation in section 6. The last section is left for summary and discussion. Dirac equation and Darboux transformations To begin with, let us introduce some notation and definitions. We will deal with quantum systems described by the stationary Dirac equation HΨ = (γ∂ x + V (x))Ψ = EΨ ,(1) where γ, V (x) ∈ C m×m and m ∈ Z + . We require H to be hermitian in C m ⊗ L 2 (R; dx), which we henceforth denote by C m ⊗ L 2 for short. It implies that there should hold γ † = −γ and V (x) † = V (x). The scalar product on C m ⊗ L 2 is defined in standard manner, so that for two m-component solutions Ψ = (ψ 1 , . . . , ψ m ) T and Φ = (φ 1 , . . . , φ m ) T of (1), the scalar product reads as (Ψ, Φ) ≡ R dxΨ † Φ = R dx m k=1 ψ * k (x)φ k (x) .(2) The corresponding probability density is P Ψ := (ψ * 1 , . . . , ψ * m ) · (ψ 1 , . . . , ψ m ) = m k=1 |ψ k | 2 ,(3) and the corresponding norm is defined through the scalar product as Ψ 2 = (Ψ, Ψ). We say that Ψ is a bound state if it fulfills (1) and has finite norm, Ψ < ∞. We also use the term "square integrable" interchangeably. Darboux transformations Now, let us briefly review the Darboux transformation for Dirac equation (1). We refer to [29] and [31] for more details. Darboux transformation is typically represented by a differential operator that mediates intertwining relations between two Dirac Hamiltonians H and H, LH = HL .(4) When an eigenstate Ψ of H is known, then (4) implies that we can find an eigenstate LΨ of H corresponding to the same eigenvalue, except the cases where LΨ = 0. Conjugating (4), one can check that L † does the converse provided that both H and H are hermitian. Although L † reverts the action of L, it is not the inverse of L. Let us show how L and H can be found such that (4) is satisfied for a given Hamiltonian H. Let us suppose that we have two stationary Dirac equations of the form HΦ ≡ (γ∂ x + V (x)) Φ = EΦ ,(5)H Φ ≡ γ∂ x + V (x) Φ = E Φ ,(6) with γ † = −γ an arbitrary invertible m × m constant matrix and V † = V and V † = V the corresponding matrix potentials. We suppose that the Hamiltonian H is given explicitly whereas L and V are to be fixed such that (4) is satisfied. We make an anstatz for L in terms of the first-order differential operator of the form 2 L = ∂ x − U x U −1 = U ∂ x U −1 ,(7) where the m × m matrix U is to be specified, and we have used the subindex notation f x = ∂ x f . This formula is inspired by supersymmetry in nonrelativistic quantum mechanics and was introduced in [29]. Eq. (7) facilitates better insight into the structure of the Darboux transformation. Indeed, substituting (7) into (4) and comparing the coefficients of the terms with the same order of derivatives, we get the following two equations V = V + [γ, U x U −1 ], V x − U x U −1 V = −γ(U x U −1 ) x − V U x U −1 .(8) The first equation identifies the new potential V in terms of V and U whereas the second one determines the matrix U . If we substitute the first equation in (8) into the second one, multiply the resulting equation by U −1 to the left, and integrate with respect to x, we obtain (γ∂ x + V ) U = U Λ , Λ = diag( 1 , . . . , m ) .(9) Here the constant matrix Λ emerges as an integration constant. For the sake of simplicity, we assume it to be diagonal. From (9), it is clear that U is composed by the eigensolutions Φ k of H, HΦ k = k Φ k , for k = 1, . . . , m. Explicitly we have U = (Φ k , . . . , Φ m ) , Φ k = (φ 1;k , . . . , φ m;k ) T , k = 1, . . . , m .(10) The eigensolutions Φ k (sometimes called "see" solutions in the literature) are not necessarily square integrable. Notice that there holds LΦ k = 0 by definition. The definition (10) completes the construction of H and L. Indeed, when we select the eigenstates Φ k in (10), we can compose the matrix U . Then we can find the new Hamiltonian H and the intertwining operator L in terms of this matrix, see (7) and (8). We have to stress that the construction is formal. It is not guaranteed that the action of L preserves the required boundary conditions, i.e. it is not guaranteed that both Φ and LΦ correspond to physical states. Additionally, it is not guaranteed that the new potential term V is hermitian, and it may even have additional singularities when det(U ) = 0 for some x. The properties of H are determined by choice of U , which should be fixed such that the new Hamiltonian has the required form. This task can be very demanding in general and it will be one of the main topics of this article. When both H and H are hermitian, they also satisfy HL † = L † H. We can see from (7) that L † annihilates the matrix U = (U −1 ) † , L † ≡ − U ∂ x U −1 = −(U −1 ) † ∂ x U † ,(11)U ≡ ( Φ 1 , . . . , Φ m ) , H Φ k = k Φ k , k ∈ {1, . . . , m}.(12) The states Φ k are frequently called "missing" eigenstates in the literature. It is not guaranteed that these states are physically acceptable. It has to be checked that they comply with the required boundary conditions. In this article, we will deal with the situation where the energy levels k are absent in the spectrum of H, i.e. Φ k in (10) are not square integrable, nevertheless, Φ k have finite norm and k form discrete energies of H. Challenges of 4 × 4 Darboux transformation and physics of graphene The relations (4)-(10) provide a powerful tool to construct solvable equations of the form (1) with V as the new potential term. Yet, the physical interpretation of such equation can be complicated. We are primarily interested in physics of graphene where (1) appears in lowenergy description of quasi-particles. The dimension of the matrix coefficients is related to the number of degrees of freedom that are considered in the system. In general, there is pseudospin degree of freedom induced by presence of two atoms in the elementary cell of graphene crystal. The valley degree of freedom is related to existence of two inequivalent Dirac points in the first Brillouin zone of graphene. Finally, there is also a spin degree of freedom of the electrons. Therefore, the Hamiltonian reflecting all the properties would acquire the form of a C 8×8 matrix, i.e. m = 8 in (1). Nevertheless, only those degrees of freedom relevant for the considered interaction appear in the effective description. Throughout this manuscript, we will be primarily interested the situations where the effective Hamiltonian is given in terms of 4 × 4 matrices, i.e. m = 4 in (1). Such energy operator can describe distortion scattering or spin-orbit interaction of Dirac fermions in graphene. Let us consider the following two types of energy operators, H dis = −iσ 3 ⊗ σ 1 ∂ x + V dis =     V A −i∂ x + V W A W + −i∂ x + V * V B W − W B (W A ) * (W − ) * V A i∂ x + V (W + ) * (W B ) * i∂ x + (V ) * V B     , H soc = −iσ 0 ⊗ σ 1 ∂ x + V soc =     V − ∆ −i∂ x 0 0 −i∂ x V + ∆ −2iλ 0 0 2iλ V + ∆ −i∂ x 0 0 −i∂ x V − ∆     .(13) The Hamiltonian H dis of disorder scattering acts on the bispinors whose components are ordered as (ψ K A , ψ K B , ψ K A , ψ K B ), where ψ K(K ) A(B) is the wave function localized at the lattice A(B) and describing states from the vicinity of the Dirac point K(K ), see e.g. [13,14]. The interactions associated with W A , W B and W ± cause intervalley scattering, which may be caused by impurities or atomic defects of the crystal lattice of mechanical contact with substrate [37], [38], [39]. The Hamiltonian H soc of spin-orbit interaction acts on the bispinors that have the following ordering (ψ ↑ A , ψ ↑ B , ψ ↓ A , ψ ↓ B ). Here ψ ↑(↓) A(B) is wave function localized at the lattice A(B) and with spin up(down). The potential term ∆ describes intrinsic spin-orbit interaction and λ represents Rashba spin-orbit interaction [40], [41], [42]. We would like to utilize Darboux transformation in the analysis of the systems described by (13) by constructing solvable models with H ≡ H dis or H ≡ H soc . As we mentioned in the previous subsection, this is a nontrivial task. Let us suppose that H = γ∂ x + V is hermitian and solutions of its stationary equation are known. Then, the new potential term V should be hermitian and free of additional singularities. The latter is achieved if [γ, U x U −1 ] † = [γ, U x U −1 ], det U = 0.(14) Additionally, V should have the same structure as either V dis or V soc , V = V (x) + [γ, U x U −1 ] ∼ V dis , γ = iσ 3 ⊗ σ 1 , V soc , γ = iσ 0 ⊗ σ 1 .(15) For instance, there should hold (10), such that (14) and (15) V 12 = V 13 = V 14 = V 24 = V 34 = 0, V 11 = V 44 and V 22 = V 33 for V ∼ V soc , However, it is very nontrivial to fix U , i.e. to select the eigenstates Φ k inare satisfied. Each state Φ k is a linear combination Φ k = 4 j=1 c k;j Φ (j) k of four fundamental solutions Φ (j) k of the stationary equation (H − k )Φ (j) k = 0. Thus, there are twenty parameters c k;j and k , j, k = 1, . . . , 4, in the matrix U in general. In the relations (14) and (15), there are rational functions composed by fourth-order polynomials c j in both the numerator and denominator. The eigenvalue k is involved in the explicit form of the fundamental solutions Φ (j) k . These parameters have to be fine-tuned such that (14) and (15) are satisfied. We are not aware of any systematic way that would allow us to do it properly, or, in other words, to keep sufficient control over the form of V . For the reasons mentioned above, direct use of Darboux transformation does not seem to be quite effective for construction of solvable systems where, besides Hermiticity, the Hamiltonian should take a specific form, e.g. in (13). Nevertheless, let us show how the major problems can be circumvented. Reduction scheme and Darboux transformations There is a reducible class of Hamiltonians H that can be brought into a block-diagonal form by implementing a unitary transformation generated by a unitary matrix U as U −1 HU = S 1 ⊗ h 1 + S 2 ⊗ h 2 , U † = U −1 .(16) where the matrices S 1 and S 2 are defined as S 1 := 1 0 0 0 , S 2 := 0 0 0 1 , S j S k = δ j,k S j .(17) We suppose that the reduced operators h 1 and h 2 are hermitian h j = −iσ 1 ∂ x + v j , v j = v j a j a * j w j , v j † = v j , j = 1, 2,(18)so that v j (x), w j (x) ∈ R and a j (x) ∈ C, j = 1, 2. Reducibility of H simplifies the solutions of (1) considerably. For instance, if one can find the spinors ξ E and χ E such that (h 1 − E)ξ E = 0, (h 2 − E)χ E = 0, ξ E = (ξ 1;E , ξ 2;E ) T , χ E = (χ 1;E , χ 2;E ) T ,(19) then we can define Φ E as Φ E = U (1, 0) T ⊗ ξ E + (0, 1) T ⊗ χ E = U ξ E χ E (20) = U(ξ 1;E , ξ 2;E , χ 1;E , χ 2;E ) T ,(21) that satisfies (H − E)Φ E = 0. The operator H dis in (13) is reducible provided that V = −V, W + = −e −2iα (W − ) † , W B = e −iα c B , W A = e −iα c A ,(22)with c A ≡ c A (x) and c B ≡ c B (x) real-valued functions. Indeed, if we define the unitary matrix U ≡ U dis as U dis = √ 2 2     0 1 0 −e −iα 1 0 −e −iα 0 0 −e iα 0 −1 e iα 0 1 0     ,(23) then there holds U −1 dis H dis U dis = (S 1 ⊗ h 1 + S 2 ⊗ h 2 ) ,(24) where v 1 = c B + V B V † − e iα W − V − e −iα (W − ) † −c A + V A , v 2 = −c B + V B V † + e iα W − V + e −iα (W − ) † c A + V A .(25) Hermiticity of v 1 and v 2 follows from (22). It is worth noticing that the components of v 1 and v 2 are linearly independent. It means that for any hermitian h 1 and h 2 , we can revert (16) and construct H dis whose components of the potential matrix satisfy (22). The Hamiltonian H soc is reducible without any additional requirements. Fixing the unitary transformation U ≡ U soc as U soc = √ 2 2     1 0 −e −iα 0 0 1 0 −e −iα 0 e iα 0 1 e iα 0 1 0     , α = π/2,(26) then we get U −1 soc H soc U soc = (S 1 ⊗ h 1 + S 2 ⊗ h 2 ) ,(27) where v 1 = V + ∆ 0 0 V − ∆ + λ , v 2 = V + ∆ 0 0 V − ∆ − λ .(28) We can construct the spin-orbit interaction Hamiltonian H soc as U soc ( S 1 ⊗ h 1 + S 2 ⊗ h 2 ) U −1 soc where v 1 and v 2 are as in (28). Dealing with a 2 × 2 Darboux transformation is much simpler than the 4 × 4 transformation, as it is easier to guarantee the hermiticity of the new Hamiltonian while preserving the required form [29]. We can use the latter to generate new Hamiltonians h j = −iσ 1 ∂ x + v j = −iσ 1 ∂ x + v j + [σ j , ∂ x U j U −1 j ](29)= −iσ 1 ∂ x + v j a j a * j w j , j = 1, 2.(30) such that L j h j = h j L j , L j = U j ∂ x U −1 j , ∂ x (U −1 j h j U j ) = 0, j = 1, 2.(31) In this form, we can directly compose the Hamiltonian H dis as H dis = U dis S 1 ⊗ h 1 + S 2 ⊗ h 2 U −1 dis (32) = −iσ 3 ⊗ σ 1 ∂ x +      V A V W A W + V * V B W − W B W * A ( W − ) * V A V ( W + ) * W * B V * V B      ,(33) where the components of potential term are given explicitly as V A := w 1 + w 2 2 , V B := v 1 + v 2 2 , V = − V = a * 1 + a * 2 2 , W − := e −iα 2 ( a 1 − a 2 ) ,(34)W A := e −iα 2 ( w 1 − w 2 ) , W B := − e −iα 2 ( v 1 − v 2 ) , W + := − e −iα 2 ( a * 1 − a * 2 ) .(35) On the other hand, the construction of H soc from h j is less straightforward. As we observed in (28), the potential terms of h j associated with H soc have to be diagonal matrices. Therefore, we need to generate v j in (30) with a j = 0, j = 1, 2. Depending on the properties of the initial Hamiltonian, we can get H soc by a convenient choice of the matrix U . Let us fix the initial operator h 1 as h 1 = −iσ 1 ∂ x + ω 3 σ 3 , ω 3 (x) ∈ R.(36) The Hamiltonian anticommutes with σ 2 , {h 1 , σ 2 } = 0. We compose the matrix U 1 = (ξ 1 , σ 2 ξ 1 ) from an eigenstate ξ 1 of h 1 , (h 1 − 1 )ξ 1 = 0. Then the new Hamiltonian reads as [29] h 1 = −iσ 1 ∂ x + ω 3 σ 3 , ω 3 = ω 3 + 2 ξ T 1 σ 2 (∂ x ξ 1 ) ξ T 1 ξ 1 .(37) The potential term v 1 = ω 3 σ 3 has the required diagonal form now. We can follow the same steps to geth 2 with a diagonal potential term. Nevertheless, it is unclear how 11 , which is also required in (28). Therefore, we can implement the Darboux transformation only partially to get h 1 whereas we can fix h 2 as to get v 1 = v 2 such that ( v 1 ) 11 = ( v 2 )h 2 = −iσ 1 ∂ x + ω 3 σ 0 + 2(λ(x) − ω 3 )S 2(38) such that (28) is satisfied by construction. Then we can construct H soc as H soc = −iσ 0 ⊗ σ 1 ∂ x + V soc = U soc S 1 ⊗ h 1 + S 2 ⊗ h 2 U −1 soc ,(39) where ( H soc − E) Φ = 0 is (at least) partially solvable as we suppose that the eigenstates of h 1 are known, but h 2 and its eigenstates are still unknown. The potential V soc reads as, V soc =     ω 3 0 0 0 0 λ − ω 3 −iλ 0 0 iλ λ − ω 3 0 0 0 0 ω 3     .(40) Comparing with (13), we can conclude that the Rashba term is proportional to λ, whereas electrostatic field V and the intrinsic spin-orbit coupling term ∆ are V = 1 2 λ, ∆ = ω 3 − λ 2 .(41) We will discuss this situation later in the text. Let us discuss the correspondence between 2 × 2 Darboux transformation of reduced systems and 4 × 4 Darboux transformation of the Hamiltonians H dis and H soc . First, we focus on the case of H dis . The reduced hamiltonians satisfy the intertwining relations (31). The operators H dis and H dis are given by (24) and (33). We can construct the following operator L dis = I 4×4 ∂ x − (∂ x U )U −1 = U dis (S 1 ⊗ L 1 + S 2 ⊗ L 2 )U −1 dis ,(42) where the matrix 4 × 4 matrix U can be written as U = U dis (S 1 ⊗ U 1 + S 2 ⊗ U 2 )U −1 dis .(43) Then, using the orthogonality property S j S k = δ j,k S j together with (31), (24) and (33), one can check that that there holds L dis H dis = H dis L dis .(44) Now, the case of H soc and H soc can be treated in similar manner. The intertwining operator for the partial Darboux transformation can be defined as L soc = U soc (S 1 ⊗ L 1 + S 2 ⊗ σ 0 )U −1 soc .(45) Notice that L 2 has been replaced by the identity operator σ 0 . The operator satisfies L soc H soc = H soc L soc .(46) It has to be kept in mind that in this case, we require ( v 1 ) 11 = (v 2 ) 11 . A few comments are in order. It is worth noticing that L dis is a special case of 4 × 4 Darboux transformation (7). In contrary, the intertwining operator L soc lies outside this class as it cannot be written as U ∂ x U −1 for a 4 × 4 matrix U . When the Darboux transformation L of a reducible Hamiltonian (16) is reducible, i.e. it can be written like in (42) or (45), then the new operator H is also reducible. When L cannot be transformed into the corresponding block diagonal form by the unitary transformation, reducibility of H is not granted. We will discuss the non-reducible Darboux transformations later in the text. So far, we have shown that the reduction scheme can be used for construction of a solvable 4 × 4 Hamiltonian that possesses the required form. We found that whenever we get hermitian 2 × 2 hamiltonians h 1 and h 2 as in (30), we can construct H dis with the potential as in (33). In this case, the reducible Darboux transformation L dis defined in (42) is form-preserving, see (44). The Hamiltonian H soc with spin-orbit interaction can be constructed from two 2 × 2 operators whose potentials satisfy (39). Here, Darboux transformation is only partial as we use it to get h 1 . The operator h 2 has to be fixed such that ( v 1 ) 11 = (v 2 ) 11 . Finally, let us notice that although we focused on the specific operators H dis and H soc , the results are applicable for any reducible Hamiltonian that satisfies (16). 2 × 2 reflectionless models via Darboux transformation We focus on the construction of new exactly solvable models based on the free particle system described by 2 × 2 Hamiltonian. Depending on the explicit form of the Darboux transformation, the new systems may inherit physical characteristics of the original system, such as reflection-free scattering. In this section, we will discuss construction and main properties of these systems. We consider a matrix Hamiltonian of dimension 2 × 2 with a constant matrix potential; that is, its matrix elements are independent of the position and time coordinates, h = −iσ 1 ∂ x + v a a * w , v, w ∈ R , a ∈ C .(47) The eigenvalue equation hψ E = Eψ E ,(48) has the following fundamental set of solutions, ψ E = ψ 1;E ψ 2;E = e −i(Re a)x cosh(κ E x) i w−E (Im a cosh(κ E x) + κ E sinh(κ E x)) , ψ E = ψ 1;E ψ 2;E = e −i(Re a)x i v−E (−(Im a) cosh(κ E x) + κ E sinh(κ E x)) cosh(κ E x) ,(49) where κ 2 E = (Im a) 2 − (v − E)(w − E) .(50) The solutions (49) have exponential-like behavior for real κ E , whereas they are oscillating for complex κ E . The parameter κ E is real for E that lies within the following interval, E ∈ ( − , + ) , ± = (v + w) ± (v − w) 2 + 4(Im a) 2 2 .(51) The solutions in (49) are not normalizable, independently on the value of κ E . The Hamiltonian h has translational symmetry, so that the spinors ψ E (x+α) and ψ E (x+α), α ∈ C, are solutions of (48) as well. We will use this fact later in the text. We can proceed to the construction of the new models with the use of Darboux transformation. As we discussed in section 3, the intertwining operator as well as the new Hamiltonian are defined in terms of the matrix U that satisfies h U = U diag( 1 , 2 ), see (10). We fix 1 and 2 as 1 , 2 ∈ ( − , + ) so that the corresponding eigenstates (49) are exponentially expanding for large |x|. For convenience 3 , we construct the seed matrix U by combining solutions in the form ψ 1 and ψ 2 . That is, U = ψ 1 , ψ 2 = ψ 1; 1 ψ 1; 2 ψ 2; 1 ψ 2; 2 = e −i(Re a)x × cosh(z 1 ) i v− 2 (−(Im a) cosh(z 2 ) + κ 2 sinh(z 2 )) i w− 1 ((Im a) cosh(z 1 ) + κ 1 sinh(z 1 )) cosh(z 2 ) ,(52) with κ j = κ E→ j , for j = 1, 2, and z 1 ≡ z 1 (x) = κ 1 x + δ 1 , z 2 ≡ z 2 (x) = κ 2 x + δ 2 .(53) The determinant of U takes the explicit form det(U ) = e −2i(Re a)x cosh(z 1 ) cosh(z 2 ) (v − 2 )(w − 1 ) D(x) ,(54) where D(x) = (v − 2 )(w − 1 ) + (κ 1 tanh(z 1 ) + Im a) (κ 2 tanh(z 2 ) − Im a) .(55) From (54)-(55), it is clear that the zeros of det(U ) correspond to those of the function D(x). By fixing v < w, Im a > 0 and 1 , 2 ∈ (v, w), it is possible to analytically show that D(x) is a nodeless function if the following condition holds 4 (w − 1 )( 2 − v) + Im a 2 > κ 1 κ 2 + (Im a)(κ 1 + κ 2 ) . From all the previous considerations, we can implement the Darboux transformation of Sec. 2, L = U ∂ x U −1 ,(57) and construct the new Hamiltonian h = h − [U U −1 , γ] = −iσ 1 ∂ x + v(x) = −iσ 1 ∂ x + v a a * w ,(58) where the components of the potential are Contrary to the initial free particle Hamiltonian h, the operator h admits two bound states ψ 1 and ψ 2 associated with the energies 1 and 2 , respectively, see (12). After some calculation, we get v = w + 2 − 1 − 2( 1 − 2 ) (w − 1 )(v − 2 ) D(x) , w = v − 2 + 1 + 2( 1 − 2 ) (w − 1 )(v − 2 ) D(x) , a = Im a + ( 1 − 2 ) (Im a)(v − w + 1 − 2 ) + (v − 2 )κ 1 tanh(z 1 ) + (w − 1 )κ 2 tanh(z 2 ) D(x) .(59)ψ 1 = e −i(Re a)x sech(z 1 ) N 1 D(x) 1 i (v− 2 ) (− Im a + κ 2 tanh(z 2 )) , ψ 2 = e −i(Re a)x sech(z 2 ) N 2 D(x) i (w− 1 ) (Im a + κ 1 tanh(z 1 )) 1 ,(60) with N j the corresponding normalization factors. As long as 1 = 2 or κ j = 0, both ψ 1 and ψ 2 vanish exponentially for large |x|. Since D(x) = 0 on the whole real line, both spinors in (60) have finite norm. In turn, if 1 = 2 = ∈ ( − , + ), the potential v corresponds to a constant matrix so that none of the missing states (60) are square integrable. On the other hand, for 1 = + (or 1 = − ) and 2 = ± , we have κ 1 = 0 and ψ 1 does not have finite norm. We thus only get ψ 2 as the bound state solution. Analogous results are obtained for 2 = ± . The probability density P j (x) associated to the missing eigensolutions is shown in Fig. 2 for several values of the respective parameters. From the latter, it can be noticed that the probability density becomes a symmetric function for Im a = δ 1 = δ 2 = 0. The presence of the phases δ k and Im a induces an asymmetry on the probability densities, as depicted in Figs. (2b)-(2c). The intertwining operator L provides a one-to-one mapping of the scattering states 5 . It can serve well in the analysis of the scattering properties of the new system. For illustration, let us consider is an incoming plane wave solution ψ E , The intertwining operator L transforms the scattering state into spinor ψ E = L ψ E such that h ψ E = Eψ E , ψ E = e κ E x 1, i(E − v) a + κ E T , E ∈ ( − , + ).(61)h ψ E = E ψ E .(62) The potential term of h tends to a constant matrix for large |x|. Therefore, it can be expected that the scattering states can be identified with plane waves for x → ±∞. This is indeed the case. There holds lim x→±∞ U x U −1 = w ± where w ± are constant 4 × 4 matrices whose explicit form is calculated in the Appendix. The operator L acts asymptotically as L ψ E → (∂ x + w ± )ψ E = e κ E x (κ E σ 0 + w ± ) 1, i(E − v) a − iκ E T , x → ±∞.(63) The action of the operator L on the scattering state ψ E does not change its momentum. As there is no back-scattered wave, the potential barrier represented by v in (58) is reflectionless for any energy E ∈ ( − , + ). We can also write L ψ E | x→+∞ = (κ E σ 0 + w + )(κ E σ 0 + w − ) −1 L ψ E | x→−∞ .(64) It is worth noticing that Darboux transformation of the free-particle system described by 2 × 2 Dirac Hamiltonian was discussed in [35]. The Hamiltonian (47), in comparison to the one used in [35], is more general and lacks the chiral symmetry. Therefore our current results extend those of [35]. Reflectionless models of distortion scattering and spin-orbit interaction In this section, we construct solvable systems described by H dis or H soc with the use of (33) or (39), where we identify h 1 and h 2 with the reflectionless Hamiltonians discussed in the previous section. Let us start with the case of the Hamiltoinan for distortion scattering. Reflectionless distortion Hamiltonian Let us recall that the reducible Hamiltonian H dis can be written in terms of two 2 × 2 Dirac operators h 1 and h 2 , see (33), H dis = U dis S 1 ⊗ h 1 + S 2 ⊗ h 2 U −1 dis .(65) We shall identify the reduced operators h j , j = 1, 2, with h in (58). The potential term v of the later Hamiltonian depends on the parameters v, a, w, 1 and 2 , δ 1 and δ 2 . These parameters can acquire different values in h 1 and h 2 . We define the two operators in the following manner, h 1 = h| v→v 1 ,a→a 1 ,w→w 1 = −iσ 1 ∂ x + v 1 a 1 a * 1 w 1 , h 2 = h| v→v 2 ,a→a 2 ,w→w 2 , 1 → 3 , 2 → 4 ,δ 1 →δ 3 ,δ 2 →δ 4 = −iσ 1 ∂ x + v 2 a 2 a * 2 w 2 .(66) We select 1 , 2 , 3 and 4 such that κ (1) 1(2) and κ (2) 3(4) are real, where κ (j) E = | Im a j | 2 − (v j − E)(w j − E), j = 1, 2. Then the potential terms of both h 1 and h 2 are regular and asymptotically constant, see (59) with an appropriate substitution from (66). The eigenstates of h 1 and h 2 can be obtained from (48) with the use of the intertwining operator (57). The missing states associated with h 1 and h 2 can be deduced from (60). They are ξ 1 = ψ 1 | v→v 1 ,a→a 1 ,w→w 1 , ξ 2 = ψ 2 | v→v 1 ,a→a 1 ,w→w 1 ,(67) and χ 3 = ψ 1 | v→v 2 ,a→a 2 ,w→w 2 , 1 → 3 , 2 → 4 ,δ 1 →δ 3 ,δ 2 →δ 4 ,(68)χ 4 = ψ 2 | v→v 2 ,a→a 2 ,w→w 2 , 1 → 3 , 2 → 4 ,δ 1 →δ 3 ,δ 2 →δ 4 .(69) and satisfy h 1 ξ 1,2 = 1,2 ξ 1,2 , h 2 χ 3,4 = 3,4 χ 3,4 . Now, we can construct the Hamiltonian H dis by inserting the explicit form of h 1 and h 2 from (66) into (33). As we argued in the end of the preceeding section, the potential V dis is reflectionless as it does not induce any backscattering. It is rather straightforward to get the explicit form of its components. For instance, the first diagonal component V A can be obtained as V A = w| v→v 1 ,a→a 1 ,w→w 1 2 + w| v→v 2 ,a→a 2 ,w→w 2 , 1 → 3 , 2 → 4 ,δ 1 →δ 3 ,δ 2 →δ 4 2 ,(71) where w is given in (59). The other elements of V dis can be obtained in exactly the same manner. Additionally, there hold the following relations between the elements, Therefore, we prefer to illustrate their behavior in figure instead of presenting their explicit forms. In Fig. 3 we depict the behavior of the matrix elements of the reflectionless distortion interaction V dis . For simplicity, we have fixed α = 0. We thus show the shape of the real-valued matrix inputs V A , V B , and W A in Fig. 3b, whereas, in Fig. 3c, we depict the imaginary part of W + and V . Note that the real part of W + and V take constant values since α = 0. Moreover, W B and W − differ from W A and W + , respectively, by a constant shift. We thus omit W B and W − in Fig. 3. V B := − V A + w 1 + w 2 + v 1 + v 2 2 , W B = W A + e −iα 2 (w 2 + v 2 − w 1 − v 1 ) ,(72)W − := W + + e −iα Re(a 1 − a 2 ) .(73) The distortion Hamiltonian has up to four 6 bound states that are based on the missing state of h 1 and h 2 , see (67) and (69), respectively. They can be defined as Φ 1,2 = U dis (1, 0) T ⊗ ξ 1,2 Φ 3,4 = U dis (0, 1) T ⊗ χ 3,4 .(74) By construction, their probability density is identical to (67) or (69), e.g. Φ † 1,2 Φ 1,2 ∼ ( ξ 1 ) † ξ 1 . Therefore, Fig. 3a serves well for illustration of density of probability of (74), while Fig. 3b-3c depicts the behavior of the elements of V dis . Spin-orbit coupling case In this section, we focus on systems with spin-orbit interaction described by the corresponding Hamiltonian H soc in (13). In order to exploit the Darboux transformation, we construct the new Hamiltonian H soc via (39), H soc = U soc S 1 ⊗ h 1 + S 2 ⊗ h 2 U −1 soc . As we discussed in section 3, the potential matrices v 1 and v 2 of h 1 and h 2 , respectively, should be diagonal. Additionally, there should hold that ( v 1 ) 11 = (v 2 ) 11 . Likewise in the previous subsection, we identify h 1 with h in (58), h 1 = h| v→v 1 ,a→a 1 ,w→w 1 , v 1 , w 1 , a 1 , δ 1 , δ 2 ∈ R.(75) Additionally, we fix the parameters of the potential term of h 1 as follows w 1 = −v 1 , δ 1 = δ 2 = 0 , a 1 = 0 , 2 = − 1 , 1 ∈ (0, v 1 ) , v 1 > 0.(76) Then the operator h 1 acquires the form h 1 = −iσ 1 ∂ x + v 1 σ 3 , v 1 (x) = v 1 − 2κ 2 1 v 1 + 1 cosh κ 1 x ,(77) where κ 1 = v 2 1 − 2 1 . Let us notice that for this specific choice of parameters, the seed solutions ξ 1 and ξ − 1 , which form the matrix U in (52), satisfy ξ − 1 ∼ σ 2 ξ 1 . This is in line with the construction presented in (36)- (37). The operator h 2 in (38) reads as h 2 = −iσ 1 ∂ x + v 1 σ 0 + 2(λ(x) − v 1 )S 2 ,(78) where λ(x) is a real function. In dependence on its explicit form, the eigenstates of h 2 might not be calculable analytically. Nevertheless the eigenstates of h 1 are known. Now, we can use (39) to get the Hamiltonian H soc of spin-orbit interaction. The electrostatic field V , intrinsic spin-orbit interaction ∆ and the Rashba term λ read as V (x) = 1 2 λ(x), λ(x) = λ(x), ∆(x) = v 1 (x) − λ(x) 2 .(79) The Hamiltonian H soc is partially-exactly solvable as for each energy E, we can find an eigenstate Φ E = (1, 0) T ⊗ ξ E , where ( H soc − E) Φ E = 0, ( h 1 − E) ξ E = 0.(80) Notice that these states are independent on the actual choice of λ(x). In particular, there are two localized states, Φ 1 = (1, 0) T ⊗ ξ 1 , Φ − 1 = (1, 0) T ⊗ ξ − 1 ,(81) where ξ ± 1 can be obtained from (60) together with (76). Let us consider the particular case λ(x) = v 1 (x) in h 2 . This leads to the eigenvalue problem h 2 χ E = (−iσ 1 ∂ x + v 1 (x)σ 0 )χ E = E χ E(82) the solutions of which can be determined with ease and are given by χ E = e −iσ 1 x v 1 (s)ds c 1 e iE x (1, 1) T + c 2 e −iE x (1, −1) T , c 0 , c 1 ∈ C.(83) The Hamiltonian h 2 is also reflectionless; indeed, it is known fact that electrostatic barriers are penetrated by Dirac fermions of any energy without back-scattering. This phenomenon is called Klein tunneling. The formula (83) suggests a simple explanation of this phenomenon [33]: the Hamiltonian (82) is unitarily equivalent to free particle, h 2 = e −iσ 1 x v 1 (s)ds (−iσ 1 ∂ x )e iσ 1 x v 1 (s)ds .(84) Therefore, the corresponding Hamiltonian of spin-orbit coupling H soc , whose components of the potential are V (x) = v 1 (x) 2 , λ(x) = v 1 (x), ∆(x) = v 1 (x) 2(85) is exactly solvable and reflectionless. Non-Hermiticity and non-reducible Darboux transformation In this section, we briefly discuss some of the issues that appear when dealing with generic 4 × 4 Darboux transformation, non-hermiticity of the new Hamiltonian in particular. For illustration, we fix the Hamiltonian H in the following manner, H = S 1 ⊗ h 1 + S 2 ⊗ h 2 .(86) Here, h 1 and h 2 are generic hermitian operators as defined in (18). Let us consider the most general matrix U in the following, non-diagonal, block form: U = U 1 U 3 U 4 U 2(87) with U j , for j = 1, 2, 3, 4, such that they satisfy h 1 U 1 = U 1 Λ 1 , h 1 U 3 = U 3 Λ 2 , h 2 U 4 = U 4 Λ 1 , h 2 U 2 = U 2 Λ 2 ,(88) where Λ 1 = diag( 1 , 2 ) and Λ 1 = diag( 3 , 4 ) are constant 2 × 2 diagonal matrices. There holds HU = U Λ 1 0 0 Λ 2(89) with 0 the null 2 × 2 matrix, so that U leads to the Hamiltonian H = γ∂ x + V , with V = V + i[γ, U x U −1 ], γ = −iσ 0 ⊗ σ 1 , and V † = V . To this end, we conveniently rewrite the inverse of U in a non-diagonal block form, U −1 = U 1 U 3 U 4 U 2 , where U U −1 = I and U 1 = (U 1 − U 3 U −1 2 U 4 ) −1 , U 3 = (U 4 − U 2 U −1 3 U 1 ) −1 , U 4 = (U 3 − U 1 U −1 4 U 2 ) −1 , U 2 = (U 2 − U 4 U −1 1 U 3 ) −1 .(90) In this form, we may determine the most general expression for the matrix potential as V = V + i [σ 1 , (∂ x U 1 )U 1 + (∂ x U 3 )U 4 ] [σ 1 , (∂ x U 1 )U 3 + (∂ x U 3 )U 2 ] [σ 1 , (∂ x U 4 )U 1 + (∂ x U 2 )U 4 ] [σ 1 , (∂ x U 4 )U 3 + (∂ x U 2 )U 2 ] ,(91) provided that all matrices U j are invertible. The latter formula suggests that V is hermitian as long as the diagonal blocks are hermitian, whereas the anti-diagonal blocks must be one the adjoint of the other. Those conditions are quite restrictive and cannot be guaranteed in general. As a particular example to illustrate non-Hermitian constructions, let us consider the 4 × 4 free particle Hamiltonian H = S 1 ⊗ h 1 + S 2 ⊗ h 2 , with h j given in (66), together with U 4 = 0. The latter choice is always feasible as the null matrix corresponds to the trivial solution to the eigenvalue equation associated with U 4 . The remaining matrices U 1 , U 2 , and U 3 are defined in terms of the general one introduced in (52) through the reparametrizations U 1 = U |v→v1,a→a1,w→w1 1 → 1 , 2 → 2 δ 1 →δ 1 ,δ 2 →δ 2 , U 2 = U |v→v2,a→a2,w→w2 1 → 3 , 2 → 4 δ 1 →δ 3 ,δ 2 →δ 4 , U 3 = U |v→v2,a→a2,w→w2 1 → 3 , 2 → 4 δ 1 →δ 3 ,δ 2 →δ 4 .(92) In this case, we get det(U ) = det(U 1 ) det(U 2 ), where the individual 2 × 2 determinants can be each proved to be different from zero. See discussion in Sec. 4. In this way, U is invertible and takes the form U −1 = U −1 1 −U 1 −1 U 3 U −1 2 0 U −1 2 , (U −1 ) † = (U −1 1 ) † 0 −(U −1 2 ) † U † 3 (U −1 1 ) † (U −1 2 ) † .(93) whereas the Hamiltonian becomes H = H + i [σ 1 , (∂ x U 1 )U −1 1 ] [σ 1 , (∂ x U 3 )U −1 2 − (∂ x U 1 )U −1 1 U 3 U −1 2 ] 0 [σ 1 , (∂ x U 2 )U −1 2 ] .(94) In the latter, the diagonal blocks correspond to the potential matrix generated by 2 × 2 Darboux transformation, see (58), which are hermitian. Nevertheless, the non-vanishing upperantidiagonal block breaks hermiticity of the Hamiltonian H. We can impose the condition (∂ x U 3 )U −1 2 − (∂ x U 1 )U −1 1 U 3 U −1 2 = f 0 (x)σ 0 + f 1 (x)σ 1 , with f 0 (x) and f 1 (x) some functions, possibly null, to be determined. Nevertheless, fixing hermiticity of H this way would make the Hamiltonian reducible. Therefore, by departing from a separable Hamiltonian, it seems that a tight relationship exists between the hermiticity and the separability of the Hamiltonian obtained via the Darboux transformation. Thus, the concept of reducible Darboux transformations applied on reducible Hamiltonians gives us an additional insight on how to construct appropriate Darboux transformations such that the new model is free of singularities and hermitian, a task not so clear from the non-separable setup. Let us briefly discuss some aspects of Darboux transformation when H is non-Hermitian. To begin with, it is worth noticing that, although the intertwining relation (4) still holds true when H is non-Hermitian, its adjoint ceases to related H and H as we get instead L † H † = HL † . That is, L † maps eigensolutions of H † into eigensolutions of H. Since (U −1 ) † is annihilated by L † , it provides us with the missing eigenstates of H † . Explicitly, we have H † Φ k = k Φ k , (U −1 ) † ∝ (Φ 1 , . . . , Φ 4 ), k ∈ {1, 2, 3, 4}.(95) Comparing with the explicit form of U in (93), we can see that the missing eigenstates Φ 3 and Φ 4 coincide with Φ 3 and Φ 4 of the separable case in Sec. 5.1. The difference emerges in the eigensolutions Φ 1 and Φ 2 that get modified by presence of U 3 in U . Such a behavior is depicted in the probability densities presented in Fig. 4, where it is clear that only P 3 (green) and P 4 (red) are the same as their separable counterparts presented in Fig. 3. It is worth to mention that, despite the lack of Hermiticity, the eigenvalues added to the new Hamiltonian H † are all-real, and their corresponding eigenfunctions have finite-norm. Here, we shall not discuss the properties mathematical apparatus behind non-Hermitian structures such as pseudo-Hermiticity, metric operators, bi-orthogonality, or PT-symmetry, as those deserve attention by their own. Rather, by means of this particular example, we want to point out the apparent connection between Hermiticity and separability. Concluding remarks In the article, we discussed possible problems and prospects related with the use of Darboux transformation in the context of physical systems described by 4 × 4 Dirac Hamiltonians. We Figure 4: Probability densities P j (x) associated with the missing eigensolutions Φ j for 1 (blue), 2 (orange), 3 (green), and 4 (red). The parameters, introduced in (92), have been fixed as v 1 = 3, v 2 = 2.5, w 1 = −2, w 2 = −1.5, a 1 = i, a 2 = 1, 1 = 1.25, 2 = 0.25, 3 = 0.75, 4 = −0.5, δ 1 = 1, δ 2 = −1, and δ 3 = δ 3 = δ 4 = δ 4 = 0 reviewed the general framework of Darboux transformation in section 2, showing explicitly the problems that appear when we require it to produce energy operators of specific form. As follows from (14) and (15) , the difficulty to get the new operator in the required form increases rapidly with dimension of the involved matrices. In this sense, the 2 × 2 Darboux transformation is the easiest to deal with. In section 3, we focused on a specific class of 4 × 4 Dirac operator that are reducible, i.e. they can be brought into block-diagonal form by a unitary transformation (16). We showed that both the Hamiltonians of distortion scattering and spin-orbit interaction belong to this class, see (24) and (27). This observation paves the way to use 2 × 2 Darboux transformation in construction of these 4 × 4 Dirac Hamiltonians such that the form of the potential term is granted by construction. In section 4, we discuss in detail 2 × 2 Darboux of free-particle system. We derive a class of systems that posses bound states and are reflectionless. We used these results in section 5, where reflectionless Hamiltonians with spin-orbit coupling and distortion scattering are constructed. The section 6 was devoted to discussion of non-reducible Darboux transformations. It is worth mentioning that both the Hamiltonian of distortion scattering and of spin-orbit interaction in (13) are just two members of the family of reducible Dirac operators. The concept of reducibility as reflected in (16) has much broader applicability. We also found that the reducibility allows us to define partial Darboux transformations (45) that lie out of the usual definition (7). We believe that these concepts and their application in description of physical systems are worth of further investigation, which, however, goes beyond the scope of the current article. Figure 1 : 1Components ω(x) (a) and Im( a * (x)) introduced in (58)-(59). In both cases, we have considered factorization energies 1 = −1 and = 2, together with v = −2 and w = 5. Moreover, the following set of parameters were used: {δ 1 = 0, δ 2 = 0, A = 0} (black-solid), {δ 1 = 4, δ 2 = −4, a = 0} (blue-dashed), and {δ 1 = 0, δ 2 = 0, a = 2i} (red-dotted).Here, z 1 and z 2 are linear functions of x, see (53), whereas D(x) is defined in (54). To illustrate the properties of h, inFig. 1we depict the diagonal component w and the anti-diagonal component Im a. Figure 2 : 2Probability densities P 1 (x) (blue) and P 2 (orange) associated to the missing states in (60). For all the cases, we have used the factorization energies 1 = −1 and 2 = 2, together with v = −2 and w = 5. The rest of parameter have been selected as {δ 1 = 0, δ 2 = 0, a = 0} (a), {δ 1 = 4, δ 2 = −4, a = 0} (b), and {δ 1 = 0, δ 2 = 0, a = 2i} (c). Figure 3 : 3(a) Probability densities P j (x) associated with the missing eigensolutions Φ j for 1 (blue), 2 (orange), 3 (green), and 4 (red). (b) Matrix potential entries V A (blue-solid), V B (red-dashed), and W A (black-dotted), together with (c) Im W + (blue-solid) and Im V (reddashed). In all cases, we have fixed the complex-phase α = 0 and the remaining parameters, previously introduced in (66), as v 1 = 3, v 2 = 2.5, w 1 = −2, w 2 = −1.5, a 1 = i, a 2 = 1, 1 = 1.25, 2 = 0.25, 3 = 0.75, 4 = −0.5, δ 1 = 1, δ 2 = −1, δ 3 = δ 4 = 0. By 2 × 2 operator we denote an operator acting on C 2×2 ⊗ H where H is a Hilbert space. Strictly speaking, the intertwining operator should write L = Im×m∂x −UxU −1 with Im×m the identity matrix of dimension m. Nevertheless, for simplicity, we dropout the identity matrix form the notation. This choice provides more compact formulas. It is also easier to discuss invertibility of the matrix U .4 This condition is sufficient but not necessary. Numerical analysis confirms that there is a wide range of parameters where the condition is not satisfied but D(x) is still node-less. A similar condition could be found for w < v, Im a < 0 and 1, 2 ∈ (w, v). L annihilates the two eigenstates from the interior of the energy band. It depends on the actual values of j , j = 1, 2, 3, 4, see discussion below (60). AppendixWe shall find asymptotic form of the matrix U x U −1 , where U is given in (52). We havewhere the matrix elements and the denominator are as followsand D(x) is given in(55). For the sake of completeness we put the formula here againThe c 1 and c 2 constants are given asNow, we calculate the limit x → ±∞ of the matrix (∂ x U )U −1 ,1 ± Im aκ 1 ∓ κ 2 (Im a ± κ 1 )) i c 1 ± c 2 ± i a κ 1 (Im a ∓ κ 2 ) ,where D ± = −Ima 2 ± κ 2 (Im a ± κ 1 ) ∓ Im aκ 1 + (v − 2 )(w − 1 ). The quantum theory of the electron. P Dirac, Proc. R. Soc. Lond. A. R. Soc. Lond. A117610P. Dirac. "The quantum theory of the electron," Proc. R. Soc. Lond. A 117, 610 (1928). J D Bjorken, S D Drell, Relativistic Quantum Mechanics. New YorkMcGraw-HillJ. D. Bjorken and S. D. Drell. 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We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computing is reduced to a separate integral over the unit sphere S 2 . In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space is very versatile and adjusts in a very simple manner, to several cases that involve energy dissipation due to local microreversibility (inelastic interactions) or elastic model of slowing down process. Our simulations are benchmarked with the available exact self-similar solutions, exact moment equations and analytical estimates for homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in [13] and generalized to a wide range of related models in[12]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard-spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods in [49; 26; 46; 35] and rigourously proven in [34] and [15].

10.1016/j.jcp.2008.09.033

0710.5308

4cadb31b6211cd2b973e956ea4d4fea9dd15fa48

Spectral -Lagrangian methods for Collisional Models of Non -Equilibrium Statistical States 25 Mar 2008 Irene M Gamba Dept. of Mathematics & Institute of Computational Engineering and Sciences Institute of Computational Engineering and Sciences University of Texas Austin University of Texas Austin Harsha Sri Dept. of Mathematics & Institute of Computational Engineering and Sciences Institute of Computational Engineering and Sciences University of Texas Austin University of Texas Austin Tharkabhushanam Dept. of Mathematics & Institute of Computational Engineering and Sciences Institute of Computational Engineering and Sciences University of Texas Austin University of Texas Austin Spectral -Lagrangian methods for Collisional Models of Non -Equilibrium Statistical States 25 Mar 2008Preprint submitted to Elsevier 25 March 2008arXiv:0710.5308v2 [math-ph]Spectral MethodBoltzmann Transport EquationConservative/ Non-conservative deterministic MethodLagrangian optimizationFFT We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann Transport Equation for Variable Hard Potential (VHP) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computing is reduced to a separate integral over the unit sphere S 2 . In addition, the conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space is very versatile and adjusts in a very simple manner, to several cases that involve energy dissipation due to local microreversibility (inelastic interactions) or elastic model of slowing down process. Our simulations are benchmarked with the available exact self-similar solutions, exact moment equations and analytical estimates for homogeneous Boltzmann equation for both elastic and inelastic VHP interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in [13] and generalized to a wide range of related models in[12]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard-spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods in [49; 26; 46; 35] and rigourously proven in [34] and [15]. Introduction In a microscopic description of a rarefied gas, all particles are assumed to be traveling in a straight line with a fixed velocity until they enter into a collision. In such dilute flows, binary collisions are often assumed to be the main mechanism of particle interactions. The statistical effect of such collisions can be modeled by collision terms of the Boltzmann or Enskog transport equation type, where the kinetic dynamics of the gas are subject to the molecular chaos assumption. The nature of these interactions could be elastic, inelastic or coalescing. They could either be isotropic or anisotropic, depending on their collision rates as a function of the scattering angle. In addition, collisions are described in terms of inter-particle potentials and the rate of collisions is usually modeled as product of power laws for the relative speed and the differential cross section, at the time of the interaction. When the rate of collisions is independent of the relative speed, the interaction is referred to as of Maxwell type. When it corresponds to relative speed to a positive power less than unity, they are referred to as Variable Hard Potentials (VHP) and when the rate of collisions is proportional to the relative speed, it is referred to as hard spheres. The Boltzmann Transport Equation (an integro-differential transport equation) describes the evolution of a single point probability distribution function f (x, v, t) which is defined as the probability of finding a particle at position x with velocity (kinetic) v at time t. The mathematical and computational difficulties associated to the Boltzmann equation are due to the non localnon linear nature of the collision operator, which is usually modeled as a multi linear integral form in d-dimensional velocity space and unit sphere S d−1 . From the computational point of view, of the well-known and well-studied methods developed in order to solve this equation is an stochastic based method called "Direct Simulation Monte-Carlo" (DSMC) developed initially by Bird [2] and Nanbu [48] and more recently by [54; 55]. This method is usually employed as an alternative to hydrodynamic solvers to model the evolution of moments or hydrodynamic quantities. In particular, this method have been shown to converge to the solution of the classical Boltzmann equation in the case of mono atomic rarefied gases [57]. One of the main drawbacks of such methods is the inherent statistical fluctuations in the numerical results, which becomes very expensive or unreliable in presence of non-stationary flows or non equilibrium statistical states, where more information is desired about the evolving probability distribution. Currently, there is extensive work from Rjasanow and Wagner [55] and references therein, to determine accurately the high-velocity tail behavior of the distribution functions from DSMC data. Implementations for micro irreversible interactions such as inelastic collisions have been carefully studied in [35]. In contrast, a deterministic method computes approximations of the probability distribution function using the Boltzmann equation, as well as approximations to the observables like density, momentum, energy, etc.,. There are currently two deterministic approaches to the computations of non-linear Boltzmann, one is the well known discrete velocity models and the second a spectral based method, both implemented for simulations of elastic interactions i.e. energy conservative evolution. Discrete velocity models were developed by Broadwell [20] and mathematically studied by Illner, Cabannes, Kawashima among many authors [41; 42; 21]. More recently these models have been studied for many other applications on kinetic elastic theory in [7; 24; 44; 59; 39]. These models have not adapted to inelastic collisional problems up to this point according to our best knowledge. Spectral based models, which are the ones of our choice in this work, have been developed by Pareschi, Gabetta and Toscani [32], and later by Bobylev and Rjasanow [17] and Pareschi and Russo [52]. These methods are supported by the ground breaking work of Bobylev [4] using the Fourier Transformed Boltzmann Equation to analyze its solutions in the case of Maxwell type of interactions. After the introduction of the inelastic Boltzmann equation for Maxwell type interactions and the use of the Fourier transform for its analysis by Bobylev, Carrillo and one of the authors here [6], the spectral based approach is becoming the most suitable tool to deal with deterministic computations of kinetic models associated with Boltzmann non-linear binary collisional integral, both for elastic or inelastic interactions. More recent implementations of spectral methods for the non-linear Boltzmann are due to Bobylev and Rjasanow [19] who developed a method using the Fast Fourier Transform (FFT) for Maxwell type of interactions and then for Hard-Sphere interactions [18] using generalized Radon and X-ray transforms via FFT. Simultaneously, L. Pareschi and B. Perthame [51] developed similar scheme using FFT for Maxwell type of interactions. Later, I. Ibragimov and S. Rjasanow [40] developed a numerical method to solve the space homogeneous Boltzmann Equation on a uniform grid for a Variable Hard Potential interactions with elastic collisions. This particular work has been a great inspiration for the current work and was one of the first initiating steps in the direction of a new numerical method. We mention that, most recently, Filbet and Russo [27], [28] implemented a method to solve the space inhomogeneous Boltzmann equation using the previously developed spectral methods in [52; 51]. Afore mentioned work in developing deterministic solvers for non-linear BTE have been restricted to elastic, conservative interactions. Finally, Mouhout and Pareschi [47] are currently studying the approximation properties of the schemes. Part of the difficulties in their strategy arises from the constraint that the numerical solution has to satisfy conservation of the initial mass. To this end, the authors propose the use of a periodic representation of the distribution function to avoid aliasing. There is no conservation of momentum and energy in [28], [27] and [47]. Both methods ( [28], [27], [47]), which are developed in 2 and 3 dimensions, do not guarantee the positivity of the solution due to the fact that the truncation of the velocity domain combined with the Fourier method makes the distribution function negative at times. This last shortcoming of the spectral approach remains in our proposed technique; however we are able to handle conservation in a very natural way by means of Lagrange multipliers. We also want to credit an unpublished calculation of V. Panferov and S. Rjasanow [50] who wrote a method to calculate the particle distribution function for inelastic collisions in the case of hard spheres, but there were no numerical results to corroborate the efficiency of the method. Our proposed approach is slightly different and it takes a less number of operations to compute the collision integral. Our current approach, based on a modified version of the work in [17] and [40], works for elastic or inelastic collisions and energy dissipative non-linear Boltzmann type models for variable hard potentials. We do not use periodic representations for the distribution function. The only restriction of the current method is that it requires that the distribution function at any time step be Fourier transformable. The required conservation properties of the distribution function are enforced through a Lagrange multiplier constrained optimization problem with the desired conservation quantities set as the constraints. Such corrections to the distribution function to make it conservative are very small but crucial for the evolution of the probability distribution function according to the Boltzmann equation. This Lagrange optimization problem gives the freedom of not conserving the energy, independent of the collision mechanism, as long momentum is conserved. Such a technique plays a major role as it gives the option of computing energy dissipative solutions by just eliminating one constraint in the corresponding optimization problem. The current method can be easily implemented in any dimension. A novel aspect of the presented approach here lays on a new method that uses the Fourier Transform as a tool to simplify the computation of the collision operator that works, both for elastic and inelastic collisions. It is based on an integral representation of the Fourier Transform of the collision kernel as used in [17]. If N is the number of discretizations in one direction of the velocity domain in d-dimensions, the total number of operations required to solve for the collision integral are of the order of N 2d log(N) + O(N 2d ). And this number of operations remains the same for elastic/ inelastic, isotropic/ anisotropic VHP type of interactions. However, when the differential cross section is independent of the scattering angle, the integral representation kernel is further reduced by an exact closed integrated form that is used to save in computational number of operations to O(N d log(N)). This reduction is possible when computing hard spheres in d+2 dimensions or Maxwell type models in 2-dimensions. Nevertheless, the method can be employed without much changes for the other case. In partic-ular the method becomes O(P d−1 N d log(N)), where P , the number of each angular discretizations is expected to be much smaller than N used for energy discretizations. Such reduction in number of operations was also reported in [28] with O(Nlog(N)) number of operations, where the authors are assuming N to be the total number of discretizations in the d-dimensional space (i.e. our N d and P of order of unity). Our numerical study is performed for several examples of well establish behavior associated to solutions of energy dissipative space homogeneous collisional models under heating sources that secure existence of stationary states with positive and finite energy. We shall consider heating sources corresponding to randomly heated inelastic particles in a heat bath, with and without friction; elastic or inelastic collisional forms with anti-divergence terms due to dynamically (self-similar) energy scaled solutions [34; 15] and a particularly interesting example of inelastic collisions added to a slow down linear process that can be derived as a weakly coupled heavy/light binary mixture. On this particular case, when Maxwell type interactions are considered, it is shown that [13; 14; 12], on one hand dynamically energy scaled solutions exist, they have a close, explicit formula in Fourier space for a particular choice of parameters and their corresponding anti Fourier transform in probability space exhibit a singularity at the origin and power law high energy tails, while remaining integrable and with finite energy. On the other hand they are stable within a large class of initial states. We used this particular example to benchmark our computations by spectral methods by comparing the dynamically scaled computed solutions to the explicit one self similar one. Convergence and error results of the Fourier Transform Lagrangian method, locally in time, are currently being developed [36], and it is expected that the proposed spectral approximation of the free space problem will have optimal algorithm complexity using the non-equispaced FFT as obtained by Greengard and Lin [38] for spectral approximation of the free space heat kernel. Implementation of the space inhomogeneous case are also currently being considered. The spectral-Lagrangian scheme methodology proposed here can be extended to cases of Pareto tails, opinion dynamics and N player games, where the evolution and asymptotic behavior of probabilities are studied in Fourier space as well. [53; 12]. The paper is organized as follows. In section 2, some preliminaries and description of the various approximated models associated with the elastic or inelastic Boltzmann equation are presented. In section 3, the actual numerical method is discussed with a small discussion on its discretization. In section 4, the special case of spatially homogeneous collisional model for a slow down process derived from a weakly coupled binary problem with isotropic elastic Maxwell type interactions is considered wherein an explicit solution is derived and shown to have power-like tails in some particular cases corresponding to a cold thermostat problem. Section 5 deals with the numerical results and examples. Finally in section 6, direction of future work is proposed along with a summary of the proposed numerical method. Preliminaries The initial value problem associated to space homogeneous Boltzmann Transport Equation modeling the statistical (kinetic) evolution of a single point probability distribution function f (v, t) for Variable Hard Potential (VHP) interactions is given by ∂ ∂t f (v, t) = Q(f, f )(v, t) = w∈R d ,σ∈S d−1 [J β f ( ′ v, t)f ( ′ w, t) − f (v, t)f (w, t)] B(|u|, µ) dσdw f (v, 0) = f 0 (v) , (2.1) where the initial probability distribution f 0 (v) is assumed integrable and J β = ∂(v ′ ,w ′ ) ∂(v,w) is Jacobian of post with respect to pre collisional velocities which depend the local energy dissipation [22]. The problem may or may not have finite initial energy E 0 = R d f 0 (v)|v| 2 dv and the velocity interaction law, written in center of mass and relative velocity coordinates is u = v − w : the relative velocity v ′ = v + β 2 (|u|σ − u), w ′ = w − β 2 (|u|σ − u) , µ = cos(θ) = u · σ |u| : the cosine of the scattering angle , B(|u|, µ) = |u| λ b(cos θ) with 0 ≤ λ ≤ 1, ω d−2 π 0 b(cos θ) sin d−2 θdθ < K : Grad cut-off assumption β = 1 + e 2 : the energy dissipation parameter , (2.2) where the parameter e ∈ [0, 1] is the restitution coefficient corresponding from sticky to elastic interactions, where J β = J 1 = 1. We denote by ′ v and ′ w the pre-collision velocities corresponding to v and w. In the case of micro-reversible (elastic) collisions one can replace ′ v and ′ w with v ′ and w ′ respectively in the integral part of (2.1). We assume the differential cross section function b( u·σ |u| ) is integrable with respect to the post-collisional specular reflection direction σ in the d − 1 dimensional sphere, referred as the Grad cut-off assumption, and that b(cos θ) is renormalized such that S d−1 b( u · σ |u| ) dσ = ω d−2 π 0 b(cos θ) sin d−2 θ dθ = ω d−2 1 −1 b(µ)(1 − µ 2 ) (d−3)/2 dµ = 1 , (2.3) where the constant ω d−2 is the measure of the d − 2 dimensional sphere and the corresponding scattering angle is θ is defined by cos θ = σ·u |u| . The parameter λ regulates the collision frequency as a function of the relative speed |u|. It accounts for inter particle potentials defining the collisional kernel and they are referred to as Variable Hard Potentials (VHP) whenever 0 < λ < 1, Maxwell Molecules type interactions (MM) for λ = 0 and Hard Spheres (HS) for λ = 1. The Variable Hard Potential collision kernel then takes the following general form: B(|u|, µ) = C λ (σ)|u| λ ,(2.4) with C λ (σ) = 1 4π b(θ), λ = 0 for Maxwell type of interactions; C λ (σ) = a 2 4 , λ = 1 for Hard Spheres. In addition, if C λ (σ) is independent of the scattering angle we call the interactions isotropic. Otherwise we referred to them as anisotropic Variable Hard Potential interactions. For classical case of elastic collisions, it has been established that the Cauchy problem for the space homogeneous Boltzmann equation has a unique solution in the class of integrable functions with finite energy (i.e. C 1 (L 1 2 (R d ))), it is regular if initially so, and f (., t) converges in L 1 2 (R d ) to the Maxwellian distribution M ρ,V,E (v) associated to the d + 2-moments of the initial state f (v, 0) = f 0 (v) ∈ L 1 2 (R d ). In addition, if the initial state has Maxwellian decay, this property will remain with a Maxwellian decay globally bounded in time ( [33]), as well as all derivatives if initial so (see [1]). Depending on their nature, collisions either conserve density, momentum and energy (elastic) or density and momentum (inelastic) or density (elastic -linear Boltzmann operator), depending on the number of collision invariants the operator Q(f, f )(t, v) has. In the case of the classical Boltzmann equation for rarefied (elastic) mono-atomic gases, the collision invariants are exactly d + 2, that is, according to the Boltzmann theorem, the number of polynomials in velocity space v that generate φ(v) = A + B · v + C|v| 2 , with C ≤ 0. In particular, one obtains the following conserved quantities density ρ(t) = v∈R d f (v, t)dv momentum m(t) = v∈R d vf (v, t)dv (2.5) energy E(t) = 1 2ρ(t) v∈R d |v| 2 f (v, t)dv . Of significant interest from the statistical view point are the evolution of moments or observables, at all orders. They are defined by the dynamics of the corresponding time evolution equation for the velocity averages, given by ∂ ∂t M j (t) = v∈R d f (v, t)v ∨ j dv = v∈R d Q(f, f )(v, t)v ∨ j dv , (2.6) where, v ∨ j = the standard symmetric tensor product of v with itself, j times. Thus, according to (2.6), for the classical elastic Boltzmann equation, the first d + 2 moments are conserved, meaning, M j (t) = M 0,j = v∈R d f 0 (v)v ∨ j dv for j = 0, 1; and E(t) = tr(M 2 )(t) = E 0 = v∈R d f 0 (v)|v| 2 dv. Higher order moments or observables of interest are Momentum Flow M 2 (t) = R d vv T f (v, t)dv Energy Flow r(t) = 1 2ρ(t) R d v|v| 2 f (v, t)dv Bulk Velocity V (t) = m(t) ρ(t) Internal Energy E(t) = 1 2ρ (tr(M 2 ) − ρ|V | 2 ) Temperature T (t) = 2E(t) kd (2.7) with k− Boltzmann constant. We finally point out that, in the case of Maxwell molecules (λ = 0), it is possible to write recursion formulas for higher order moments of all orders ( [5] for the elastic case, and [6] in the inelastic case) which, in the particular case of isotropic solutions depending only on |v| 2 /2, take the form m n (t) = R d |v| 2n f (v, t)dv = e −λnt m n (0)+ n−1 k=1 1 2(n + 1) 2n + 2 2k + 1 B β (k, n − k) t 0 m k (τ ) m n−k (τ ) e −λn(t−τ ) dτ ; with λ n = 1 − 1 n + 1 [β 2n + n k=0 (1 − β) 2k ] , B β (k, n − k) = β 2k 1 0 s k (1 − β(2 − β)s) n−k ds , (2.8) for n ≥ 1, 0 ≤ β ≤ 1, where λ 0 = 0, m 0 (t) = 1, and m n (0) = R d |v| 2n f 0 (v)dv. Boltzmann collisional models with heating sources A collisional model associated to the space homogeneous Boltzmann transport equation (2.1) with grad cutoff assumption (2.2), can be modified in order to accommodate for an energy or 'heat source' like term G(f (t, v)), where G is a differential or integral operator. In these cases, it is possible to obtain stationary states with finite energy as for the case of inelastic interactions. In such general framework, the corresponding initial value problem model is ∂ ∂t f (v, t) = ζ(t) Q(f, f )(v, t) + G(f (t, v)) , f (v, 0) = f 0 (v) , (2.9) where the collision operator Q(f, f )(v, t) is as in (2.1) and G(f (t, v)) models a 'heating source' due to different phenomena. The term ζ(t) may represent a mean field approximation that allows from proper time rescaling. See [6] and [15] for several examples for these type of models and additional references. Following the work initiated in [15] and [14] on Non-Equilibrium Stationary States (NESS), our computational approach we shall present several computational simulations of non-conservative models for either elastic or inelastic collisions associated to (2.9) of the Boltzmann equation with 'heating' sources. In all the cases we have addressed one can see that stationary states with finite energy are admissible, but they may not be Maxwellian distributions. Of this type of model we show computational output for three cases. First one is the pure diffusion thermal bath due to a randomly heated background [58; 49; 34], in which case G 1 (f ) = µ ∆f,(2.10) where µ > 0 is a constant. The second example relates to self-similar solutions of equation (2.9) for G(f ) = 0 [45; 25], but dynamically rescaled by f (v, t) = 1 v d 0 (t)f ṽ(v, t),t(t) ,ṽ = v v 0 (t) , (2.11) where v 0 (t) = (a + ηt) −1 ,t(t) = 1 η ln(1 + η a t), a, η > 0. (2.12) Then, the equation forf(ṽ,t) coincides (after omitting the tildes) with equation (2.9),for G 2 (f ) = −η div(vf ), η > 0 . (2.13) Of particular interest of dynamical time-thermal speed rescaling is the case of collisional kernels corresponding to Maxwell type of interactions. Since the second moment of the collisional integral is a linear function of the energy, the energy evolves exponentially with a rate proportional to the energy production rate, that is d dt E(t) = λ 0 E(t), or equivalently E(t) = E(0) e λ 0 t ,(2.14) with λ 0 the energy production rate. Therefore the corresponding rescaled variables and equations (2.11) and (2.9),(2.13) for the study of long time behavior of rescaled solutions are f (v, t) = E − d 2 (t)f v E 1 2 (t) = (E(0)e λ 0 t ) − d 2f (v (E(0)e λ 0 t ) − 1 2 ) , (2.15) andf satisfies the self-similar equation (2.9) G 2 ′ (f ) = −λ 0 xf x , where x = vE − 1 2 (t) is the self-similar variable . (2.16) We note that it has been shown that these dynamically self-similar states are stable under very specific scaling for a large class of initial states [12]. The last source type we consider is given by a model, related to weakly coupled mixture modeling slowdown (cooling) process [14] given by an elastic model in the presence of a thermostat given by Maxwell type interactions of particles of mass m having the Maxwellian distribution M T (v) = m (2πT ) d/2 e −m|v| 2 2T , with a constant reference background or thermostat temperature T (i.e. the average of M T dv = 1 and |v| 2 M T dv = T ). Define Q L (f )(v, t) . = w∈R d ,σ∈S d−1 B L (|u|, µ)f ( ′ v, t)M T ( ′ w) − f (v, t)M T (w)] dσdw . (2.17) Then the corresponding evolution equation for f (v, t) is given by ∂ ∂t f (v, t) = Q(f, f )(v, t) + ΘQ L (f )(v, t) f (v, 0) = f 0 (v) . (2.18) where Q(f, f ), defined as in (2.1), is the classical collision integral for elastic interactions (i.e. β = 1), so it conserves density, momentum and energy. The second integral term in (2.18) is a linear collision integral which conserves just the density (but not momentum or energy) since u = v − w the relative velocity v ′ = v + m m + 1 (|u|σ − u), w ′ = w − 1 m + 1 (|u|σ − u) . (2.19) The coupling constant Θ depends on the initial density, the coupling constants and on m. The collision kernel B L of the linear part may not be the same as the one for the non-linear part of the collision integral, however we assume that the Grad cut-off assumption (2.3) is satisfied and that, in order to secure mass preservation, the corresponding differential cross section functions b N and b L , the non-linear and linear collision kernels respectively, satisfy the renormalized condition S d−1 b N ( u · σ |u| ) + Θb L ( u · σ |u| ) dσ = 1 + Θ . (2.20) This last model describes the evolution of binary interactions of two sets of particles, heavy and light, in a weakly coupled limit, where the heavy particles have reached equilibrium. The heavy particles set constitutes the background or thermostat for the second set of particles. It is the light particle distribution that is modeled by (2.18). Indeed, Q(f, f ) corresponds to all the collisions which the light particles have with each other and the second linear integral term corresponds to collisions between light and heavy particles at equilibrium given by a classical distribution M T (v). In this binary 3-dimensional, mixture scenario, collisions are assumed to be isotropic, elastic and the interactions kernels of Maxwell type. In the particular case of equal mass (i.e. m = 1), the model is of particular interest for the development of numerical schemes and simulations benchmarks. Even though the local interactions are reversible (elastic), it does not conserve the total energy. In such a case, there exists a special set of explicit, in spectral space, self-similar solutions which are attractors for a large class of initial states. When considering the case of Maxwell type of interactions in three dimensions i.e. B(|u|, µ) = b(µ) with a cooling background process corresponding to a time temperature transformation, T = T (t) such that T (t) → 0 as t → 0, the models have self similar asymptotics [14; 12] for a large class of initial states. Such long time asymptotics corresponding to dynamically scaled solutions of (2.18), in the form of (2.16), yields interesting behavior in f (v, t) for large time, converging to states with power like decay tails in v. In particular, such solution f (v, t) of (2.18) will lose moments as time grows, even if the initial state has all moments bounded. (see [14; 12] for the analytical proofs). Collision Integral Representation One of the pivotal points in the derivation of the spectral numerical method for the computation of the non-linear Boltzmann equation lays in the representation of the collision integral in Fourier space by means of the weak form. Since for a suitably regular test function ψ(v), the weak form of the collision integral takes the form (suppressing the time dependence in f) v∈R d Q(f, f )ψ(v)dv = (w,v)∈R d ×R d , σ∈S d−1 f (v)f (w)B(|u|, µ)[ψ(v ′ )−ψ(v)]dσdwdv , (2.21) with v ′ = v + β 2 (|u|σ − u). In particular, taking ψ(v) = e −iζ·v /( √ 2π) d , where ζ is the Fourier variable, we get the Fourier Transform of the collision integral through its weak form as Q(ζ) = 1 ( √ 2π) d v∈R d Q(f, f )e −iζ·v dv = (w,v)∈R d ×R d , σ∈S d−1 f (v)f (w) B(|u|, µ) ( √ 2π) d [e −iζ·v ′ − e −iζ·v ]dσdwdv . (2.22) We will use . = F (.) -the Fourier transform and F −1 for the classical inverse Fourier transform. Plugging in the definitions of collision kernel B(|u|, µ) = C λ (σ)|u| λ (which in the case of isotropic collisions would just be the Variable Hard Potential collision kernel) and v ′ Q(ζ) = 1 ( √ 2π) d u∈R d G λ,β (u, ζ) v∈R d f (v)f (v − u)e −iζ·v dvdu = u∈R d G λ,β (u, ζ)[f (v)f (v − u)] du ,(2.23) where G λ,β (u, ζ) = σ∈S d−1 C λ (σ)|u| λ [e −i β 2 ζ·(|u|σ−u)) − 1]dσ = |u| λ e i β 2 ζ·u σ∈S d−1 C λ (σ)e −i β 2 |u|ζ·σ dσ − ω 2 . (2.24) Note that (2.24) is valid for both isotropic and anisotropic interactions. For the former type, a simplification ensues due to the fact the C λ (σ) is independent of σ ∈ S d−1 : G λ,β (u, ζ) = C λ ω d−2 |u| λ e i β 2 ζ.u sinc( β|u||ζ| 2 ) − 1 . (2.25) Thus, it is seen that the dependence on σ i.e. the integration over the unit sphere S d−1 is completely done independently and there is actually a closed form expression for this integration, given by (2.25) in the case of isotropic collisions. In the case of anisotropic collisions, the dependence of C λ on σ is again isolated into a separate integral over the unit sphere S d−1 as given in (2.24). The above expression can be transformed for elastic collisions β = 1 into a form suggested by Rjasanow and Ibragimov in their paper [40]. The corresponding expression for anisotropic collisions is given by (2.24). Further simplification of (2.23) is possible by observing that the Fourier transform inside the integral can be written in terms of the Fourier transform of f (v) since it can also be written as a convolution of the Fourier transforms. Let h(v) = f (v − u) Q(ζ) = u∈R d G λ,β (u, ζ)[f (v)h(v)] du = u∈R d G λ,β (u, ζ) 1 ( √ 2π) d (f * ĥ)(ζ)du = u∈R d G λ,β (u, ζ) 1 ( √ 2π) d ξ∈R df (ζ − ξ)f(ξ)e −iξ·u dξdu = 1 ( √ 2π) d ξ∈R df (ζ − ξ)f(ξ)Ĝ λ,β (ξ, ζ)dξ, (2.26) whereĜ λ,β (ξ, ζ) = u∈R d G λ,β (u, ζ)e −iξ·u du. Let u = re, e ∈ S d−1 , r ∈ R For d = 3, G λ,β (ξ, ζ) = r e r 2 G(re, ζ)e −irξ·e dedr = 16π 2 C λ r r λ+2 [sinc( 2β|ζ| 2 )sinc(r|ξ − β 2 ζ|) − sinc(r|ξ|)]dr . Since the domain of computation is restricted to Ω v = [−L, L) 3 , u ∈ [−2L, 2L) 3 ⇒ r ∈ [0, 2 √ 3L] G λ,β (ξ, ζ) = 16π 2 C λ 2 √ 3L 0 r λ+2 [sinc( 2β|ζ| 2 )sinc(r|ξ − β 2 ζ|) − sinc(r|ξ|)]dr . (2.27) A point worth noting is that the above formulation (2.26) results in O(N 2d ) number of operations, where N is the number of discretizations in each velocity direction. Also, exploiting the symmetric nature in particular cases of the collision kernel one can reduce the number of operations to O(N d logN). Numerical Method Discretization of the Collision Integral Coming to the discretization of the velocity space, it is assumed that the two interacting velocities and the corresponding relative velocity v , w, and w ∈ [−L, L) d , (3.1) while ζ ∈ [−L ζ , L ζ ) d ,(3.2) where the velocity domain L is chosen such that u = v − w ∈ [−L, L) d through an assumption that supp(f ) ∈ [−L, L) d . For a sufficiently large L, the computed distribution will not lose mass, since the initial momentum is conserved (there is no convection in space homogeneous problems), and is renormalized to zero mean velocity. We assume a uniform grid in the velocity space and in the fourier space with h v and h ζ as the respective grid element sizes. h v and h ζ are chosen such that h v h ζ = 2π N , where N = number of discretizations of v and ζ in each direction. Time Discretization To compute the actual particle distribution function, one needs to use an approximation to the time derivative of f . For this, a second-order Runge-Kutta scheme or a Euler forward step method were used. Since a non-dimensional Boltzmann equation is computed, for numerical computations the value of time step dt is chosen such that it corresponds in its dimensional form to 0.1 times the time between consecutive collisions (which depends on the collision frequency). During the standard process of non-dimensionalization of the Boltzmann Equation, such a reference quantity (time between collisions) comes up. With time discretizations taken as t n = ndt, the discrete version of the Runge-Kutta scheme is given by f 0 (v j ) = f 0 (v j ) f (v j ) = f t n (v j ) + dt 2 Q λ,β [f t n (v j ), f t n (v j )] f t n+1 (v j ) = f t n (v j ) + dtQ λ,β [f (v j ),f (v j )] .(3. 3) The corresponding Forward Euler scheme with smaller time step is given bỹ f(v j ) = f t n (v j ) + dtQ(f t n , f t n ) . (3.4) Conservation Properties -Lagrange Multipliers Since the calculation of Q λ,β (f, f )(v) involves computing Fourier Transforms with respect to v, we extensively use Fast Fourier Transform. Note that the total number of operations in computing the collision integral reduces to the order of 3N 2d log(N) + O(N 2d ) for (2.23) and O(N 2d ) for (2.26). Observe that, choosing 1/2 ≤ β ≤ 1, the proposed scheme works for both elastic and inelastic collisions. As a note, the method proposed in the current work can also be extended to lower dimensions in velocity space. In the current work, due to the discretizations and the use of Fourier Transform, the accuracy of the proposed method relies heavily on the size of the grid and the number of points taken in each velocity/ Fourier space directions. Because of this it is seen that the computed Q λ,β [f, f ](v) does not really conserve quantities it is supposed to i.e. ρ, m, e for elastic collisions, ρ for Linear Boltzmann Integral and ρ, m for inelastic collisions. Even though the difference between the computed (discretized) collision integral and the continuous one is not great, it is nevertheless essential that this issue be resolved. C (d+2)×M =        ω j v i ω j |v j | 2 ω j        and a (d+2)×1 = ρ m1 m2 m3 e T be the vector of conserved quantities. Using the above vectors, the conservation can be written as a constrained optimization problem: ( * )      f − f 2 2 → min Cf = a; C ∈ R d+2×M , f ∈ R M , a ∈ R d+2 . To solve ( * ), one can employ the Lagrange multiplier method. Let λ ∈ R d+2 be the Lagrange multiplier vector. Then the scalar objective function to be optimized is given by i.e. retrieves the constraints. Solving for λ, CC T λ = 2(a − Cf) . L(f, λ) = M j=1 |f j − f j | 2 + λ T (Cf − a) .(3.8) Now CC T is symmetric (CC T ) T = CC T and because C is the integration matrix, CC T is positive definite. By linear algebra, the inverse of CC T exists. In particular one can compute the value of λ by λ = 2(CC T ) −1 (a − Cf) . (3.9) Substituting λ into (3.6), f =f + C T (CC T ) −1 (a − Cf) . (3.10) Using equation for forward Euler scheme (3.4), the complete scheme is given by (f t n (v j ) = f n j ) ∀j: f j = f n j + dtQ(f n j , f n j ) f n+1 j =f j + C T (CC T ) −1 (a − Cf) . (3.11) So, f n+1 j = f n j + dtQ(f n j , f n j ) + C T (CC T ) −1 (a − Cf ) = f n j + dtQ(f n j , f n j ) + C T (CC T ) −1 (a − a − dtCQ(f n j , f n j )) = f n j + dtQ(f n j , f n j ) − dtC T (CC T ) −1 CQ(f n j , f n j ) = f n j + dt[I − C T (CC T ) −1 C]Q(f n j , f n j ) , (3.12) with I -N × N identity matrix. Letting Λ N (C) = I − C T (CC T ) −1 C with I - N × N identity matrix, one obtains f n+1 j = f n j + dtΛ N (C)Q(f n j , f n j ) ,(3.13) where we expect the required observables are conserved and the solution approaches a stationary state, since lim n→∞ Λ N (C) Q(f n j , f n j ) ∞ = 0 . Identity (3.13) summarizes the whole conservation process. As described previously, setting the conservation properties as constraints to a Lagrange multiplier optimization problem ensures that the required observables are conserved. Also, the optimization method can be extended to have the distribution function satisfy more (higher order) moments from (2.8). In this case, a(t) will include entries of m n (t) from (5.1). We point out that for the linear Boltzmann collision operator used in the mixture problem conserves density and not momentum(unless one computes isotropic solutions) and energy. For this problem, the constraint would just be the density equation. For inelastic collisions, density and momentum are conserved and for this case the constraint would be the energy and momentum equations. And for the elastic Boltzmann operator, all three quantities (density, momentum and energy) are conserved and thus they become the constraints for the optimization problem. The behavior of the conservation correction for Pseudo-Maxwell Potentials for Elastic collisions will be numerically studied in the numerical results section. This approach of using Lagrangian constraints in order to secure moment preservation differs from the one proposed in [27], [28] for spectral solvers. 4 Self-Similar asymptotics for a general elastic or inelastic BTE of Maxwell type or the cold thermostat problem -power law tails As mentioned in introduction, a new interesting benchmark problem for our scheme is that of a dynamically scaled solutions or self-similar asymptotics. More precisely, we present simulation where the computed solution in properly scaled time approaches a self similar solution. This is of interest because of the power tail behavior i.e. higher order moments of the computed solution are bounded. For the completeness of this presentation, the analytical description of such asymptotics is given in the following two sub sections. Self-Similar Solution for a non-negative Thermostat Temperature We consider the Maxwell type equation from (2.18) related to a space homogeneous model for a weakly coupled mixture modeling slowdown process. The content of this section is dealt in detail in [14] for a particular choice of zero background temperature (cold thermostat). For the sake of brevity, we refer to [14] for details. However, a slightly more general form of the self-similar solution for non zero background temperature is derived here from the zero background temperature solution. Without loss of generality for our numerical test, we assume the differential cross sections b L for collision kernel of the linear and b N , the corresponding one for the nonlinear part, are the same, both denoted by b( k.σ |k| ), satisfying the Grad cut-off conditions (2.3). In particular, condition (2.20) is automatically satisfied. In [14], Fourier transform of the isotropic self-similar solution associated to problem in (2.18) will take the form: φ(x, t) = ψ(xe −µt ) = 1 − a(xe −µt ) p , as xe −µt → 0, with p ≤ 1 , (4.1) where x = |ζ| 2 /2 and µ and Θ are related by µ = 2 3p 2 and Θ = (3p + 1)(2 − p) 3p 2 . Note that p = 1 corresponds to initial states with finite energy. It was shown in [14] for T = 0 (i.e. cold thermostat), the Fourier transform of the self-similar, isotropic solutions of (2.18) is given by φ(x, t) = 4 π ∞ 0 1 (1 + s 2 ) 2 e −xe −2t 3 as 2 ds ,(4.2) and its corresponding inverse Fourier transform, both for p = 1, µ = 2 3 and Θ = 4 3 (as computed in [14]) is given by f ss 0 (|v|, t) = e t F 0 (|v|e t/3 ) with F 0 (|v|) = 4 π ∞ 0 1 (1 + s 2 ) 2 e −|v| 2 /2s 2 (2πs 2 ) 3 2 ds. (4.3) Remark: It is interesting to observe that, as computed originally in [9], for p = 1 3 or p = 1 2 in (4.2) yields Θ = 0, and one can construct explicit solutions to the elastic BTE with infinite initial energy. It is clear now that in order to have self-similar explicit solutions with finite energy one needs to have this weakly couple mixture model for slowdown processes, or bluntly speaking the linear collisional term added to the elastic energy conservative operator. Finally, in order to recover the self-similar solution for the original equilibrium positive temperature T (i.e. hot thermostat case) for the linear collisional term, we denote, including time dependence for convenience, φ 0 (x, t) = φ(x, t) T hermostat=0 and φ T (x, t) = φ(x, t) T hermostat=T so that φ T (x, t) = φ 0 (x, t)e −T x . (4.4) Note that the solution constructed in (4.2) is actually φ 0 (x, t). Then the selfsimilar solution for non zero background temperature, denoted by φ T (x, t) satisfies In particular, letT = e −2t/3 as 2 + T then, taking the inverse Fourier Transform, we obtain the corresponding self-similar state, according to (2.15), in probability space 3 2 ds. (4.6) Then, letting t → ∞, sinceT = T + as 2 e −2t 3 → T , yields f ss T (|v|, t) = e t F T (|v|e t/3 ) with F T (|v|) = 4 π ∞ 0 1 (1 + s 2 ) 2 e −|v| 2 /2T (2πT )F T (|v|) → t→∞ 4 π 1 (2πT ) 3 2 e −|v| 2 /2T ∞ 0 1 (1 + s 2 ) 2 ds = M T (v) , (4.7) since 4 π ∞ 0 1 (1 + s 2 ) 2 ds = 2 π s 1 + s 2 + arctan(s) | ∞ 0 = 1. (4.8) So, the self-similar particle distribution f ss T (v, t) approaches a rescaled Maxwellian distribution with the background temperature T , that is according to (2.15), Remark: As pointed out in the previous remark, such asymptotic behavior, for finite initial energy, is due to the balance of the binary term and the linear collisional term in (2.18). f ss T (|v|, t) = e t F T (|v|e t/3 ) ≈ e t (2πT ) In addition, very interesting behavior is seen on F T (|v|) as T → 0 (cold thermostat problem), where the particle distribution approaches a distribution with power-like tails (i.e. a power law decay for large values of |v|) and an integral singularity at the origin. Indeed, in [14] an asymptotic behavior is derived for F 0 (|v|) from In particular the self-similar particle distribution function F (|v|), v ∈ R 3 , behaves like 1 |v| 6 as |v| → ∞, and as 1 |v| 2 as |v| → 0, which indicates a very anomalous, non-equilibrium behavior as function of velocity; but, nevertheless, remains with finite mass and kinetic temperature. This asymptotic effect can be described as an overpopulated (with respect to Maxwellian), large energy tails and infinitely many particles at zero energy. This interesting, unusual behavior is observed in problems of soft condensed matter [37]. We shall see, then in the following section, that our solver captures these states with spectral accuracy and consequently the self similar solutions are attractors for a large class of initial states. These numerical tests are a crucial aspect of the spectral Lagrangian deterministic solver used to simulate this type of non-equilibrium phenomena, where all these explicit formulas for our probability distributions allow us to carefully benchmark the proposed numerical scheme. Self-Similar asymptotics for a general problem The self-similar nature of the solutions F (|v|) for a general class of problems, for a wide range of values for the parameters β, p, µ and Θ, was addressed in [12] with much detail. Three different behaviors have been clearly explained. Of particular interest for our present numerical study are the mixture problem with a cold background and the inelastic Boltzmann cases. Interested readers are referred to [12]. For the purpose of our presentation, let φ = F [f ] be the Fourier transform of the probability distribution function satisfying the initial value problem (2.1)-(2.2) or (2.9). Let's denote by Γ(φ) = F [Q + (f, f )] the Fourier transform of the gain part of the collisional term associated with the initial value problem. It was shown in that the operator Γ(φ), defined over the Banach space of continuous bounded functions with the L ∞ -norm (i.e. the space of characteristic functions, that is the space of Fourier transforms of probability distributions), satisfies the following three properties [12]: 1 -Γ(φ) preserves the unit ball in the Banach space. 2 -Γ(φ) is L-Lipschitz operator, i.e. there exists a bounded linear operator L in the Banach space, such that |Γ(u 1 ) − Γ(u 2 )|(x, t) ≤ L(|u 1 − u 2 |(x, t)), ∀ u i ≤ 1; i = 1, 2 . -Γ(φ) is invariant under transformations (dilations) e τ D Γ(u) = Γ(e τ D u) , D = x ∂ ∂x , e τ D u(x) = u(xe τ ), τ ∈ R + . (4.11) In the particular case of the initial value problem associated to Boltzmann type of equations for Maxwell type of interactions, the bounded linear operator that satisfies property 2, is the one that linearizes the Fourier transform of the gain operator about the state u = 1. Next, let x p be the eigenfunction corresponding to the eigenvalue λ(p) of the linear operator L associated to Γ, i.e. L(x p ) = λ(p)x p . Define the spectral function associated to Γ given by µ(p) = λ(p)−1 p defined for p > 0. It was shown in [12] that µ(0+) = +∞ (i.e. p = 0 is a vertical asymptote) and that for the problems associated to the initial value problems (2.1)-(2.2) or (2.9), there exists a unique minimum for µ(p) localized at p 0 > 1, and that µ(p) → 0 − as p → +∞. Then, the existence of self-similar states and convergence of the solution to the initial value problem to such self-similar distribution function was described in [12] in the following four statements: u 0 + µ(p) x p u ′ 0 = Γ(u 0 ) + O(x p+ǫ ), such that p + ǫ < p 0 , (4.12) (i.e. µ(p); µ ′ (p) < 0). Then, there exists a unique, non-negative, self-similar solution f ss (|v|, t) = e − d 2 µ(p)t F p (|v|e − 1 2 µ(p)t ) , with F (F p (|v|)) = w(x), x = |k| 2 /2 s.t. µ(p)x p w ′ (x) + w(x) = Γ(w). (iii) Self similar asymptotics -Section 9 and Theorem 11.1 in [12]: There exists a unique (in the class of probability measures) solution f (|v|, t) satisfying f (|v|, 0) = f 0 (|v|) ≥ 0, with R d f (|v|)dv = 1, such that, for x = |k| 2 2 , and F [f 0 (|v|)] = 1 − a x p + O(x p+ǫ ), x → 0, 0 ≤ p ≤ 1 with p + ǫ < p 0 . Then, for any given 0 ≤ p ≤ 1, there exists a unique non-negative self-similar where µ(p) is the value of spectral function associated to the linear bounded operator L as described above. i.e. m q = R d F p (|v|)|v| 2q dv ≤ ∞; 0 ≤ q ≤ p. However, if p = 1 (finite energy case) then, the boundedness of moments of any order larger than 1, depend on the conjugate value of µ(1) by the spectral function µ(p). That means m q ≤ ∞ only for 0 ≤ q ≤ p * , where p * ≥ p 0 > 1 is the unique maximal root of the equation µ(p * ) = µ(1). solution f (p) ss (|v|, t) = e − d 2 µ(p)t F p (|v|e − 1 2 µ(p)t ) such that f (|v|, t) → t→∞ e − d 2 µ(p)t F p (|v|e − 1 2 µ(p)t ) . Remark 1: When p = 1, µ(1) is the energy dissipation rate, and E(t) = e µ(1)t the kinetic energy evolution function. So, E(t) d/2 f (vE(t), t) → F 1 (|v|). Remark 2: We point out that condition (4.12) on the initial state is easily satisfied by taking a sufficiently concentrated Maxwellian distribution as shown in [12], and as done for our simulations in the next section. However, rescaling with a different rate, it is not possible to pick up the nontrivial limiting state f ss , since f (|v|e 1 2 ηt , t) → t→∞ e − d 2 ηt δ 0 (|v|); η > µ(1) ,(4.15) and f (|v|e 1 2 ηt , t) → t→∞ 0; µ(p min ) < µ(1 + δ) < η < µ(1) . (4.16) These results are also true for any p ≤ 1. For the general space homogeneous (elastic or inelastic) Boltzmann model of Maxwell type or the corresponding mixture problem, the spectral function µ(p) is given in Figure 1. Numerical Results We benchmark the new proposed numerical method to compute several examples of 3 − D in velocity and time for initial value problems associated with non-conservative models where some analysis is available, as are exact moment formulas for Maxwell type of interactions as well as qualitative analysis for solutions of VHS models. We shall plot our numerical results versus the exact available solutions in several cases. Because all the computed problems converge to an isotropic long time state, we choose to plot the distribution function in only one direction, which is chosen to be the one with the initial anisotropies. All examples considered in this manuscript are assumed to have isotropic, VHS collision kernels, i.e. differential cross section independent from scattering angle. We simulate the homogeneous problem associated to the following problems for different choices of the parameters β and λ, and the Jacobian J β and heating force term G(f ). Maxwell type of Elastic Collisions Consider the initial value problem (2.1), (2.2), with B(|u|, µ) = 1 4π |u| λ . In (2.1), (2.2), the value of the parameters are β = 1, J β = 1 and λ = 0 with the pre-collision velocities defined from (2.2). In this case, for a general initial state with finite mass, mean and kinetic energy, there is no exact expression for the evolving distribution function. However there are exact expressions for all the statistical moments (observables). Thus, the numerical method is compared with the known analytical moments for different discretizations in the velocity space. The initial states we take are convex combinations of two shifted Maxwellian distributions. So consider the following case of initial states with unit mass R 3 f 0 (v)dv = 1 given by convex combinations of shifted Maxwellians f (v, 0) = f 0 (t) = γM T 1 (v − V 1 ) + (1 − γ)M T 2 (v − V 2 ); with 0 γ 1 where M T (v − V ) = 1 (2πT ) 3/2 e −|v−V | 2 (2T ) . Then, taking γ = 0.5 and mean fields for the initial state determined by V 1 = [−2, 2, 0] T , V 2 = [2, 0, 0] T ; T 1 = 1 , T 2 = 1 , enables the first five moment equations corresponding to the collision invariants to be computed from those of the initial state. All higher order moments are computed using the classical moments recursion formulas for Maxwell type of interactions (2.8). In particular, it is possible to obtain the exact evolution of moments as functions of time. Thus ρ(t) = ρ 0 = 1 and V (t) = V 0 = [0, 1, 0] T . By a corresponding moment calculation as in (2.8), the complete evolution of the second moment tensor (2.7) is given by M(t) =        5 −2 0 −2 3 0 0 0 1        e −t/2 + 1 3        8 0 0 0 11 0 0 0 8        (1 − e −t/2 ) , and the energy flow (2.7) r(t) = 1 2        −4 13 0        e −t/3 + 1 6        0 43 0        (1 − e −t/3 ) − 1 6        12 4 0        (e −t/2 − e −t/3 ) , and the kinetic temperature is conserved, so T (t) = T 0 = 8 3 . (5.1) The above moments along with their numerical approximations for different discretizations in velocity space are plotted in Figures 5.1. In Figure 3, the evolution of the computed distribution function into a Maxwellian is plotted for N = 40. In order to check the conservation accuracy of the method, let f u -unconserved distribution given as input to the conservation routine and f cconserved distribution resulting from the conservation routine. With a convex combination of two Gaussians as input, the numerical method is allowed to run and f c − f u ∞ is plotted for all times for different values of N in figure 4. As expected, for t approaching the final time, the largest value of N gives the smallest conservation correction. Maxwell type of Elastic collisions -Bobylev-Krook-Wu (BKW) Solution An explicit solution to the initial value problem (2.1) for elastic, Maxwell type of interactions (β = 1, λ = 0) was derived in [3] and independently in [43] for initial states that have at least 2 + δ-moments bounded. It is not of self-similar type, but it can be shown to converge to a Maxwellian distribution. This solution takes the form where K = 1 − e −t/6 and η =initial distribution temperature. It is interesting that it is negative for small values of t. So in order to obtained a physically meaning probability distribution, f must be non-negative. This is indeed the case for any K 3 5 or t t 0 ≡ 6ln( 5 f (v, t) = e −|v| 2 /(2Kη 2 ) 2(2πKη 2 ) 3/2 ( 5K − 3 K + 1 − K K 2 |v| 2 η 2 ) ,(5. Hard-Sphere Elastic Collisions In (2.1), (2.2), we have β = 1, J β = 1 and λ = 1 with the post-collision velocities defined from (2.1). Unlike Maxwell type of interactions, there is no explicit expression for the moment equations and neither is there any explicit solution expression as in the BKW solution scenario. For Hard Sphere isotropic collisions, the expected behavior of the moments is similar to that of the Maxwell type of interactions case except that in this case, the moments somewhat evolve to the equilibrium a bit faster than in the former case i.e. figure 6. Also plotted is the time evolution of the distribution function starting from the convex combination of Maxwellians as described in a previous subsection in Figure 7. Inelastic Collisions This is the scenario wherein the utility of the proposed method is clearly seen. can obtain the ODE governing the evolution of the kinetic energy K(t) K ′ (t) = β(1 − β)( |V | 2 2 − K(t)) ,(5. 3) where V -conserved (constant) bulk velocity of the distribution function. This gives the following solution for the kinetic energy as computed in (2.8) K(t) = K(0)e −β(1−β)t + |V | 2 2 (1 − e −β(1−β)t ) ,(5.4) where K(0) = kinetic energy at time t = 0. As we have an explicit expression for the kinetic energy evolving in time, this analytical moment can be compared with its numerical approximation for accuracy and the corresponding graph is given in Figure 8. Also the general evolution of the distribution in an inelastic collision environment is also shown in Figure 8. In the conservation routine (constrained Lagrange multiplier method), energy is not used as a constraint and just density and momentum equations are used for constraints. Figure 8 shows the numerical accuracy of the method even though the energy (plotted quantity) is not being conserved as part of the constrained optimization method. Inelastic Collisions with Diffusion Term Here we simulate, the equations (2.9), (2.10). Here we simulate a model corresponding to inelastic interactions in a randomly excited heat bath with constant temperature η. The evolution equation for kinetic temperature as a function of time is given by: The above equation gives a closed form expression for the time evolution of the kinetic temperature and can be expressed as follows: dT dt = 2η − ζ 1 − e 2 24 v∈R 3 w∈R 3 σ∈S 2 (1 − µ)B(|u|, µ)|u| 2 f (v)f (w) dσdwdv ,T (t) = T 0 e −ζπC 0 (1−e 2 )t + T M M ∞ [1 − e −ζπC 0 (1−e 2 )t ] ,(5.7) where T 0 = 1 3 v∈R 3 |v| 2 f (v)dv and T M M ∞ = 2η ζπC 0 (1 − e 2 ) . As it can be seen from the expression for T, in the absence of the diffusion term (i.e. η = 0) and for e = 1 (inelastic collisions), the kinetic temperature of the distribution function decays like an exponential just like in the previous section. So, the presence of the diffusion term pushes the temperature to an equilibrium value of T M M ∞ > 0 even in the case of inelastic collisions. Also note that if the interactions were elastic and the diffusion coefficient positive then, T M M ∞ = +∞, so there would be no equilibrium states with finite kinetic temperature. These properties were shown in [34] and similar time asymptotic behavior is expected in the case of hard-sphere interactions where T HS ∞ > 0 is shown to exist. However, the time evolution of the kinetic temperature is a non-local integral (5.5) does not satisfy a close ODE form (5.6). The proposed numerical method for the calculation of the collision integral is tested for these two cases. We compared with the analytical expression (5.7) for different initial data, the corresponding computed kinetic temperatures for Maxwell type interactions in Figure 9. The asymptotic behavior is observed in the case of hard-sphere interactions in Figure 10. The conservation properties for this case of inelastic collisions with a diffusion term are set exactly like in the previous subsection (inelastic collisions without the diffusion term). (T M M ∞ > T 0 ) (T M M ∞ < T 0 ) Maxwell type of Elastic Collisions -Slow down process problem Next, consider (2.18) with β = 1, J β = 1 and B(|u|, µ) = 1 4π i.e. isotropic collisions. The second term is the linear collision integral which conserves only density and the the first term is the classical collision integral from 1 (2πT ) 3/2 , with T the constant thermostat temperature. In particular, any initial distribution function converges to the background distribution M T . Such behavior is well captured by the numerical method. Indeed, Fig. 11 corresponds to an initial state of a convex combination of two Maxwellians. In addition, from (4.6): f ss T (v, t) = (2) π 5/2 ∞ 0 1 (1 + s 2 ) 2 e −|v| 2 /2T T 3 2 dsT = T + as 2 e −2t 3 , which is the finite energy solution for p = 1, a = 1, µ = 2 3 , θ = 4 3 in (4.2), i.e. p = 1 in (4.13) and (4.14). As t → ∞, the time rescaled numerical distribution is compared with the analytical solution f ss T for a positive background temperature T and it converges to a Maxwellian M T . From Figure 11, it can be seen that the numerical method is quite accurate and the computed distribution is in very good agreement with the analytical self-similar distribution f ss T from (4.6). Similar agreement is observed for different constant values of T approaching 0 ( Figure 11). The interesting asymptotics (4.10) corresponding to power-like tails and infinitely many particles at zero energies occur only when T = 0 as shown in (4.10) and (4.10). Since letting T = 0 in the scheme created an instability, we proposed the following new methodology to counter this effect. We let, instead, T = ζe −αt ensuring that the thermostat temperature vanishes for large time and set T = ζe −αt + as 2 e −2t 3 ,(5.8) where the role of α is very important and a proper choice needs to be made. In our simulations, we take ζ = 0.25 and the values of α need to be chosen exactly as α = µ(1) = 2/3, the energy dissipation rate as described in section 4.2 to recover the asymptotics as in (4.10). Remark: Due to the exponential time rescaling of Fourier modes, our procedure to compute self-similar solutions in free space may also be viewed as a non-uniform grid of Fourier modes that are distributed according to the continuum spectrum of the associated problem. This choice plays the equivalent role to the corresponding spectral approximation of the free space problem of the heat kernel, that is, the Green's function for the heat equation, which happens to be a similarity solution as well, due to the linearity of the problem in this case. In particular, we expect optimal algorithm complexity using such non-equispaced Fast Fourier Transform, as obtained by Greengard and Lin [38] for spectral approximation of the free space heat kernel. This problem will be addressed in a forthcoming paper. The following plots elucidate the fact that power-like tails are achieved asymptotically with a decaying T . For a decaying background temperature as in (5.8), Figure 12 shows evolution of a convex combination of Maxwellians to a self-similar (blow up for zero energies and power-like for high energies) behavior. Figure 13 plots the computed distribution along with a Maxwellian with temperature of the computed solution. This illustrates that the computed selfsimilar solution is largely deviated from a Maxwellian equilibrium. In order to better capture the power-like effect using this numerical method, we set T = ζe −2t/3 = ζe −µt , see (5.8), where µ is related the spectral properties of the Fourier transformed equation as described in section 4.2 on the slow down process problem with µ = µ(1) the energy dissipation rate. Thus, as it was computed in [14] and revised in section 4 of this paper, we know that for initial states with finite energy, p = 1 and the corresponding energy dissipation rate is µ(1) = µ = 2/3 is positive. In particular p * = 1.5 is the conjugate of p = 1 of the spectral curve m q in Theorem 4.1 part (i). In addition the rescaled probability will converge to the moments of the self-similar state (4.13), (4.14), that is e −qt2/3 v∈R 3 f (v)|v| 2q dv → m q , and we know any moment m q is unbounded for q > p * = 1.5. We have plot in Fig. 14 the evolution of e −qt2/3 v∈R 3 f (v)|v| 2q dv for q = 1, 1.3, 1.45, 1.5, 1.55, 1.7, 2.0, computed for different values of N = 10, 14, 16, 18, 22, 26. It can be seen that as time progresses (and as the thermostat temperature T decreases to 0), the approximated numerically computed moments to m q , q ≥ 1.5 start to blow up as predicted. The value q = 1.5 is the threshold value, as any moment m q>1.5 (t) → ∞. The expected spectral accuracy, as the value of N increases, improves the growth zone of such moments for larger final times. The reason for such effect is that since the truncation of Fourier modes that results in the truncation in velocity domain, makes the distribution function to take small negative values for large velocities contributing to numerical errors that may cause m q to peak and then relax back. In particular, larger order moments of the computed self-similar asymptotics with the negative oscillating parts on large energy tails, result in the large negative moment values for the above mentioned values of N crating large negative errors. However it is noticed that the negative oscillation values of f (t, v) coincide with large velocity values used in getting its q-moments approximating m q , for q > 1.5, and that such error is reduced in time for larger number of Fourier modes. Finally we point out that a FFTW package has been used. We have noticed in our numerics that for the specific choice of values N = 6, 10, 14, 18, 22, 26, ..., 6 + 4k; k = 0, 1, 2, 3, ..., the approximating moments to m q (t) start to take negative values very quickly, as seen in Fig. 15 for N = 16 and 20, making the numerical solution inadmissible since analytically m q (t) > 0, ∀t. Such effect may be due to the particular choice of the FTTW solver. Conclusions and Future Work In conclusion, the presented numerical method works for elastic and inelastic variable hard potential interactions. This is first of its kind as no additional modification is required to compute for elastic and inelastic collisions. In comparison with the known analytical results (moment equations for elastic BTE, BKW self-similar solution, attracting Bobylev-Cercignani-Gamba self-similar solutions for elastic collisions in a slow down process), the computed ones are found to be very close. The method employs a Fast Fourier Transform for faster evaluation of the collision integral. Even though the method is implemented for a uniform grid in velocity space, it can even be implemented for a nonuniform velocity grid. The only challenge in this case is computing the Fast Fourier Transform on such a non-uniform grid. There are available packages for this purpose, but such a non-uniform FFT can also be implemented using certain high degree polynomial interpolation and this possibility is currently being explored. The integration over the unit sphere is avoided completely and only a simple integration over a regular velocity grid is needed. Even though a trapezoidal rule is used as an integration rule, other integration rules like a Gaussian quadrature can be used to get better accuracy. For time discretization, a simple second-order Runge Kutta scheme is used. The proposed method has a big advantage over other non-deterministic methods as the exact distribution function can actually be computed instead of just the averages. Implementations of this scheme for the space inhomogeneous case is currently developed by the authors by means of splitting algorithms in advection and collision components. Next step in this direction would be to implement the method for a practical 1 and 2 − D space inhomogeneous problems such shock tube phenomena for specular and diffusive boundary conditions, and Rayleigh-Benard instability or a Couette flow problem. To remedy this, a simple constrained Lagrange multiplier method is employed where the constraints are the required conservation properties. Let M = N d , the total number of discretizations of the velocity space. Assume that the classical Boltzmann collision operator is being computed. So ρ, m = (m1, m2, m3) and e are conserved. Let ω j be the integration weights where j = 1, 2, ..., M. Let f = f 1f2 . .f M T be the distribution vector at the computed time step and f = f 1 f 2 . .f M T be the corrected distribution vector with the required moments conserved. Let can actually be solved explicitly for the corrected distribution value and the resulting equation of correction be implemented numerically in the code. Taking the derivative of L(f, λ) with respect to f j , j = 1, ..., M and λ i , i = 1, ..., d + 2 i.e. gradients of L, ∂L ∂f j = 0; j = 1, ... e −(|v| 2 e 2t/3 )/2T +t , as t → ∞ .(4.9) F (4.3), for large and small values of |v|, leading to (2 [1 + 2|v| 2 ln(|v|) + O(|v| 2 )], for |v| → 0. (4.10) (i) [12] -Lemma 5.1 (existence): There exists a unique isotropic solution f (|v|, t) to the initial value problem (2.1)-(2.2) or (2.9) for Maxwell type interactions, in the class of probability measures, satisfying f (|v|, 0) = f 0 (|v|) ≥ 0, R d f 0 (|v|)dv = 1 such that for the Fourier transform problem x = |k| 2 2 , u 0 = F [f 0 (|v|)] = 1 + O(x), as x → 0, (ii) Self similar states -Theorem 7.2: f (|v|, t) has self-similar asymptotics in the following sense: Taking the Fourier transform of the initial state to satisfy (p)t , t) → t→∞ F p (|v|) , (4.14) Fig. 1 . 1Spectral Function µ(p) for a general homogeneous Boltzmann collisional problem of Maxwell type (iv) Power tail behavior of the asymptotic limit: If µ(p) < 0, then the selfsimilar limiting function F p (|v|) does not have finite moments of all orders. In addition, if 0 ≤ p ≤ 1 then all moments of order less than p are bounded; Fig. 2 . 2Maxwell type of Elastic collisions: Momentum Flow M 11 , M 12 , M 22 , M 33 , Energy Flow r 1 , r 2 Fig. 3 . 3Maxwell type of Elastic collisions: Evolution of the Distribution functionFig. 4. Maxwell type Elastic Collisions: Conservation Correction for Elastic Collisions Fig. 5 . 5BKW, ρ, E(t) conserved Fig. 6 . 6No other deterministic method can compute the distribution function in the case of inelastic collisions (isotropic), but the current method computed this 3−D evolution without much complication and with the exactly same number of operations as used in an elastic collision case. This model works for all sorts of Variable Hard Potential interactions. Consider the special case of Maxwell (λ = 0) type of inelastic (β = 1) collisions in a space homogeneous Boltzmann Equation in (2.1), (2.2). Let φ(v) = |v| 2 be a smooth enough test function. Using the weak form of the Boltzmann equation with such a test function one Hard Sphere, Elastic: Momentum Flow M 11 , M 12 , M 22 , M 33 , Energy Flow r 1 , r 2 Fig. 7 . 7Hard-Sphere, Elastic: Evolution of the Distribution function, N = 32 Fig. 8 . 8Inelastic: Kinetic Energy (left) & f (v, t) (right)which, in the case of inelastic Maxwell type of interactions according to (2.8),(5.5) becomes dT dt = 2η − ζπC 0 (1 − e 2 )T . (5.6) Fig. 9 . 16 Fig. 10 . 91610Maxwell type of Inelastic collisions, Diffusion Term for N = Hard-Sphere, Inelastic Collisions, Diffusion Term, T HS ∞ < T 0 for N = 16 125 Fig. 11 . 12511. f ss T : T = 0.25 Computed Vs. f ss T : T = 0.Maxwell type collisions, Slow down process with Θ = 4/3, µ = 2/3, N = 24 Fig. 12 .Fig. 13 . 1213Slow Computed distribution Vs. Maxwellian with temperature of the computed distribution Fig. 14 . 14m q (t) for T = e −2t/3 Fig. 15 . 15m q (t) for T = e −2t/3 ) ∼ 5.498. In order to test the accuracy of our solver, set the initial distribution function to be the BKW solution, the numerical approximation to the BKW solution and the exact solution are plotted for different values of N at various time steps inFigure 5. AcknowledgementsThe authors would like to thank Sergej Rjasanow for discussions about the conservation properties of the numerical method and for other comments. Both authors are partially supported under the NSF grant DMS-0507038. Support from the Institute of Computational Engineering and Sciences and the University of Texas Austin is also gratefully acknowledged. Propagation of l 1 and l ∞ maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic boltzmann equation. R J Alonso, I M Gamba, Journal of Mathematiques Pures et Appliquées. To appear inR. J. Alonso, I. M. 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The d-wave superconductor CeCoIn 5 has been proposed as a strong candidate for supporting the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state near the low-temperature boundary of its upper critical field. Neutron diffraction, however, finds spin-density wave (SDW) order in this part of the phase diagram for field in the a-b plane, and evidence for the SDW disappears as the applied field is rotated toward the tetragonal c axis. It is important to understand the interplay between the SDW and a possible FFLO state in CeCoIn 5 , as the mere existence of an SDW does not necessary exclude an FFLO state. Here, based on a model constructed on the basis of available experiments, we show that an FFLO state competes with an SDW phase. The SDW state in CeCoIn 5 is stabilized when the field is directed close to the a-b plane. When the field is rotated toward the c axis, the FFLO state emerges, and the SDW phase disappears. In the FFLO state, the nodal planes with extra quasiparticles (where the superconducting order parameter is zero) are perpendicular to the field, and in the SDW phase, the quasiparticle density of states is reduced. We test this model prediction by measuring heat transported by normal quasiparticles in the superconducting state. As a function of field, we observe a reduction of thermal conductivity for field close to the a-b plane and an enhancement of thermal conductivity when field is close to the c axis, consistent with theoretical expectations. Our modeling and experiments, therefore, indicate the existence of the FFLO state when field is parallel to the c axis. arXiv:1902.04797v1 [cond-mat.supr-con]

10.1103/physrevlett.124.217001

1902.04797

819c1cb26fdc10619c991e5ccf52d9e8acb35da6

Interplay of the spin density wave and a possible Fulde-Ferrell-Larkin-Ovchinnikov state in CeCoIn 5 in rotating magnetic field (Dated: February 14, 2019) Shi-Zeng Lin Theoretical Division CNLS Los Alamos National Laboratory T-4, 87545Los AlamosNew MexicoUSA Duk Y Kim MPA-CMMS Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Center for Integrated Nanostructure Physics Institute for Basic Science 16419SuwonRepublic of Korea Eric D Bauer MPA-CMMS Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Filip Ronning MPA-CMMS Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA J D Thompson MPA-CMMS Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Roman Movshovich MPA-CMMS Los Alamos National Laboratory 87545Los AlamosNew MexicoUSA Interplay of the spin density wave and a possible Fulde-Ferrell-Larkin-Ovchinnikov state in CeCoIn 5 in rotating magnetic field (Dated: February 14, 2019) The d-wave superconductor CeCoIn 5 has been proposed as a strong candidate for supporting the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state near the low-temperature boundary of its upper critical field. Neutron diffraction, however, finds spin-density wave (SDW) order in this part of the phase diagram for field in the a-b plane, and evidence for the SDW disappears as the applied field is rotated toward the tetragonal c axis. It is important to understand the interplay between the SDW and a possible FFLO state in CeCoIn 5 , as the mere existence of an SDW does not necessary exclude an FFLO state. Here, based on a model constructed on the basis of available experiments, we show that an FFLO state competes with an SDW phase. The SDW state in CeCoIn 5 is stabilized when the field is directed close to the a-b plane. When the field is rotated toward the c axis, the FFLO state emerges, and the SDW phase disappears. In the FFLO state, the nodal planes with extra quasiparticles (where the superconducting order parameter is zero) are perpendicular to the field, and in the SDW phase, the quasiparticle density of states is reduced. We test this model prediction by measuring heat transported by normal quasiparticles in the superconducting state. As a function of field, we observe a reduction of thermal conductivity for field close to the a-b plane and an enhancement of thermal conductivity when field is close to the c axis, consistent with theoretical expectations. Our modeling and experiments, therefore, indicate the existence of the FFLO state when field is parallel to the c axis. arXiv:1902.04797v1 [cond-mat.supr-con] Introduction.-A key property of a superconductor is how it responds to an external magnetic field. Generally, magnetic field suppresses superconductivity. For superconductors with singlet Cooper pairing, magnetic field destroys superconductivity in two ways. Firstly, the Lorentz force that an external magnetic field exerts on the two electrons of a Cooper pair have opposite directions. These forces tear a Cooper pair apart, thereby suppressing superconductivity via orbital limiting. Secondly, a magnetic field tends to polarize electron spins via Zeeman coupling, reducing the electrons' energy and leading to an enhanced Pauli susceptibility. Under certain circumstances, superconductivity can be destroyed by this mechanism of Pauli limiting even when Cooper pairs form with electrons of opposite spin alignment. While the maximal magnetic field that most known superconductors can sustain is defined by orbital limiting, some superconductors have been identified whose upper critical field is determined largely by Pauli limiting. These superconductors also can stabilize a spatially modulated superconducting state, known as the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state, before superconductivity is suppressed entirely [1,2]. In the FFLO state, Cooper pairs acquire a nonzero momentum, and the superconducting order parameter vanishes locally in space along nodal planes. This results in an excess of quasiparticles, which can significantly modify physical properties of the superconductors. Theoretically, it is clear that the formation of an FFLO state is unavoidable when a clean Pauli limited superconductor is subjected to a strong magnetic field. However, the experimental detection of an FFLO state remains a challenge despite some encouraging experimental evidence [3][4][5][6][7][8][9][10][11][12][13], mainly because the superconducting order parameter cannot be measured directly. Often, competing effects render the interpretation of the ex-perimental data difficult. The discovery of the heavy-fermion superconductor CeCoIn 5 with tetragonal crystal structure and a transition temperature T c = 2.3 K has provided an exciting playground to search for the FFLO state. High quality CeCoIn 5 with a large electron mean free path l ≈ 10ξ can be achieved, where ξ is the superconducting coherence length. Various measurements have revealed that the upper critical field in CeCoIn 5 is mainly limited by the Pauli mechanism. Initial experimental measurements have shown the existence of a new phase inside the superconducting phase for field both along the crystal c and a axis, see Ref. 14 for a review. For instance, a double-peak structure has been observed in NMR spectra in the high-field and low-temperature corner of the superconducting phase diagram [4]. In the case of H a, neutron-scattering measurements have identified this new phase as a spin-density-wave (SDW) state that coexists with the superconducting state. In the SDW, magnetic moments with magnitude 0.1 µ B (µ B is the Bohr magneton) are aligned along the crystalline c axis due to crystal field effects. A similar SDW is observed in the slightly Hg-doped CeCoIn 5 [15]. This SDW is induced by magnetic field and disappears when superconductivity is destroyed by field. The SDW phase is suppressed (eventually entirely) when the field is rotated away from the a-b plane [16]. The double peak structure in NMR spectra for H a can be explained in terms of the SDW phase. However the state responsible for similar NMR features for H c requires further study. The appearance of an SDW only inside the superconducting phase comes as a surprise, because it is generally believed that SDW competes with superconductivity for the density of states at the Fermi surface. Several scenarios have been put forward to account for the stabilization of the SDW. These scenarios highlight the importance of the vortex lattice [17], Pauli pair breaking [18,19], the FFLO state [20,21], a superconducting pairing density wave [22][23][24], and Fermi surface nesting improved by the magnetic field [25][26][27]. It was shown in Ref. 28 that an in-plane magnetic field enhances the transverse magnetic susceptibility in Pauli-limited d-wave superconductors, which can result in a divergence of the dressed susceptibility. As a consequence, the magnetic fluctuations condense and static SDW order sets in. This magnon condensation picture is favored by inelastic neutron-scattering measurements [29][30][31]. The emergence of an SDW does not rule out the existence of an FFLO state. In the case when the SDW competes with the FFLO state, one can suppress the SDW phase by rotating the field out of the a-b plane (as is the case in CeCoIn 5 ), which in turn can favor the FFLO state. This requires understanding the interplay between the SDW and the FFLO states, with a proper model that is relevant for CeCoIn 5 . In this Letter, we study the relation between SDW and FFLO states using a theoretical model described below. We find that the SDW competes with the FFLO phase. When H a, an SDW phase emerges inside the superconducting phase due to magnon condensation triggered by the magnetic field. When H is rotated towards the c axis, the SDW phase is disfavored and the FFLO state appears. Because of the coupling between superconductivity and magnetism, the SDW order induces weak modulation in the superconducting order parameter and vice versa. Guided by the theory, we performed thermalconductivity measurements in CeCoIn 5 for a magnetic field applied within the a-c plane as a function of angle away from the c-axis. The data are consistent with expectations from the model. The synergy between modeling and experiment suggests the presence of an FFLO phase in CeCoIn 5 when H c. Model.-We construct a model Hamiltonian based on the following experimentally established facts. (1) Various experiments have shown that CeCoIn 5 is close to an SDW instability [29][30][31][32][33][34][35][36], which is consistent with the fact that a field of order of 11 T is sufficient to trigger an SDW instability. (2) The SDW is formed by gapping quasiparticles in nodes of the superconducting d x 2 −y 2 order parameter. The SDW ordering wave vector is Q = (0.44, ±0.44, 0.5). (3) The SDW moments have strong Ising anisotropy and lie along the c axis. (4) The Fermi surface relevant for superconductivity is a warped cylinder. Based on these facts, we can construct the following mean-field Hamiltonian in two dimensions: H = i, j,σ t i j c † iσ c jσ − µ i,σ c † iσ c iσ + i, j (∆ i j c † i↑ c † j↓ + ∆ * i j c j↓ c i↑ ) − i h i s z,i − g ab µ B H x s x,i − g c µ B H z s z,i .(1) where t i j describes electron hopping on a square lattice with the dispersion (k) = 2t 1 [cos(k x ) + cos(k y )] + 4t 2 cos(k x ) cos(k y ) + 2t 3 (NN), the second NN (along the diagonal), and the third NN (along the bond) hopping amplitudes, respectively, and their strengths have been estimated by using density functional theory [37]. The electron density for up and down spins is n i↑ = l |u i↑,l | 2 f (E l ) and n i↓ = l |v i↓,l | 2 f (−E l ), where f (E l ) is the Fermi distribution function and u i↑,l , v i↓,l , E l are the l-th eigen vector and eigen energy of the Bogoliubov-de Gennes equation associated with Eq. (1) [38]. We fix the electronic density at n = 0.72 by tuning the chemical potential µ. (a) (b) (c) (d) The self-consistent equation for the pairing potential ∆ i j and molecular field h i are ∆ i j = V 4 l u i↑,l v * j↓,l + u j↑,l v * i↓,l tanh(E l /2T ) and h i = − j J i j (n j↑ − n j↓ ). The spin of conduction electrons is s i = α β c † iα σ α β c iβ . There is a strong anisotropy in the electron g-factor [39], and we take g c /g ab = 3. Here µ B is the Bohr magneton. To stabilize SDW order at Q, we use J i j = J 1 (NN antiferromagnetic interaction) and the third NN (competing) interaction J 3 = −J 1 /4 cos(2πQ x ). To ensure d-wave pairing symmetry, we choose the NN pairing potential ∆ i j with V = 4.5t 1 in the calculations. The d-wave order parameter is given by ∆ d = (∆ i,i+x + ∆ i,i−x − ∆ i,i+ŷ − ∆ i,i−ŷ )/4, wherex andŷ are the unit vectors in the x and y direction, respectively [40]. Similar models were introduced to describe the emergence of an SDW state in Pauli-limited superconductors in high field [25,26], as well as the phase diagram for Nd-doped CeCoIn 5 [27,41]. This model describes the competition between an SDW and d-wave superconductivity, which depends on values of V and J 1 [42]. When J 1 /t 1 > 3.6, SDW order develops and coexists with superconducting order [41]. To model CeCoIn 5 , the system is tuned to an SDW instability by setting J 1 /t 1 = 3.2. The phase diagram at zero temperature T = 0, when field is rotated in the a-c plane, is displayed in Fig. 1. For H a, the model correctly describes the development of the SDW phase inside the superconducting state at high field. When field is canted towards the c axis, the SDW state is suppressed, and the FFLO state appears [43]. The transition from the uniform superconducting state to the FFLO (SDW) state is of first (second) order, while the transition from the FFLO state (SDW) to the normal state is of second (first) order. The spatial distributions of ∆ d and SDW orders are shown in Fig. 2. In the FFLO state, the modulation of ∆ d induces modulation of the SDW; while in the SDW phase, modulation of SDW order generates weak modulation in ∆ d . Because of the competition between SDW and superconductivity, the maxima in ∆ d corresponds to the minima in the SDW order, and vice versa. In our 2D calculations, the wavevector of the FFLO, Q FFLO , is confined to the plane. For an s-wave superconductor with a vortex lattice, Q FFLO is parallel to the vortex lines and H [44,45]. For d-wave, one would also expect that Q FFLO H, otherwise there would be modulation of the superconducting order parameter with two different periods in the a-b plane. This is energetically disfavored both by Pauli and orbital pair breaking effects. This picture is supported by numerical calculation based on a quasiclassical theory for a single vortex in a d-wave superconductor [46]. For Q FFLO H, the thermal conductivity will be enhanced when thermal current J ⊥ H due to the excess of quasiparticles around nodal planes in the FFLO state. In contrast, some of nodal quasiparticles of the dwave superconductivity are gapped out in the SDW state and one expects a reduction of the thermal conductivity. These the crystal because the electronic mean free path is mainly limited by vortex scattering even with very large inter-vortex distances at low fields. For H close to the a-b plane, the thermal conductivity κ drops sharply with decreasing field just below H c2 . This drop originates from the development of the SDW and concomitant pair-density-wave (PDW) orders which gap out some of d-wave nodal quasiparticles [47] [48]. As field is rotated toward the c axis, the drop decreases, signaling the suppression of SDW order. The threshold angle where this sharp drop disappears is approximately 70 • , which is consistent with previous measurements [16]. For H close to the c axis, a significant increase of κ is observed as H approaches H c2 from below. We ascribe this enhancement to a signature of the FFLO state, in accord with the model calculation. Such enhancement of thermal conductivity below H c2 is not expected for a Pauli-limited superconductor with a first-order superconducting transition. Instead, thermal conductivity below H c2 should be rather field independent up to H c2 , and display a sharp step at H c2 . A contour plot of κ in the complete field and angle range is depicted in Fig. 4. The behavior of κ near H c2 is consistent with the expectation from our model. Discussions and summary.-We stress the importance of Ising-like magnetic anisotropy in the model. The presence of this anisotropy naturally explains why the SDW phase only occurs for field close to the a-b plane within the magnon condensation picture [28]. It also leads to the competition between the SDW and FFLO states, as shown in Fig. 1. Our approach and results are different from a model with isotropic magnetic fluctuations [20], where it was proposed that the SDW phase is stabilized by Andreev bound states localized around FFLO nodal planes. We too observe an induced SDW oscillation in the FFLO state [see Fig. 2], but with a much weaker amplitude compared to that for H [100], where the full SDW develops. In addition, the picture that the SDW state for H [100] is induced by the FFLO state is not supported by recent neutron-scattering measurements, which reveal that the SDW state is induced by closing the magnon gap [29][30][31]. A number of measurements reveal the presence of a quantum critical point around H c2 = 5 T when H [001] [33,49,50]. Strong quantum fluctuations should suppress the thermal conductivity near H c2 . There must exist another mechanism which counters this suppression and leads to the enhancement of κ, as observed experimentally. Model calculations show that κ increases with H near H c2 in orbital-limited superconductors [51]. The orbital-limited critical field H orb c2 in CeCoIn 5 is about three times larger than the Pauli limited critical field H P c2 , and therefore, the experimentally measured superconducting critical field H c2 as well. The effect of orbital limiting on thermal conductivity near H c2 , can be gleaned from calculations for the case of orbital limiting alone around a field of about 1 3 H c2 [52,53]. The variation of thermal conductivity with field is very slow in this region, in contrast to the sharp increase in the thermal conductivity near H c2 shown in Fig. 3. Therefore, orbital limiting cannot be causing the observed increase in κ. The vortex lattice undergoes a number of structural transitions as a function of field strength when H [001], including transition from square to rhombic to triangular between 3 T and H c2 = 5 T [54]. The vortex lattice transition is a gradual process. For instance, the apex angle of a unit cell for triangle vortex lattice is 60 • and that for a square lattice is 90 • . The evolution from triangular to the square lattices corresponds to the continuous change of the apex angle from 60 • to 90 • [55]. The smooth deformation of the vortex lattice, while the vortex lines are kept perpendicular to heat current, is therefore unlikely to give rise to a dramatic increases of κ. The Pauli pairing breaking effect, therefore, should be dominant in CeCoIn 5 at fields near H c2 . While our modeling and experiments suggest the existence of the FFLO state when the field is orientated close to the c axis, two experimental observations deserve further attention. First, the region of enhanced κ near H c2 is wider than that identified as the FFLO region by NMR measurements [4]. However, a similar effect was observed for the field dependence of thermal conductivity in CeCoIn 5 at the SDW transition for field in the a-b plane [47]. There, the onset of the reduction in thermal conductivity at very low temperature was observed to take place at 9 T, whereas specific heat and neutron scattering measurements firmly place the SDW transition at 10 T. This may be due to fluctuations of the order parameter. Second, we are not able to resolve hysteresis in κ near H c2 when the direction of the field sweep is reversed. Such hysteresis is expected from the first order phase transition between the uniform superconducting state and the FFLO state. The lack of hysteresis may be due to a weak first order nature of the transition. To summarize, combined modeling and thermalconductivity measurements suggest the existence of an FFLO state in CeCoIn 5 for field aligned along the c axis. The FFLO state competes with an SDW phase, and their relative stability can be tuned by rotating field in the a-c plane. Additional neutron scattering measurements with a scattering plane that includes the c axis may be promising in resolving the modulated susceptibility caused by the FFLO state. We expect the FFLO modulation wave vector to lie along the c axis when magnetic field is applied along the c axis. We note that the anomalous drop in the vortex lattice form factor, observed by the neutron scattering for field close to H c2 , was ascribed to the formation of an FFLO state [54]. The authors would like to thank Takanori Taniguchi and Daniel Mazzone for helpful discussions. Computer resources for numerical calculations were supported by the Institutional Computing Program at LANL. The work by SZL was carried out under the auspices of the U.S. DOE Contract No. DE-AC52-06NA25396 through the LDRD program. Experimental work was supported by the LDRD program at LANL. * [email protected] † [email protected] FIG. 1 . 1[cos(2k x ) + cos(2k y )] − µ. Here t 1 , t 2 = −0.5t 1 and t 3 = −0.4t 1 are the nearest neighbor (color online) Theoretical phase diagram at T = 0 when field is rotated in the a-c plane. The phase diagram is constructed based on the FFLO and SDW order parameters. The orange (blue) line denotes a second (first) order phase transition. SC -the superconducting state. FIG. 2 . 2(color online) Profiles of superconducting and SDW order parameter ∆ d (a, c) and M i = n i↑ − n i↓ (b, d). The upper (lower) two panels correspond to the FFLO (SDW) state at g ab µ B H = 0.4t 1 and H c (g ab µ B H = 0.6t 1 and H a). FIG. 3 .FIG 3(color online) Thermal conductivity with J [010] as a function of magnetic field at different angles. θ represents the angle between the field and the crystalographic c axis. Here T = 0.09 K. expectations are borne out by the thermal-conductivity measurement shown below. Thermal conductivity measurements.-We performed thermal-conductivity measurements on CeCoIn 5 with magnetic field rotating from H [001] to H [100] at T = 0.09 K. To minimize the effect of the vortex lattice on the thermal conductivity, we chose configuration where the vortex lattice is always perpendicular to the thermal current during field rotation, i.e. J [010] and field in the 010 plane. Single crystal CeCoIn 5 (0.2 × 2.5 × 0.05 mm 3 ) was grown from an excess indium flux, and the thermal conductivity was measured by the standard steady-state method. The results are shown in Fig. 3. κ initially decreases with field due to vortex scattering. This behavior indicates a high quality of . 4. 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We use a semi-analytic model of Lyα emitters (LAEs) to constrain the reionization history. By considering two physically motivated scenarios in which reionization ends either early (ERM, z i ≈ 7) or late (LRM, z i ≈ 6), we fix the global value of the IGM neutral fraction (e.g. χ HI = 3 × 10 −4 , 0.15 at z = 6.56 for the ERM and LRM, respectively) leaving only the star formation efficiency and the effective escape fraction of Lyα photons as free parameters. The ERM fits the observed LAE luminosity function (LF) at z = 5.7 and 6.56 requiring no redshift evolution or mass dependence of the star formation efficiency, and LAE star formation rates (SFR) of 3 <Ṁ ⋆ /M ⊙ yr −1 < 103, contributing ≈ 8% of the cosmic SFR density at z = 5.7. The LRM requires a physically uncomfortable drop of ≈ 4.5 times in the SFR of the emitters from z = 6.5 to 5.7. Thus, the data seem to imply that the Universe was already highly ionized at z = 6.56. The mass-dependent Lyα transmissivity is 0.36 < ∼ T α < ∼ 0.51 (ERM) and T α < ∼ 0.26 (LRM) at z = 6.56. The LF data at z = 4.5 imply an extra Lyα line damping factor of ≈ 0.25 possibly due to dust; the presence of a (clumpy) dust component with E(B − V ) < ∼ 0.28 is also required to reproduce the observed large Lyα equivalent widths at the same redshift. Additional useful information can be extracted from the line profile (weighted) skewness, found to be S W = 10 − 17Å for the two reionization models, which shows an interesting L α − χ HI anti-correlation, holding under the model assumptions. The shortcomings of the model and strategies to overcome them are discussed.

10.1111/j.1365-2966.2008.13721.x

0807.2975

1079732bec335fd319cf07dc74aa8ccf523cadb0

Signatures of reionization on Lyα emitters 2002 Pratika Dayal SISSA/International School for Advanced Studies Via Beirut 2-4 Trieste34014Italy Andrea Ferrara SISSA/International School for Advanced Studies Via Beirut 2-4 Trieste34014Italy Simona Gallerani Institute of Physics Eötvös University Pázmány P. s. 1/A1117BudapestHungary Signatures of reionization on Lyα emitters Mon. Not. R. Astron. Soc 0002002Received 2007 December 24; in original form 2007 December 24(MN L A T E X style file v2.2)line:profiles -galaxies:high redshift -luminosity function -intergalactic medium -cosmology:theory We use a semi-analytic model of Lyα emitters (LAEs) to constrain the reionization history. By considering two physically motivated scenarios in which reionization ends either early (ERM, z i ≈ 7) or late (LRM, z i ≈ 6), we fix the global value of the IGM neutral fraction (e.g. χ HI = 3 × 10 −4 , 0.15 at z = 6.56 for the ERM and LRM, respectively) leaving only the star formation efficiency and the effective escape fraction of Lyα photons as free parameters. The ERM fits the observed LAE luminosity function (LF) at z = 5.7 and 6.56 requiring no redshift evolution or mass dependence of the star formation efficiency, and LAE star formation rates (SFR) of 3 <Ṁ ⋆ /M ⊙ yr −1 < 103, contributing ≈ 8% of the cosmic SFR density at z = 5.7. The LRM requires a physically uncomfortable drop of ≈ 4.5 times in the SFR of the emitters from z = 6.5 to 5.7. Thus, the data seem to imply that the Universe was already highly ionized at z = 6.56. The mass-dependent Lyα transmissivity is 0.36 < ∼ T α < ∼ 0.51 (ERM) and T α < ∼ 0.26 (LRM) at z = 6.56. The LF data at z = 4.5 imply an extra Lyα line damping factor of ≈ 0.25 possibly due to dust; the presence of a (clumpy) dust component with E(B − V ) < ∼ 0.28 is also required to reproduce the observed large Lyα equivalent widths at the same redshift. Additional useful information can be extracted from the line profile (weighted) skewness, found to be S W = 10 − 17Å for the two reionization models, which shows an interesting L α − χ HI anti-correlation, holding under the model assumptions. The shortcomings of the model and strategies to overcome them are discussed. INTRODUCTION The Epoch of Reionization (EoR) marks the second major change in the ionization state of the universe after recombination and is directly linked to structure formation. Reionization begins when the first structures form within dark matter halos and emit neutral hydrogen ionizing photons. In addition to changing the ionization state, these first structures also affect subsequent structure formation due to various radiative, mechanical and chemical feedback effects. Thus, to probe reionization, one needs an excellent understanding of initial density perturbations and their growth, as well as simulations that can trace the evolution of structure formation. One of the major challenges of reionization models is to be able to simultaneously account for the considerable, and often apparently conflicting, amount of data accumulated by experiments exploiting QSO absorption line spectra (Fan et al. 2006), cosmic microwave background radiation (Page et ⋆ E-mail: [email protected] (PD) al. 2007, Spergel et al. 2007) and high redshift galaxy surveys (Bouwens et al. 2006, Stark et al. 2007). The emerging picture ) is one in which hydrogen reionization is an extended process starting at z ≈ 15 and being 90% complete by z = 8. Reionization is initially driven by metal-free stars in low mass (M < 10 8 M⊙) halos; the conditions for the formation of these objects are soon erased by the combined action of chemical and radiative feedbacks at z < 10. Given the many assumptions necessarily made by reionization models, the above scenario needs constant confrontation with freshly acquired data sets. In this sense, it has been suggested (Malhotra & Rhoads 2004Santos 2004;Haiman & Cen 2005;Mesinger, Haiman & Cen 2004;Dijkstra, Wyithe & Haiman 2007;Dijkstra, Lidz & Wyithe 2007;Mesinger & Furlanetto 2007) that a class of high redshift galaxies, the Lyman Alpha Emitters (LAEs) can be suitably used to put additional constraints on the reionization history: the Lyman break and the strength, width and asymmetry of the observed Lyα line make the detection of LAEs unambiguous. The strength of the method is based on the sensitivity of Lyα photons to even tiny amounts of H I in the intergalactic medium (IGM). At redshifts z ∼ 5, the optical depth to Lyα photons is very large. Let e be the electron charge, f the oscillator strength (0.4162), λα the wavelength of Lyα in its rest frame (1216Å), me the electron mass, c the speed of light, H(z) the Hubble parameter at the required redshift, nHI the global neutral hydrogen density and nH the global mean hydrogen density at that redshift. Note that nHI = χHInH where χHI is the fraction of neutral hydrogen at the redshift under consideration. Further, Ω b represents the baryonic density parameter and Ωm is the total (baryonic + dark) matter density parameter of the universe, Ωm = Ω b + Ω dm . Then, τα = πe 2 f λα mecH(z) nH nHI nH , where πe 2 f λα mecH(z) nH = 1.76 × 10 5 h −1 Ω −1/2 m Ω b h 2 0.022 1 + z 8 3/2 . Hence, even a H I fraction of 10 −4 can lead to a significant attenuation of the Lyα line. The observed (i.e. transmitted) Lyα luminosity, Lα, can then be used to infer the ionization state of the IGM at redshifts close to those of the emitter and hence to reconstruct, at least piecewise, the cosmic reionization history. This simple picture is complicated by a number of important physical effects. First of all, Lyα photons from the stars have to propagate through and escape from the interstellar medium of the LAE. During their travel they are multiply scattered by H I atoms (thus being either removed from or added to the line of sight [LOS]) and possibly absorbed by dust grains (Neufeld 1991;Tasitsiomi 2005;Hansen & Oh 2006;Finkelstein et al. 2007). These processes modify both the emerging Lyα luminosity and the shape and equivalent width of the line. Second, the ionizing radiation from the same stars builds regions of ionized IGM around the emitters, whose size depends on the star formation rate, age, escape of ionizing photons from the galaxy and the stellar Initial Mass Function (IMF; the case of very massive stars has been explored, for example, by . As a result, the flux redwards of the Lyα line can escape, attenuated only by the red damping wing of the Gunn-Peterson absorption (Miralda-Escudé 1998;Madau & Rees 2000). To a first approximation, the spatial scale imposed by the Gunn-Peterson damping wing on the size of the H II region corresponds to a redshift separation of ∆z ≈ 0.01, i.e. about 200 kpc (physical) at z = 10. The effects of the damping wing fade away if the emitter is powerful enough to create a large enough H II region and/or if the universe is already reionized when the emitter turns on. Alternatively, one would observe the damping wing if there were even a small fraction of neutral hydrogen left inside the sphere and/or if a H I cloud is present along the LOS to the source. All the above effects combine to shape the observed LAE Luminosity Function (LF), which has been now measured (Rhoads et al. 2000;Taniguchi et al. 2005;Shimasaku et al. 2006;Iye et al. 2006;Kashikawa et al. 2006;Murayama et al. 2007;Ota et al. 2007;Dawson et al. 2007) with different degrees of accuracy up to z ≈ 7. Such tremendous progress has been made possible by the increase of survey fields and available samples. The current observational situation can be summarized as follows. All studies seem to converge toward the conclusion that there is very little indication of evolution of the LF moving from z = 3 to z = 5.7. Beyond that epoch there seems to be evidence of a decline in the LF, with L * at z = 6.6 being about 50% of that at z = 5.7. Such a high luminosity steepening of the LF can be produced by a number of different physical effects. A rapid evolution of the IGM ionization state can be invoked (Kashikawa et al. 2006) if the overlapping phase of reionization ended around z = 6; however, the net effect of reionization on the observed Lyα luminosity of the most luminous (and presumably massive) LAEs is unclear. If these objects are expected to live in more dense and hence more neutral environments, they are also more heavily clustered (McQuinn et al. 2007). The two effects might not change appreciably the size of their H II regions. Alternatively, the observed evolution could be simply a result of the evolution of the mass function of dark matter halos housing the LAEs (Dijkstra, Wyithe & Haiman 2007;Dijkstra, Lidz & Wyithe 2007). Finally, extinction due to dust, which is expected to be more prominent in actively star forming galaxies, may act as a sink for Lyα photons in the most luminous LAEs. As of now, it is difficult to firmly assess which of these explanations is more robust. Fortunately, other aspects of the data, such as the line shape and equivalent width, might allow one to make progresses. Here we try to assess to what extent the reionization history can affect the shape of the LF and the observed properties of individual LAEs. Our approach is similar in spirit to some of those mentioned above, but it has the strength of being based on reionization models that simultaneously account for all the available data beyond LAEs including Lyα/Lyβ Gunn-Peterson opacity, electron scattering optical depth, Lyman Limit Systems, cosmic SFR history and the number density of high-redshift sources. 1 . THE MODEL In this section we describe the physical features of the model we have developed to derive the various properties of LAEs which will then be compared with observations. Several steps are required in order to carry out this task which are described in detail in the following. These include the use of the Sheth-Tormen mass function to obtain the redshift dependence of the number density of dark matter halos, the star formation prescriptions required to build the luminosity function, the production rate of H I ionizing photons and the intrinsic Lyα luminosity, the size of the Strömgren sphere built by LAEs and the H I density profile within it and in the general IGM, for which we use a previously developed reionization model. 1 Throughout the paper, we use the best-fit cosmological parameters from the 3-year WMAP data (Spergel et al. 2007), i.e., a flat universe with (Ωm, Ω Λ , Ω b h 2 , h)=(0.24, 0.76, 0.022, 0.72). The parameters defining the linear dark matter power spectrum are σ 8 = 0.82, ns = 0.95, dns/d ln k = 0. We use a value of σ 8 much higher that quoted from WMAP3 (0.76) as the combination of WMAP3 and SDSS data give σ 8 ∼ 0.78 (0.86) for low (high) resolution Lyα forest data (Viel et al. 2006). Mpc is comoving unless otherwise specified. The mass function We start with the well known Sheth-Tormen mass function, Sheth & Tormen (1999), which is used to calculate the number density of dark matter halos of mass between M and M + dM at any redshift z, represented by n(M, z)dM , as n(M, z)dM = A 1 + 1 ν ′2q 2 πρ M dν ′ dM e −ν ′2 /2 dM,(1) where ν ′ = √ aν. In eq.1, A, a and q are modifications to the original Press-Schechter mass function, Press & Schechter (1974), to make it agree better with simulations. Here, A ≈ 0.322, q = 0.3 and a = 0.707. As in the Press-Schechter mass function, ν = δc D(z)σ(M ) , D(z) = g(z)/[g(0)(1 + z)], g(z) = 2.5Ωm[Ω 4/7 m − ΩΛ + (1 + Ωm/2)(1 + ΩΛ/70)] −1 . Here, δc(= 1.69) is the critical overdensity for spherical collapse and D(z) is the growth factor for linear fluctuations, Carroll, Press & Turner (1992). Further, the variance of the mass M contained in a radius R is given by σ 2 (R) = 1 2π 2 k 3 P (k)W 2 (kR) dk k .(2) In eq.2, W (kR) = 3(sin(kR) − kR cos(kR)) is the window function that represents the Fourier transform of a spherical top hat filter of radius R, P (k) = Apk n T 2 (k) is the power spectrum of the density fluctuations, extrapolated to z = 0 using linear theory where Ap is the amplitude of the density fluctuations calculated by normalizing σ(M ) to σ8 which represents the variance of mass in a sphere of size 8h −1 Mpc at z = 0. The term T (k) is a transfer function which represents differential growth from early times (Bardeen et al. 1986). T (k) = 0.43q −1 ln(1 + 2.34q) [1 + 3.89q + (16.1q) 2 + (5.46q) 3 + (6.71q) 4 ] 1/4 , (3) where q = k(Ωmh 2 ) −1 . Once the mass function is obtained, a SFR recipe (Sec 2.2) is used to obtain the intrinsic Lyα luminosity for any halo on the mass function in Sec 2.3, thereby providing the intrinsic Lyα luminosity function. The attenuation of the intrinsic Lyα luminosity by the IGM, as calculated in Sec 2.4, then allows the mass function to be translated into the observed Lyα luminosity function. The ionizing photon rate The baryonic mass, M b , contained within a halo of mass M h can be expressed as M b = Ω b Ωm M h . We assume that a fraction f * of this baryonic matter forms stars over a timescale t * = ǫ dc tH , where ǫ dc is the duty cycle and tH is the Hubble time at z = 0. Thus, we can write the star formation rate (SFR) asṀ * = f * ǫ dc 1 tH Ω b Ωm M h .(4) Using the population synthesis code Starburst99 (Leitherer et al. 1999) we obtain the hydrogen ionizing photon rate, Q, emitted by galaxies having a given SFR, assuming a metallicity Z = 0.05Z⊙. Determining the metallicity of the LAEs proves very challenging, as for most of the cases, only the Lyα line can be detected from these objects. To guess their metallicity, we use the results from studies of LBGs (Lyman Break Galaxies) and DLA (Damped Lyα) systems, which indicate values of 0.05 − 0.10Z⊙, which justifies our assumption, Pettini (2003). We use a Kroupa IMF with a slope of 1.3 between 0.1 and 0.5M⊙ and 2.35 between 0.5 and 100M⊙. Using the fact that Q scales linearly with SFR, we can calculate Q for the desired SFR. Intrinsic Lyα line Star formation in LAEs produces photons with energy > 1 Ryd. These photons ionize the interstellar H I , leading to the formation of free electrons and protons inside the emitter. Due to the high density of the ISM, these then recombine on the recombination time scale, giving rise to a Lyα emission line. Let fesc be the fraction of H I ionizing photons that escape the galaxy without causing any ionizations, fα the fraction of Lyα photons that escape the galaxy without being destroyed by dust, να be the frequency of Lyα in the rest frame of the galaxy (1216Å) and h be the Planck constant. Then, the intrinsic Lyα luminosity, L int α , from the galaxy can be expressed as L int α = 2 3 Q(1 − fesc)fαhνα.(5) It has been calculated that there is a two-thirds probability of the recombination leading to a Lyα line and a one-third probability of obtaining photons of frequencies different from the Lyα (Osterbrock 1989). This gives rise to the factor of two-thirds in eq.5. For (1 − fesc)fα = 1, the intrinsic Lyα luminosity and the SFR are related by the following L int α = 2.80 × 10 42 erg s −1 SFR M⊙yr −1 Modeling the Lyα line to be Doppler broadened, the complete line profile is L int α (ν) = 2 3 Q(1 − fesc)fαhνα 1 √ π∆ν d exp −(ν−να ) 2 /∆ν 2 d ,(6) where ∆ν d = (vc/c)να, vc is the rotation velocity of the galaxy and c is the speed of light. The minimum rotation velocity of the galaxy would be equal to the rotation velocity of the host halo, v h . However, for more quiescent star formation, for realistic halo and disk properties, vc can have values between v h and 2v h (Mo, Mao & White 1998;Cole et al. 2000). We use the middle value between these limits in our model, so that vc = 1.5v h . To illustrate, as M h increases from 10 10 to 10 12 M⊙, vc increases from 102 to 475 km s −1 at z ∼ 6.6. We calculate the velocity of the halo assuming that the collapsed region has an overdensity of roughly 200 times the mean cosmic density contained in a radius r200. Then, v h , the velocity at r200 is expressed as v 2 h (z) = GM h r200 = GM h 100Ωm(z)H(z) 2 GM h 1/3 ,(7) where Ωm and H are the density and Hubble parameters, respectively, at the redshift of the emitter. To summarize, the intrinsic Lyα luminosity depends upon: the ionization rate Q, the escape fraction of H I ionizing photons fesc, the escape fraction of Lyα photons fα and the rotation velocity of the galaxy vc . In turn, Q depends on the SFR (which is a function of halo mass), the metallicity Z, and the age of the emitter t * , chosen such that the number of ionizing photons emitted per second settles to a constant value. Observed Lyα line The intrinsic Lyα line is attenuated by the neutral hydrogen present in the IGM along the line of sight toward the emitter. In this section we compute the neutral hydrogen distribution and the attenuation caused by it. Global χHI calculation We use the global value of the H I fraction χHI = nHI /nH resulting from the modeling by Gallerani, Choudhury & Ferrara (2006), further refined in Gallerani et al. (2007). The main features of the model are summarized here. Mildly non-linear density fluctuations giving rise to spectral absorption features in the Intergalactic medium (IGM) are described by a Log-Normal distribution. This has been shown to fit the observed probability distribution function of the transmitted flux between redshifts 1.7 and 5.8 by Becker, Rauch & Sargent (2007). For a given IGM equation of state, this being the temperature-density relation, the mean global H I fraction (χHI ) can be computed from photoionization equilibrium as a function of baryonic over-density (∆ ≡ ρ/ρ) and photoionization rate (ΓB) due to the ultra-violet background radiation field. These quantities must be determined from a combination of theory and observations. Gallerani et al. (2007) included two types of ultraviolet photons: from QSOs and Pop II stars. The free parameters in their model were (i) the SFR efficiency (f * ) and (ii) the escape fraction of ionizing photons from the galaxy (fesc). These were calibrated to match the redshift evolution of Lyman-limit systems, Lyα and Lyβ optical depths, electron scattering optical depth, cosmic SFR history and number density of high redshift sources. The following reionization scenarios provide a good fit to observational data: (i) Early Reionization Model (ERM), in which reionization ends at zi = 7, (f * = 0.1, fesc = 0.07), (ii) Late Reionization Model (LRM), where reionization ends at zi = 6, (f * = 0.08, fesc = 0.04). Neutral hydrogen profile The IGM is approximately in local photoionization equilibrium. Under such conditions ionizations are balanced by recombinations, nHI ΓB = nenpαB,(8) where nHI , np, ne are the number density of neutral hydrogen, protons and electrons respectively, αB is the hydrogen Case B recombination coefficient and ΓB is the ionization rate due to the background. As mentioned in Sec.2.4.1, in this work we take advantage of the results presented by Gallerani et al. (2007). Once that χHI is fixed to the their values 2 , the photoionization rate contributed by the ionizing background light produced by quasars and galaxies is given by: ΓB = (1 − χHI ) 2 χHI nH αB.(9) Moreover, the radiation from stars inside the galaxy ionizes the region surrounding the emitter, the so-called Strömgren sphere. The evolution of the Strömgren sphere is given by the following relation, (Shapiro & Giroux 1987;Madau, Haardt & Rees 1999) dVI dt − 3H(z)VI = Qfesc nHI − VI trec ,(10) where, VI is the proper volume of the Strömgren sphere, and trec = [1.17αB np] −1 is the volume averaged recombination timescale (Madau & Rees 2000). The proper radius RI = (3VI /4π) 1/3 , identifies a redshift interval ∆z between the emitter and the edge of the Strömgren sphere, given by the following: ∆z = 100(Ωmh 2 ) 1/2 (1 + z) 5/2 RI /c. Though this equation is not strictly valid at z ∼ 0, it is a good approximation at the high redshifts we are interested in (z 4.5). If ze is the redshift of the emitter, for redshifts lower than the Strömgren sphere redshift, i.e., zs = ze − ∆z, we use the χHI value from Gallerani et al. (2007). Within the Strömgren sphere, to ΓB we add the LAE photoionization rate ΓE: ΓE(r) = λ L 0 L λ 4πr 2 σL λ λL 3 λ hc dλ,(12) where L λ is the specific ionizing luminosity of the emitter (in erg s −1Å−1 ), λL is the Lyman limit wavelength (912Å) and σL is the hydrogen photoionization cross-section. Thus, inside the ionized region, χHI is computed as following: χHI(r) = 2nH αB + Γ(r) ± Γ 2 (r) + 4nH αBΓ(r) 2nH αB ,(13) where Γ(r) = ΓE(r) + ΓB. The solution must be chosen such that χHI < 1, which only happens for a negative sign before the square root. At the edge of the Strömgren sphere, we force χHI(r) to attain the global value in the IGM. Lyα optical depth and transmitted flux The transmitted Lyα luminosity is Tα = e −τα where τα is the optical depth to Lyα photons. Assuming that reionization completes at z = zi, τα can be calculated as τα(ν obs ) = z i ze σ(ν obs )nHI (z) dr dz dz, = z i ze σ0φ(ν obs )nHI (z) dr dz dz, where σ is the total absorption cross-section, σ0 = πe 2 f /(mec) and φ is the Voigt profile. For regions of low H I density, the natural line broadening is not very important and the Voigt profile can be approximated by the Gaussian core: φ ≡ φgauss = 1 √ π∆ν d exp −(ν obs −να,r ) 2 /∆ν 2 d .(14) In eq.14, να,r = c/[λα(1 + zr)] is the local Lyα frequency at a distance r from the emitter and ν obs = c/λ obs . Further, ∆ν d = b/λα, where b = 2kT /mH is the Doppler width parameter, mH is the hydrogen mass, k is the Boltzmann constant and T = 10 4 K is the IGM temperature (Santos 2004;Schaye et al. 2000;Bolton & Haehnelt 2007). For regions of high H I density, we take into account the Lorentzian damping wing of the Voigt profile. Thus, for wavelengths outside the Gaussian core, i.e. for | ν − να,r | ∆ν d , we assume the following profile (Peebles 1993): φLorentz = Λ(ν obs /να,r) 4 4π 2 (ν obs − να,r) 2 + (Λ 2 /4)(ν obs /να,r) 6 ,(15) where Λ = 8.25 × 10 8 s −1 is the decay constant for the Lyα resonance. BASIC DEPENDENCIES By using the model described in the previous section, we can compute the observed Lyα line profile: Lα = e −τα L int α = TαL int α .(16) The Lyα optical depth depends on three quantities: the star formation rate (which fixes the value of Q), the ionized region radius, and the global neutral fraction: τα = τα(Ṁ⋆, RI , χHI). Once these three parameters are given, the transmissivity is uniquely determined. Notice that RI = RI (fesc, t⋆,Ṁ⋆, χHI ). If instead we are interested in the observed Lyα luminosity, a fourth parameter needs to be specified, the "effective" Lyα photon escape fraction fesc,α = (1 − fesc)fα,(17) which expresses the physical fact that the condition to observed Lyα photons is that some ionizing photons are absorbed within the galaxy and only a fraction fα of produced Lyα photons can escape to infinity. Note that fesc,α does not affect the transmissivity as both the intrinsic and the observed luminosity depend on it and therefore it factors out. A full exploration of the physical effects of the parameters on the observed luminosity, Lα, can be performed by varying only the parametersṀ⋆, RI , χHI and fesc,α. The effects of other parameters (as, for example, metallicity, Z) can be estimated by simple scaling of the results below. To understand the impact of each of the three relevant quantities on Lα we have selected a fiducial case with parameters broadly similar to those we inferred under realistic (i.e. observationally derived) conditions for LAEs and allow them to vary in isolation taking three different values. We therefore considered 1 × fiducial + 4 × 3 = 13 different cases shown in Fig.1 and summarized in detail in Table 1. − − − 0.52 s2 54 − − − 0.50 s3 13.5 − − − 0.44 f1 − 0.9 − − 0.47 f2 − 0.1 − − 0.47 f3 − 0.03 − − 0.47 r1 − − − 2.97 0.44 r2 − − − 1.48 0.37 r3 − − − 0.74 0.27 c1 − − 3 × 10 −4 − 0.49 c2 − − 0.05 − 0.42 c3 − − 0.15 − 0.32 Star formation rate The ionizing photon rate, Q, of the emitter is directly proportional to its SFR. As a result, a larger SFR results in (a) an increase of L int α , (b) a larger ionized region around the LAE, (c) a lower value of χHI at each point within the Strömgren sphere (see eqs.12-13). The net effect is that as SFR increases, the transmission of a stronger Lyα line increases due to decreased damping by both the Gaussian core and the red damping wing. This is shown in panel (a) of Fig.1. For the fiducial case we find that 47% of the intrinsic Lyα luminosity is transmitted; this value increases with SFR, reaching 52% whenṀ⋆ = 81M⊙yr −1 , as seen from Tab.1. Effective Lyα photon escape fraction The effective Lyα photon escape fraction fesc,α scales both L int α and Lα equally, without changing either the size of the Strömgren sphere or the H I profile within it. The fraction of Lyα luminosity transmitted is hence, the same in all the cases. The variation of Lα with fesc,α is shown in panel (b) of Fig.1. Ionized region radius As the ionized region becomes larger, due to a more robust input on ionizing photons from the source, the Lyα photons reach the edge of the sphere more redshifted. Hence, the H I outside the ionized bubble is less effective in attenuating the flux. The size of the ionized region radius is therefore very important for LAEs in regions of high H I density and loses importance as the H I density decreases. We show the variation of Lα with RI 3 in panel (c) of Fig.1. from which we can readily appreciate that as RI increases (at a fixed χHI and SFR), a larger fraction of the line is transmitted due to the aforementioned effect. As, to a good approximation, RI ∝ Qfesct * χHInH 1/3 ,(18) for a fixed value of Q (SFR) and χHI , RI can vary either due to t * or fesc. These two parameters play a qualitatively different role. While the age variation can be embedded in a variation of RI only, changing the value of the escape fraction also affects L int α (see eq.6) giving rise to a physically interesting effect. In Fig.2, for illustration purposes, we fiẋ M⋆ = 27M⊙yr −1 , t * = 10 8 yr, fα = 1 and study the effect of fesc on Lα for different values of χHI . The observed Lyα luminosity decreases monotonically with fesc for low values of χHI (< 0.01), just mirroring the decreasing value of the intrinsic Lyα line. Here, the fact that the size of the Strömgren sphere built increases with increasing fesc has no effect on Lα simply because the H I density is too low to cause (red) damping wing absorption, irrespective of the size of the ionized region. For χHI 0.01, the Lα trend with fesc in not monotonic anymore (see also Santos, 2004). For example, for χHI = 0.15 the observed Lyα luminosity reaches a maximum at fesc ≈ 0.5. This can be explained by the following: for low (< 0.5) fesc values, as fesc increases, the ionized volume increases, thus leading to larger transmission. When Lα reaches its maximum (for fesc ≈ 0.5, in our example), a further fesc increase reduces the observed Lyα luminosity, as a consequence of the decreasing value of L int α . This highlights the fact that while for low values of χHI , fesc affects the observed Lyα only through the intrinsic Lyα line, for high values of χHI , the effect of fesc on the Strömgren sphere size becomes considerably important. Neutral hydrogen fraction In panel (d) of Fig.1, we study the effect of different χHI values on the Lyα line. It can be seen from Tab. 1 that the Lyα line is quite damped (Tα ∼ 0.32) for high values of χHI (= 0.15). As the value of χHI decreases, the effect of both the Gaussian core and the red damping wing start reducing, allowing more of the line to be transmitted. For χHI = 3 × 10 −4 , most of the line redwards of the Lyα wavelength escapes without being damped. This occurs because the emitter is able to (a) strongly ionize the H I within the Strömgren sphere (already ionized to a large extent even outside it) even further, and (b) build a large Strömgren sphere such that the Lyα line is not affected by the damping wing of the H I outside. We remind the reader that Lα = Lα(Ṁ⋆, fesc,α, RI , χHI ). For a continuous star formation mode, the luminosity of the source becomes rapidly independent of age (typically after 100 Myr); if, in addition, we adopt the values of χHI obtained from Gallerani et al. (2007) by matching the experimental data, we are left with two free parameters,Ṁ⋆ and fesc,α. Recalling thaṫ M⋆ ∝ f * /ǫ dc , the free parameters in our model reduce to (a) f * /ǫ dc and (b) fesc,α. COMPARISON WITH OBSERVATIONS In this section we compare the results obtained from our model to observations of the LAE LF, the UV LF, the line profile asymmetries, the equivalent widths and the cosmic SFR density. In particular, we would like to assess to what extent the study of these quantities for LAEs can be used to discriminate between the early (ERM) and late (LRM) reionization scenarios, as deduced from the study of Gallerani et al. (2007), summarized in Sec.2.4.1. Dawson et al. (2007) conducted the Large Area Lyα (LALA) survey to look for LAEs at z ∼ 4.5 and found 97 candidates; 73 of which were confirmed using DEIMOS on KECK II and the Low Resolution Imaging Spectrograph (LRIS). Shimasaku et al. (2006) identified 89 LAE candidates in the Subaru Deep Field (SDF) at z ∼ 5.7 by using the 8.2m Subaru Telescope and the following selection criteria: (a) i ′ − NB816 1; (b) NB816 26. By using the Faint Object Camera and Spectrograph (FOCAS) on Subaru and DEIMOS, 28 candidates were confirmed as LAEs. Taniguchi et al. (2005) detected 58 possible LAEs using Subaru at z ∼ 6.5 and obtained the spectra for 20 of them using the FOCAS. They found that only 9 of the above objects showed sharp cut-off at the Lyα wavelength, narrow line widths and asymmetric profiles, thus being confirmed as LAEs at z ∼ 6.5. These included the two LAEs discovered by Kodaira et al. (2005) at z = 6.541 and 6.578. Using the same selection criterion and instruments as Taniguchi et al. (2005) and including the LAEs confirmed using the Keck II DEIMOS spectrograph, Kashikawa et al. (2006) added 8 more LAEs at 6.5 to this list. Thus, the Subaru observations have a total of 17 confirmed LAEs at z ∼ 6.5. Available data Lyα Luminosity function As a first remark, it is useful to point out that if the LF evolution were to result purely from the evolution of the dark matter halos predicted by hierarchical structure formation, one would expect the comoving number density of luminous objects to increase with decreasing redshift. Although data errors are still large, it must be noted that instead there is an indication that there is no evolution of the Lyα LF between z ∼ 3 − 6 ( Dawson et al., 2007;Ouchi et al., 2007). Obviously, a number of different effects could produce this non-monotonic trend, a few examples being, SFR evolution, redshift dependent escape fractions and dust extinction, as we discuss in the following. In Fig.3, we plot the cumulative LFs at z = 4.5, 5.7 and 6.56 together with our best fit results. We now discuss the predictions of ERM and LRM separately. The ERM predicts an evolution of the hydrogen neutral fraction such that χHI = 1.3 × 10 −5 , 8.6 × 10 −5 , 3 × 10 −4 for z = 4.5, 5.7 and 6.56 respectively. Interestingly, a very good fit to the data can be obtained for the two highest redshifts with a single value of the star formation efficiency parameter f * /ǫ dc = 3.5, thus implying that the SFR for any given halo mass is not very much dependent on redshift. While a reasonable fit to the data at z = 5.7 and z = 6.56 is obtained for a single value of fesc,α ≈ 0.3; a better fit is obtained by allowing for a 40% increase of fesc,α towards larger masses. The typical LAE dark matter halo masses corresponding to the observed luminosities are in the range M h = 10 10.7−12.0 M⊙ at z = 6.56; at the same redshift the star formation ranges from 2 to 43 M⊙yr −1 . The data at z = 4.5 instead pose a challenge to the model because, assuming non-evolving values of f * /ǫ dc = 3.5 and fesc,α, the observed number density of luminous objects is lower than that predicted by the evolution of the theoretical LF. Given the relative constancy of the star formation efficiency and of the effective Lyα photons escape fraction noted for the two highest redshifts considered, the most natural explanation is in terms of increasing dust extinction. To reconcile the prediction with the data at z = 4.5 we then require that the Lyα line suffers an additional damping due to the presence of dust; which we find to be equal to 1/4.0 = 0.25, i.e fα (and hence fesc,α) decreases by a factor of 4. A strong increase of the dust content inside galaxies is expected on cosmic time scales larger than 1 Gyr (corresponding to z < ∼ 5) when evolved stars rather than corecollapse supernovae become the primary dust factories. Such a hypothesis needs to be checked carefully, as the dust would not only affect the Lyα line but also the continuum emission, finally affecting the equivalent width of the line. We will discuss these effects of dust in Sec.4.3 and 4.5. Hence, it seems that overall, a model in which reionization was completed relatively early (zi = 7) matches the data quite well. The LRM has a much slower reionization history, as is clear from the values of χHI = 1.4 × 10 −5 , 1.3 × 10 −4 , 0.15 for z = 4.5, 5.7 and 6.56 respectively. At the lowest redshifts (z = 4.5 and 5.7) this model requires exactly the same value f * /ǫ dc = 3.5 as the ERM. This does not come as a surprise of course, as χHI is so small at these epochs in both the ERM and the LRM that the observed Lyα luminosity is unaffected. However, as χHI is much larger at z = 6.56 in the LRM as compared to the ERM, a higher star formation efficiency, f * /ǫ dc = 16 is required to fit the data at z = 6.56 for the LRM. As a result the SFR of LAEs in the LRM are increased by the same amount, ranging from 11 to 197M⊙ yr −1 . As in the ERM, we use the same value of fesc,α ≈ 0.3 (increasing by 40% for larger halo masses) for z = 5.7 and 6.56, but the data at z = 4.5 again require fα to decrease by a factor of 4. A comparison between the Lyα transmissivity, Tα, for the two reionization models considered is shown in Fig.4 for z = 6.56. In both cases the transmissivity increases towards more massive halos because of their generally larger SFR; also, at a given halo mass, Tα varies from 0.36 to 0.51 for the ERM, while it varies from 0.01 to 0.26 for the LRM i.e. it is considerably smaller for the LRM. In the LRM, small LAEs are characterized by a lower Tα with respect to larger ones relative to ERM. This is because even though the SFR are higher than in the ERM, the smaller LAEs are not able to build large enough HII regions; as a result, their Lyα line is much more damped as compared to that for the larger LAEs. In conclusion, the LF data seem to require a strong increase of the SFR from z = 5.7 to 6.56 in the LRM to fit the observed LFs while a SFR that smoothly decreases with increasing redshift fits the observations for the ERM. Looking at the general trend, one finds that SFR densities decrease with increasing redshift. Hence, we find that the LF data favors the reionization scenario described by the ERM, i.e. a highly ionized (≈ 3 × 10 −4 ) Universe at z = 6.56. The Best fit parameter values for the ERM are shown in Tab.2. A caveat is that this analysis has been done for isolated emitters. As shown by McQuinn et al. 2007, clustering significantly increases the amount of Lyα luminosity that can be transmitted by an emitter by adding a boost term to the background ionization rate. We find that such a boost factor of ∼ 100 boosts the luminosity transmitted by the LAEs at z = 6.56 with χHI = 0.15 significantly and in that case, the LRM can be fit by the same parameters (fesc,α, f * /ǫ dc ) as the ERM. However, an estimate of the boost in the background requires an accurate understanding of the radial dependence of the clustering and the contribution of each emitter to the boost. We then defer this analysis to further papers where we would use LSS simulations to fix these quantities. Hence, we can not rule out the LRM completely till clustering is included and better measurements of SFR densities at z 6.56 are obtained. Shimasaku et al. (2006) transformed the z' band magnitude from the photometric sample of 89 LAE candidates into the far UV continuum at the rest frame. The UV LF was calcu-lated by dividing the number of LAEs in each 0.5 magnitude bin by the effective volume corresponding to the FWHM of the bandpass filter used (NB816). Objects fainter than the 2σ limiting magnitude (27.04 mag) in the z' band were not included in calculating the UV LF and this corresponds to the vertical line at MUV = −19.58 in Fig.5. The authors mention that the apparent flattening at MUV > 20.5 might be due to the incompleteness in the measurement of the far UV LF. Kashikawa et al. (2006) used the same methodology mentioned above to derive the rest UV continuum from their photometric sample of 58 LAEs. Their LF measurements at magnitudes fainter than MUV = −20.24 (3σ) are uncertain due to the z' band magnitudes no longer being reliable beyond this value. UV luminosity function Both the above calculations have accounted for the detection completeness of the narrow band filters. They also find that cosmic variance is not severe for the UV LF. An important point to note is that the UV LFs at z = 5.7 and z = 6.5 are in very good agreement and show no evolution between these redshifts, which is in clear contrast to the Lyα LF which shows a deficit of high luminosity LAEs at z = 6.5 as mentioned before. We derive the specific continuum luminosities using STARBURST99, adopting Z = 0.05Z⊙, an age of about 100 Myr and a Kroupa IMF (details in Sec. 2.2). The continuum luminosity is then related to the SFR by However, using this conversion and the best fit parameter values of f * /ǫ dc for the ERM as mentioned in Sec.4.2, we find that the UV LFs for both redshifts lie above the observed ones. Hence, additional dust damping of the UV LF is required to match with the observations 4 . We quantify this additional damping by introducing fc, the fraction of continuum photons that escape the LAE, unabsorbed by dust. Using a single value of fc for a specific redshift (see Tab.2), across the entire mass range considered, we find a reasonably good agreement with the observed UV LF for the bright LAEs. However, the model fails to reproduce the bending of the UV LF observed for the low luminosity emitters. This could either be due to detection incompleteness in the observations or due to the lack of a physical effect such as a halo mass dependent escape fraction of UV photons. A simple prescription for the latter would be an increasing dust content with decreasing halo mass (due to a decrease in the ejection efficiency). However, other explanations such as SFRs that decrease with decreasing halo masses can not be ruled out with this model. A full exploration of possible effects will be carried out in further works using simulations. It is interesting to note that for this model, while at the highest redshift, continuum photons are less absorbed by dust as compared to the Lyα photons, the trend reverses at lower redshifts. This could hint at dust whose inhomogeneity/clumpiness evolves with redshift. However, robust estimates of the ages, metallicites, IMF and detailed studies of dust distribution and its evolution inside LAEs are needed before such a strong claim can be made. Table 2. Best fit parameter values for the ERM to fit both the Lyα LF and UV LF. For each redshift (col 1), we mention the halo mass range required (col 2), the SFR efficiency (col 3), the associated SFR (col 4), the effective escape fraction of Lyα photons (col 5) and the escape fraction of continuum photons (col 6). Kashikawa et al. (2006) (triangles). Lines refer to model predictions at the same redshifts: z = 5.7 (dot-dashed), z = 6.56 (solid). The vertical dashed (dotted) lines represent the 2σ (3σ) limiting magnitude for z = 5.7 (z = 6.56). z M h [M ⊙ ] f * /ǫ dcṀ * [M ⊙ yr −1 ] fesc, Cosmic star formation rate density As a sanity check, using the parameters that best fit the data as discussed in Sec.4.2, we calculate the contribution of LAEs to the SFR densities at z = 4.5, 5.7 and 6.56. We compare these with the SFR densities observed by Hopkins (2004) (Table 2) for the common dust-correction case, the results for which are plotted in Fig.6. We find that for the best-fit parameters, the contribution of LAEs to the SFR density is redshift-dependent, being about 8% at z = 5.7 with SFR in the range 3 <Ṁ⋆/M⊙yr −1 < 103, and even higher at z = 4.5, although the data present a large scatter at the latter epoch. Further, two points are worth noticing about the predicted SFR density. First, the SFR density must increase strongly from z = 5.7 to 6.56 in the LRM case. Although not impossible, such behavior is certainly puzzling and not Figure 6. Contribution of LAE to the cosmic SFR density evolution from our best fit models. Points show the measurements by Hopkins (2004); the dashed (solid) line is the prediction from ERM (LRM). easy to interpret. As the dust formation timescale is about 10 Myr, if the latter is copiously produced in supernova ejecta, as pointed out by several authors (Kozasa, Hasegawa & Nomoto 1991, Todini & Ferrara 2001, Schneider, Ferrara & Salvaterra 2004, Bianchi & Schneider 2007) and recently confirmed by the extinction curves of high redshift quasars (Maiolino et al. 2004), supernova-produced dust would rapidly increase the opacity to both continuum and Lyα photons, thus causing a rapid fading of the emitter. Second, the contribution of LAEs is about 8% of the cosmic star formation rate density at z = 5.7. Thus, either the duty cycle of the actively star forming phase in these objects is of the same order, or one has to admit that only a very small fraction (∼ 1/12) of high redshift galaxies experience this evolutionary phase. In the first case, the star formation duration would last about 8% of the Hubble time at z = 5.7, i.e. 72 Myr. Lyα equivalent width From our model it is easy to derive the intrinsic rest-frame Lyα line equivalent width 5 . Since both the continuum and L int α scale linearly with SFR in our model, the intrinsic EW distribution is a δ-function at EW int ≈ 131Å. From our model, the observed EW in the rest frame of the emitter is calculated as 5 We calculate the intrinsic rest-frame EW as EW int = L int α /[Lc(1375Å)] where L int α = 2.8 × 10 42 [Ṁ * /M ⊙ yr −1 ]erg s −1 and the specific continuum luminosity is given by Lc = 2.13 × 10 40 [Ṁ * /M ⊙ yr −1 ]erg s −1Å−1 . Both the intrinsic Lyα and continuum luminosities have been derived using STARBURST99; we adopt Z = 0.05Z ⊙ and a Kroupa IMF (for details, see Sec. 2.2). EW = EW int (1 − fesc)Tα fα fc ,(19) where fc quantifies the fraction of the continuum luminosity which escapes the emitter, unabsorbed by dust. The ratio fα/fc expresses the differential extinction of the Lyα line with respect to continuum radiation due to dust grains. At z = 4.5, we have seen that we require a factor ≈ 4 suppression of the Lyα line luminosity by dust, i.e. fesc,α ≈ 0.075. As dust affects also the continuum, and hence the EW, we need to estimate the value of fc (calculated at λ = 1375Å). We find that fc ≈ 0.045 for the mean EW from our model (155Å) to be the same as the observed EW (155 A). We then use the following relations to obtain the color extinction: A λ (1375Å) = −2.5 log fc,(20)E(B − V ) = AV RV ≈ 1 4 A λ (1375Å) RV ,(21) where RV ≈ 3 and we have assumed a Galactic extinction curve. From these expressions we obtain E(B − V ) = 0.28. The value of fc implies that the continuum is extincted about 1.6 times more heavily than the Lyα line (assuming fesc ∼ 0 so that fα = 0.075). This is not inconceivable if LAE interstellar dust is inhomogenously distributed and/or clumped, as showed by Neufeld (1991). With these two values we then derive the predicted EW distribution and compare it with the Dawson et al. (2007) data in Fig.7. As mentioned before, for the best fit parameters to the LF at z = 4.5, fesc,α ≈ 0.075 and Tα ≈ 0.50. Note that fesc,α and particularly Tα depend on the LAE luminosity/mass and increase by about 45% and 20% respectively towards higher masses. The predicted EWs are concentrated in a range, 114Å < EW < 201Å (mean=155Å), whereas the observed distribution is considerably wider, spanning the range 6 − 650Å with a mean of 155Å. As explained above, the spread of the predicted EW distribution arises only from the corresponding spread of SFR (6-160 M⊙yr −1 ) required in order to match the LF at z = 4.5, via the dependence of Tα on the SFR. Calculating the rest frame EWs is easier at z = 5.7 since we have an estimate of fc from the UV luminosity function as mentioned in Tab2. We calculate the EWs using fesc,α ≈ 0.3, fc ≈ 0.25 and Tα ≈ 0.37. As for z = 4.5, fesc,α and Tα depend on the halo mass and increase by 40% and 45% respectively towards higher masses. The calculations then yield EWs that range between 56-127Å. The mean from our model (∼ 92.3Å) is much less than the mean value of 120Å, observed by Shimasaku et al., 2006. The narrow range (z=4.5) and lower mean (at z=5.7) of EWs calculated from our model can easily be explained by the fact that our model does not include inflows/outflows, assumes an age of about 100 Myr for all the emitters and a metallicity which is 1/20 of the solar value. In reality, a larger spread would be expected from the addition of physical effects lacking in this model, such as (i) gas kinematics (inflow/outflow); (ii) variations of the IMF, metallicity, and stellar populations (including PopIII stars), and (iii) young stellar ages. While inflows erase the red part of the Lyα line, thereby reducing the EW, outflows shift the line centre redwards, helping more of it to escape. Outflows can also add a bump to the red part of the line due to backscattering of Lyα photons, as shown by Verhamme, Schaerer & Maselli, 2006. A top heavy IMF produces more H I ionizing photons, as does decreasing the metallicity. Hence, both these effects increase the EW. Further, for very young emitters (∼ 10 Myr), the continuum is much less than the continuum at 100 Myr and so, the EW would be much larger for younger Figure 9. Weighted skewness of the observed Lyα line for different models. The asterisks are the data from Kashikawa et al. (2006). The dashed (solid) line correspond to the best fit ERM (LRM) at z = 6.56. emitters. All these effects need a much more dedicated study, which we defer to future work. Line profile asymmetries Additional constraints on the model can come from the information embedded in the observed line profiles, as for example the line profile asymmetry. This can be suitably quantified by the weighted skewness parameter, SW , introduced by Kashikawa et al. (2006), which we calculate for the best fit parameter values for the ERM mentioned above. We adopt the following definition for such a quantity: SW = S∆λ = S(λ10,r − λ 10,b ),(22) where λ10,r (λ 10,b ) is the wavelength redward (blueward) of the Lyα line where the flux falls to 10% of the peak value. In addition, we have that I = n i=1 fi,(23)x = n i=1 xifi I ,(24)σ 2 = n i=1 (xi −x) 2 fi I ,(25)S = n i=1 (xi −x) 3 fi Iσ 3 ,(26) where fi is the line flux in the wavelength pixel i whose coordinate is xi, and the summations are performed over the pixels covered by the Lyα line. On general grounds one would expect that the observed Lyα line shape would be more symmetric (i.e. low SW ) in reionization models characterized by a lower value of χHI. However, given the above definition, just the opposite is true. In fact, for any reason-able value of the relevant parameters (see Fig.1) the blue part of the line is heavily absorbed, thus yielding a high value of SW ; as χHI is increased, also the long-wavelength part of the line is affected by the red damping wing, making the line more symmetric around the peak. The predicted trend of SW with the observed Lyα luminosity at z = 6.56 is reported in Fig.9, for the parameters of the ERM and LRM that best fit the LF data (discussed in the previous Section). For both models, the weighted skewness of the line increases for more luminous objects; however, such dependence is steeper for the ERM than for the LRM. In general, though, the two reionization scenarios predict SW values in the range 10-17. The data from Kashikawa et al. (2006) spans the somewhat larger range 3-17, with many of the data points lying around SW = 5. Given the paucity of the observed points and the large errors associated to them, it is probably premature to draw any strong claim from these results. However, given the constant increase in the amount and quality of LAE data, it is quite possible that the line skewness could represent a very interesting tool to constrain reionization models in the near future. It has to be noted that the data show a large scatter of SW at a given value of Lα, perhaps indicating that local conditions, including gas infall/outflow, density inhomogeneities and interaction of the Lyα line with the interstellar medium of the galaxy, might play a dominant role. These can only be investigated in a statistically meaningful manner via high-resolution numerical simulations to which we will defer a forthcoming study. From the theoretical point of view it is instructive to summarize the response of the skewness to different physical conditions. As we have seen from Fig.9, SW increases with Lα (or, equivalently with SFR); this is true for any fixed value of χHI . This is because as the SFR increases, more of the Lyα line escapes forcing SW to increase as a result of the larger value of ∆λ. Further, the long -wavelength part of the observed Lyα line begins to flatten with increasing χHI due to attenuation by the red damping wing. Hence, ∆λ varies slower with Lα (SFR) for high χHI (LRM) as compared to lower values (ERM); this makes the slope of SW steeper for the ERM. A more general view of the dependence of SW on Lα (hence on SFR) and χHI is shown in Fig. 10. The plot has been obtained by dividing the observed Lyα luminosity into bins and averaging the weighted skewness over the number of LAEs in each bin at a given value of χHI . The regions with weighted skewness values equal to zero represent a lack of LAEs in that bin. The most intriguing feature of Fig.10 is a clear anticorrelation between Lα and χHI . Given the range of SFR considered (Ṁ⋆ = 2.7 − 197M⊙yr −1 ), LAEs populate progressively fainter Lyα luminosity bins as the IGM becomes more neutral. Notice that relatively luminous objects (Lα ≈ 10 42.5 erg s −1 ) would not be detected if χHI > ∼ 0.25. Within the range in which these objects are visible, the most luminous objects always show the largest SW at fixed χHI ; however, such maximum value is also seen to increase with decreasing χHI. The model does not include important effects such as inflows/outflows and interaction of the Lyα photons with the ISM, which will definitely leave an imprint on the SW and hence, weaken the Lα-χHI anti-correlation. However, a com- In future work, we will endeavor to include these effects, hence obtaining an estimate of how large a sample might be sufficient for this purpose. DISCUSSION Starting from a simple yet physical model of galaxy formation within dark matter halos coupled with a population synthesis code, we have derived the intrinsic Lyα luminosity for a LAE. We then compute the volume of the ionized region built by the source and the density profile of the neutral hydrogen within it to obtain the damping of the emitted Lyα line caused by the Gaussian core and Lorentzian wings. Using this semi-analytic model, we have first explored the physical dependence of the observed Lyα line profile on various free parameters such as the LAE star formation rate, ionized region radius, the effective escape fraction of Lyα photons and the global IGM neutral hydrogen fraction. Among other things, we pointed out the in-teresting fact that the observed Lyα luminosity, Lα, peaks at a value of fesc ∼ 0.5 if the gas is substantially neutral (χHI ∼ 0.15); for that value, the contribution of fesc to the intrinsic Lyα luminosity and the Strömgren sphere balance each other. By considering two physically motivated scenarios in which reionization occurs either early (ERM, zi ≈ 7) or late (LRM, zi ≈ 6) we have fixed the global value of the IGM neutral fraction, χHI (thus leaving the star formation efficiency and the effective escape fraction of Lyα photons as the only free parameters), and obtained both the observed Lyα line profile and the Lyα luminosity function. Finally we have compared these predictions with available data at various redshifts. Using this procedure we have been able to fit the LFs observed by Dawson et al. (2007), Shimasaku et al. (2006) and Kashikawa et al. (2006) at z = 4.5, 5.7 and 6.56 respectively for the ERM. According to this model, no redshift evolution or mass dependence of the star formation efficiency is required. On the contrary, the LRM requires an increase of a factor 16/3.5 ∼ 4.5 in the SFR efficiency from z = 5.7 to 6.56. Although not inconceivable, such an upturn of the star formation efficiency seems puzzling and at odds with the observed cosmic star formation rate density. On this basis, we are more inclined to support the ERM. In addition, we find that the evolution of the observed luminosity function from z = 5.7 to 6.5 does not imply that we are scratching the reionization surface yet. Rather, the LF evolution can be explained solely by the evolution of the underlying dark matter halo mass function between these redshifts, as has previously been discussed by Dijkstra, Wyithe & Haiman, 2007. One would however, require more observations of the SFR density and information regarding the boost added to the ionizing background due to clustering at z ∼ 6.5 to completely rule out the LRM. A reasonable fit to the data at z = 5.7 and z = 6.56 is obtained for a single value of fesc,α ≈ 0.3 (although a good fit is obtained by allowing for a (40%) increase of fesc,α towards larger masses). The data at z = 4.5 instead pose a challenge to the model, as outlined earlier, because the observed number density of luminous objects is lower than that predicted by the evolution of the theoretical LF at higher redshifts. This could imply an increase in the overall dust content of LAEs at this redshift which would lead to absorption of Lyα photons within the emitter. Such a clumpy dust component is also suggested by by the large EWs observed at z = 4.5. We obtain the UV LF at z = 5.7 and 6.5 for the best fit values of f * /ǫ dc for the ERM. We find that additional damping of the UV luminosity is needed to match the predictions with the observations and quantify this by fc, the escape fraction of continuum photons. A single value of fc for a given redshift is enough to match the high luminosity end of the UV LF but does not produce the bending required at the faint end. However, this bending might just be the result of detection incompleteness of the sample. An interesting result is that for the given IMF and metallicity, while at higher redshifts (z = 6.5), the continuum photons are less absorbed by dust as compared to Lyα photons, this trend reverses at lower redshifts(z = 4.5, 5.7). This could be explained by imhomogenously distributed/clumped dust. However, the IMF, ages and metallicies of the emitters must be fixed robustly using simulations and infall must be included in the model before such a strong claim can be made. Using the best fit value of f * /ǫ dc and escape fraction of Lyα photons (continuum photons) obtained for the ERM from the Lyα (UV) LF, we calculate the expected EWs at z = 5.7 and find that the mean (∼ 92Å) is much less than the observed value of 120Å. At z = 4.5, since there are no observations of the UV LF at present, we calculate the escape fraction of continuum photons required to match the EW mean from the model to the data. A dust extinction of E(B − V ) ≈ 0.28 brings the predicted mean Lyα EWs (≈ 155Å) in very good agreement to the observed mean (≈ 155Å). This value of dust extinction is reasonable when compared to the observational upper limit of E(B−V ) ≈ 0.4 (Lai et al., 2007). However, additional effects which vary on a galaxy to galaxy basis, such as outflows/inflows or peculiar stellar populations are required to account for the spread of EW seen in the data. The contribution of LAEs to the cosmic SFR density is small, amounting to roughly 8% at z = 5.7. Thus either the duty cycle of the actively star forming phase in these objects is of the same order, or one has to admit that only about one-twelfth of high redshift galaxies experience this evolutionary phase. Additional useful information can be extracted from the line profile by using indicators like the line weighted skewness and equivalent width. The results presented here (Sections 4.4-4.5) must be considered as very preliminary for several reasons. First the available data on both SW and EW are very scarce and of relatively poor statistical quality, as they are very difficult to obtain even from the best current observations. Second, our model contains a number of simplifications which make the comparison only meaningful at a basic level. Nevertheless, it is encouraging that the model results are broadly in agreement with the data, at least for what concerns mean values. The models presented in this study do not include feedback processes related to the energy injection by supernovae. As pointed out by Santos (2004) and Iliev et al. (2007), peculiar gas motions might affect the line profile considerably: while inflows of gas erase the Lyα line, galaxy scale outflows produced by supernova (or AGN) feedback enable more of the Lyα to escape. Yet, our models are able to fit the LF evolution in the redshift range 4.5 < z < 6.56 quite well. This might indicate that the effect of feedback might be similar for all the emitters in the halo mass range (M h = 10 10.7−12.0 M⊙ at z = 6.56). Obviously, a more firm statement can be made only after a proper inclusion of peculiar motions. The backside of this is that, if inflows are taken into account, one cannot constrain the value of χHI robustly, as noted by Santos (2004). These effects can be properly taken care of by using high-resolution numerical simulations, which we plan to use in future works. In the same spirit, a full study of the problem should also include the effects of IGM density and temperature inhomogeneities, precise values of the local metallicity and star formation rates, and require information about the spatial clustering of the emitters. These values will be fixed in subsequent papers using the results of the simulation by Tornatore, Ferrara & Schneider (2007). An even more cumbersome ingredient is represented by dust. Dust grains act as sinks of Lyα and continuum photons, thus depressing the Lyα line luminosity but possibly boosting the line EW as explained throughout the paper. To what extent and on what timescales LAEs become dustpolluted (and possibly enshrouded) remains a question to which both theory and observations can provide only coarse answers at this time. Our conclusions hint at the need for dust in order to explain the evolution of the LF toward the lowest redshift. However, lacking a precise knowledge of the mass dependence of the dust-to-gas ratios in high redshift galaxies and a deep understanding of the dust formation processes/sources, developing a fully consistent theory will keep us busy for many years to come. viding his numerical code to compute the density bias. PD would like to thank Anupam Mazumdar for innumerable discussions during the course of this paper. SG acknowledges the support by the Hungarian National Office for Research and Technology (NKTH), through the Polányi Program. Figure 1 . 1Effect of varying (a) SFR (b) fesc,α (c) R I and (d) χ HI on Lα. Refer to Tab.1 for the parameters used for each of the lines in this plot. The dashed vertical line shows the wavelength of the redshifted (emission redshift z = 6.56) Lyα line. Figure 2 . 2Dependence of Lα on fesc for different values of χ HI . Adopted parameters areṀ⋆ = 27M ⊙ yr −1 , t * = 10 8 yr, fα = 1. The solid line shows the intrinsic Lyα luminosity. Curves with symbols refer to different values of χ HI = 0.15, 0.01, 10 −3 , 3 × 10 −4 from bottom to top, respectively. Figure 3 . 3Cumulative LAE Luminosity Function for the early reionization model (ERM). Points represent the data at three different redshifts: z = 4.5Dawson et al. (2007) (squares), z = 5.7Shimasaku et al. (2006) (circles), z = 6.56Kashikawa et al. (2006) with downward (upward) triangles showing the upper (lower) limits. Lines refer to model predictions at the same redshifts: z = 4.5 (dashed), z = 5.7 (dot-dashed), z = 6.56 (solid). Figure 4 . 4Lyα transmissivity as a function of the LAE dark matter halo mass at z = 6.56 for the LRM (solid line) and ERM (dashed). Lc(1375Å) = 2.13 × 10 40 [Ṁ * /M⊙yr −1 ]erg s −1Å−1 . Figure 5 . 5UV LAE Luminosity Function for the early reionization model (ERM). Points represent the data at two different redshifts: z = 5.7Shimasaku et al. (2006) (circles), z = 6.56 Figure 7 . 7Normalized distribution of the rest frame EW for LAEs at z = 4.5. Observed values from Dawson et al. (2007) (model results) are shown by solid (dot-dashed) lines. Figure 8 . 8Normalized distribution of the rest frame EW for LAEs at z = 5.7. Observed values from Shimasaku et al. (2006) (model results) are shown by solid (dot-dashed) lines. Figure 10 . 10Dependence of S W (values are color-coded by the bar on the right) on χ HI and Lα at z = 6.56 for a set of LAEs with SFR in the range predicted by the two reionization models, i.e.Ṁ⋆ = 2.7 − 197M ⊙ yr −1 .posite Lyα line, built from the observations of a sufficiently large number of LAEs might show such an anti-correlation. 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We generalize the multiple cover formula of Y. Toda (proved by Maulik-Thomas) for counting invariants for semistable coherent sheaves on local K3 surfaces to semistable twisted sheaves over twisted local K3 surfaces. The formula has an application to prove any rank S-duality conjecture for K3 surfaces.CONTENTS Richard Thomas for valuable discussions on the Vafa-Witten invariants, and Yukinobu Toda for the discussion of the multiple cover formula for K3 and twisted K3 surfaces. Y. J. is partially supported by NSF DMS-1600997. H.-H. T. is supported in part by Simons foundation collaboration grant.2.ÉTALE GERBES AND STACKS, NOTATIONSOur main reference for stacks is[29]and [42]. The notion ofétale µ r -gerbes over schemes and C˚-gerbes on a scheme was reviewed in [21, §2]. We fix the following notations.‚ Let S be a smooth K3 surface, and p : S Ñ S a µ r -gerbe. Let X = SˆC be the local K3 surface. ‚ p : X Ñ X always represents a µ r -gerbe over a smooth scheme X. ‚ The isomorphism classes of µ r -gerbes on S or any other scheme are classified by H 2 (S, µ r ). The exact sequence

10.4310/pamq.2021.v17.n5.a10

2201.09315

9f96b36c97576cd70d406062c2e4bf8dbed8e2cc

ON MULTIPLE COVER FORMULA FOR LOCAL K3 GERBES 23 Jan 2022 Yunfeng Jiang Hsian-Hua Tseng ON MULTIPLE COVER FORMULA FOR LOCAL K3 GERBES 23 Jan 2022 We generalize the multiple cover formula of Y. Toda (proved by Maulik-Thomas) for counting invariants for semistable coherent sheaves on local K3 surfaces to semistable twisted sheaves over twisted local K3 surfaces. The formula has an application to prove any rank S-duality conjecture for K3 surfaces.CONTENTS Richard Thomas for valuable discussions on the Vafa-Witten invariants, and Yukinobu Toda for the discussion of the multiple cover formula for K3 and twisted K3 surfaces. Y. J. is partially supported by NSF DMS-1600997. H.-H. T. is supported in part by Simons foundation collaboration grant.2.ÉTALE GERBES AND STACKS, NOTATIONSOur main reference for stacks is[29]and [42]. The notion ofétale µ r -gerbes over schemes and C˚-gerbes on a scheme was reviewed in [21, §2]. We fix the following notations.‚ Let S be a smooth K3 surface, and p : S Ñ S a µ r -gerbe. Let X = SˆC be the local K3 surface. ‚ p : X Ñ X always represents a µ r -gerbe over a smooth scheme X. ‚ The isomorphism classes of µ r -gerbes on S or any other scheme are classified by H 2 (S, µ r ). The exact sequence Let S be a smooth projective K3 surface, and X = SˆC the local K3 surface. Toda's multiple cover formula for the counting invariants for semistable coherent sheaves on local K3 surface X is a powerful tool for calculating the Donaldson-Thomas invariants, see [35,Corollary 6.8,6.10], [46,Conjecture 1.3]. The formula was proved in [35] and used by Tanaka-Thomas in [43] to calculate any rank Vafa-Witten invariants for K3 surfaces. In this paper we generalize and prove a multiple cover formula for the counting invariants of semistable twisted sheaves on the local optimal K3 gerbe 1 X := SˆC. This twisted version of multiple cover formula is used in [23] to calculate the SU(r)/Z r -Vafa-Witten invariants and prove the S-duality conjecture for K3 surfaces in any rank. The twisted multiple cover formula is also useful for the calculation of Donaldson-Thomas invariants for local K3 gerbes. Let S be a smooth projective K3 surface. The isomorphism classes of µ r -gerbes on S are classified by theétale cohomology group H 2 (S, µ r ). Let S Ñ S be a µ r -gerbe on S which is given by a class [S] P H 2 (S, µ r ). Then S determines a class ϕ([S]) = α P H 2 (S, OS ) tor , where ϕ is the map in the long exact sequence¨¨¨Ñ H 1 (S, OS ) Ñ H 2 (S, µ r ) ϕ ÝÑ H 2 (S, OS ) Ѩ¨ï nduced by the short exact sequence: 1 Ñ µ r Ñ OS (¨) r ÝÑ OS Ñ 1. The cohomology H 2 (S, OS ) tor is, by definition, the cohomological Brauer group Br 1 (S), and from de Jong's theorem [9] the Brauer group Br(S) = Br 1 (S). A µ r -gerbe S Ñ S is called optimal if the class ϕ([S]) = α P H 2 (S, OS ) tor is nonzero and has order r. We call (S, α), a K3 surface S together with a Brauer class α P Br 1 (S) a twisted K3 surface as in [16]. We also call the optimal gerbe S Ñ S a twisted K3 surface since its class in Br 1 (S) is α. We always use these two notions. The twisted Mukai vectors were constructed in [17] and [51]. Let Coh(S) be the category of coherent sheaves on the gerbe S. From [44], the category has a decomposition Coh(S) = à 0ďiďr´1 Coh(S) i where Coh(S) i is the subcategory of coherent sheaves on S with µ r -weight i. The subcategory Coh(S) 1 is the category of twisted sheaves and we use Coh(S, α) or Coh tw (S) to represent this category on a twisted K3 surface. Let X := SˆC which is the local K3 gerbe. It is an optimal µ r -gerbe over X = SˆC and its class in H 2 (X, µ r ) -H 2 (S, µ r ) is also given by α. We let Coh(X, α) or Coh tw (X) be the category of twisted sheaves on a local twisted K3 surface (X, α). Let H(A tw X ) be the Hall algebra of the category A tw X = Coh tw (X) as in [7], [26]. On our gerbes S and X, we always fix a generating sheaf Ξ = À r´1 i=0 ξ i , where ξ is a fixed S and X-twisted locally free sheaf. The modified Gieseker stability is defined in [37]. We use the geometric stability in [34] or modified Gieseker stability for twisted sheaves and the moduli stack was constructed in [34] and [37]. Let Γ 0 := Z ' NS(S) ' Q measuring the Mukai vectors for S. Then for each Mukai vector v P Γ 0 there is a virtual indecomposable element ǫ ω,X (v) := ÿ ℓě1,v 1 +¨¨¨+v ℓ =v,v i PΓ 0 p ω,v i =p ω,v (m) (´1) ℓ´1 ℓ δ ω,X (v 1 ) ‹¨¨¨‹ δ ω,X (v ℓ ) where p ω,v i (m) is the reduced modified Hilbert polynomial, and δ ω,X (v) := [M ω,X (v) ã Ñ x M(X)] P H(A tw X ) is an element in the Hall algebra. Here ‹ is the Hall algebra product, ω P NS(S) is an ample divisor, and M ω,X (v) ã Ñ x M(X) is the moduli stack of semistable twisted sheaves with Mukai vector v where x M(X) is the stack of coherent twisted sheaves on X. The Joyce invariant J(v) is defined by the Poincaré polynomial of ǫ ω,X (v), see Definition 6.15. We prove a multiple cover formula for the invariant J(v). where k|v means k divides every component of v, Hilb xv/k,v/ky+1 (S) is the Hilbert scheme of xv/k, v/ky + 1 points on S, and xv/k, v/ky is the Mukai pairing. The method to prove Theorem 1.1 is as follows. In [46], Toda actually defined the curve counting invariants N n,β for the rank zero semistable sheave on a K3 surface S. In [48] Toda has proved a wall crossing formula of Pandharipande-Thomas stable pair invariants PT(X) in [39] for any Calabi-Yau threefold X in terms of the invariants N n,β and some limit stable invariants L n,β , see (5.1.4). We prove Toda's wall-crossing formula for µ r -gerbes X Ñ X over a Calabi-Yau threefold X. In [46], Toda used wall crossing formula of weak stability conditions on the derived category of coherent sheaves on a local K3 surface X = SˆC to conjecture a wall crossing formula for the stable pair invariants PT(X) in terms of the Gopakumar-Vafa invariants, see (5.2.2). Then Toda showed that for a local K3 surface X, these two formula are equivalent if the multiple cover formula [35], Maulik-Thomas proved the multiple cover formula using KKV formula for K3 surfaces [40]. For our purpose, it is not necessary to introduce the stable pair theory for twisted local K3 surfaces (which are optimal µ r -gerbes over local K3 surfaces). We only need a stable pair theory for arbitrary µ r gerbes. Thus we generalize and survey some results of stable pair theory for the cyclic µ r -gerbe S Ñ S over a smooth projective K3 surface S, and the reduced stable pair theory on the local K3 gerbe X = SˆC over X = SˆC. We also generalize Oberdieck's theorem for equating the reduced stable pair invariants to Behrend's weighted Euler characteristic forétale gerbes over surfaces, in which similar result also holds for Joyce-Song twisted stable pairs in [23]. Toda's wall crossing results for counting invariants N n,β are also generalized to cyclic gerbes. All of these results are routine generalizations for cyclic gerbes, but will have applications to the twisted multiple cover formula. Now for a twisted K3 surface (S, α), and a local twisted K3 surface (X = SˆC, α), we prove that the counting invariants N n,β for these one dimensional sheaves keep the same as for trivial µ rgerbe S 0 , and all the counting invariants for trivial µ r -gerbe S 0 are the same as the K3 surface S. This is because counting twisted one dimensional sheaves on a twisted local K3 surface is the same as counting usual untwisted one dimensional sheaves on a local K3 surface. Then we still have the multiple cover formula (1.0.1). The invariants N n,β keep the same for the optimal µ r -gerbe S Ñ S and X Ñ X. Then we prove a similar result as in [46,Theorem 1.2] for the twisted sheaves on S and X, which we call the twisted Hodge isometry theorem, see Corollary 6.21. Therefore we get the multiple cover formula in Theorem 1.1. 1.1. Outline. After we set up basic notions and notations ofétale gerbes and stacks in §2, we introduce stable pair theory of Pandharipande-Thomas onétale gerbes over surfaces in §3; and prove Oberdieck's theorem for equating the reduced stable pair invariants to Behrend's weighted Euler characteristic forétale gerbes over surfaces in §4. In §5 we generalize the wall crossing formula of Toda to Calabi-Yauétale gerbes and review the multiple cover formula of Toda. In §6 we use twisted sheaves on optimal local K3 gerbes, and study the twisted multiple cover formula for twisted sheaves. In the Appendix, we generalize Toda's counting invariants for semistable objects for K3 surfaces to twisted K3 surfaces and prove that the invariants do not depend on the stability conditions, which is used in §6. Convention. We work over complex number C throughout of the paper. We use Roman letter E to represent a coherent sheaf on a projective DM stack or anétale gerbe S, and use curly letter E to represent the sheaves on the total space Tot(L) of a line bundle L over S. We reserve rk for the rank of the torsion free coherent sheaves E. For a K3 gerbe in this paper we mean a cyclic µ r -gerbe S Ñ S where S is a K3 surface. 1 Ñ µ r Ñ OS (¨) r ÝÑ OS Ñ 1 induces a long exact sequence:¨¨Ñ H 1 (S, OS ) ψ ÝÑ H 2 (S, µ r ) ϕ ÝÑ H 2 (S, OS ) Ѩ¨Ẅ e call a µ r -gerbe p : S Ñ S "essentially trivial" if it is lying in the image of the map ψ in the above exact sequence; and "optimal" if the order |ϕ(S)| = r in H 2 (S, OS ) tor . ‚ For the surface S, the Brauer group Br(S) is by definition the group of isomorphism classes of Azumaya algebras over S; which is equal to its cohomological Brauer group Br 1 (S) := H 2 (S, OS ) tor . ‚ In theétale cohomology group H 2 (S, µ r ), we call a class g P H 2 (S, µ r ) "algebraic" if it comes from a class in H 1 (S, OS ), i.e., a line bundle over S in the above exact sequence. We call g P H 2 (S, µ r ) "non-algebraic" if its image in H 2 (S, OS ) under ϕ is nonzero. STABLE PAIR THEORY ON SOME THREEFOLD DM STACKS In this section we list some basic materials on the stable pair theory of counting curves by Pandharipande-Thomas on threefold DM stacks, and mainly focus on the threefold Calabi-Yau DM stack X := SˆC and Y := SˆE, where S Ñ S is a µ r -gerbe over a smooth projective surface S, and E is an elliptic curve over C. 3.1. Stable pair theory. Let p : X Ñ X be a smooth threefold DM stack, which is a µ r -gerbe over a smooth threefold X. Let Ξ be a generating sheaf on X. The definition and property of generating sheaves can be found in [37]. Definition 3.1. A stable pair on X is given by a morphism O X b Ξ s ÝÑ F where (1) F is a pure dimension one sheaf on X; (2) coker(s) is zero dimensional. We explain the reason to make such a definition comparing with the stable pair theory on smooth threefold X in [39]. For the µ r -gerbe X Ñ X, recall the geometric stability for a coherent sheaf E on X, which is given by the geometric Hilbert polynomial P g (E, m) = χ g (E b p˚O X (m)) := [IX : X]¨deg(Ch(E(m))¨Td X )) is defined by the geometric Euler characteristic in [34]. We assume that E is supported in dimension one, then Ch(E) = (0, 0, Ch 2 (E) =´c 2 (E), Ch 3 (E) = 1 6 c 3 (E)) P H˚(X, Q). Thus let c 1 (p˚O X (1)) = x, we calculate χ g (E b p˚O X (m)) = r ż X (´c 2 (E))mx + deg(Ch(E)¨Td X ) . Since the sheaf E is supported on a dimension cycle β P H 4 (X, Q), the second Chern class c 2 (E) =´β as a class, and we can write the above as χ g (E b p˚O X (m)) = r m ż β x + deg(Ch(E)¨Td X ) . Then fixing a K-group class for a purely one dimensional sheaf E in K 0 (X) is the same as fixing the cycle β P H 2 (X, Q) and a rational number n P Q representing Ch 3 (E). Let rk(E) := r¨ş β x. The stability of pairs (E, s) is defined as q-stable if (1) χ g (F(m)) rk(F) ă χ g (E(m)) + q(m) rk(E) , m ąą 0 for any proper subsheaf F Ă E; and (2) χ g (F(m)) + q(m) rk(F) ă χ g (E(m)) + q(m) rk(E) , m ąą 0 for any proper subsheaf F Ă E such that s factors through. A similar argument as in [39,Lemma 1.3] shows that for sufficient large q, semistablity coincides with stability and (E, s) is stable if and only if the two conditions in Definition 3.1 hold. Fix a cycle β P H 4 (X, Q), and a rational number n P Q such that n = deg(Ch(E)¨Td X ), we define the moduli stack P Ξ n (X, β) of stable pairs of X by fixing the generating sheaf Ξ. 3.2. Stable pairs on Y = SˆE. In this section we consider the stable pairs on the Calabi-Yau threefold DM stack Y = SˆE, where p : S Ñ S is a µ r -gerbe over a smooth projective K3 surface S, and E is an elliptic curve. A curve class in H 2 (S, Q) Ă H 2 (Y, Q) is given by a line bundle in Pic(S), if d P Z ě0 is a nonnegative integer, we let (β, d) P H 2 (Y, Q) be the class (β, d) := ι S˚( β) + ι E˚( d[E]) where ι S : S ã Ñ Y and ι E : E ã Ñ Y are inclusions. The Calabi-Yau threefold DM stack p : Y = SˆE Ñ Y := SˆE is also a µ r -gerbe over Y and its class in H 2 (Y, µ r ) is given by the class of the gerbe [S] P H 2 (S, µ r ). We consider the moduli stack P α (Y, (β, d)) of stable pairs (E, s) on Y with curve class (β, d) and K-group class α P K 0 (Y). Form the commutative diagram: (3.2.1) S ι S / / p SˆE p S ι S / / SˆE, , SˆE SˆE p o o p E / / p S E S S p o o Lemma 3.2. The stable pair (E, s) in P α (Y, (β, d)) only depends on the curve class (β, d) and a rational number n given by n = ż Y Ch(p˚E)¨Td Y . Proof. We calculate ż Y Ch(α)¨Td SˆE = ż Y (´c 2 (α) + c 3 (α))(1 + c 1 (Y)) + 1 12 c 1 (Y) 2 + c 2 (Y) = ż Y ((´c 2 (α))¨c 1 (Y) + c 3 (α)) = ż Y ((ι S˚( β) + ι E˚( d[E]))(c 1 (S) + c 1 (E)) + c 3 (α)) = ż Y ((ι S˚( β)¨c 1 (E) + ι E˚( d[E])¨c 1 (S)) + c 3 (α)) = ż Y ι S˚( β)¨c 1 (E) + ż Y ι E˚( d[E])¨c 1 (S) + ż Y c 3 (α) = 1 r ż Y p˚(ι S˚( β)¨c 1 (E)) + ż Y p˚(ι E˚( d[E])¨c 1 (S)) + ż Y p˚c 3 (α) Here we let p˚[E] = p˚α = ι S˚β + ι E˚( d[E]) P K 0 (Y) and β = p˚β P H 2 (S, Q), and the last equality is from the µ r -gerbe structure. Thus according to diagrams in Diagram (3.2.1), the above integral is equal to ż Y Ch(p˚E)¨Td Y which we define as the rational number n. Remark 3.3. The number n = ş Y Ch(p˚E)¨Td Y is actually an integer by Riemann-Roch theorem. In the following we let P n (Y, (β, d)) be the moduli stack of stable pairs (E, s) on Y with curve class (β, d) and rational number n above. 3.3. The elliptic curve E-action. We also consider the elliptic curve E action on Y = SˆE by the translation of the factor E, m Y : EˆY Ñ Y by: (x, (s, e)) Þ Ñ (s, e + x). Denote by P := P n (Y, (β, d)). Let t x for x P E be the translation of Y determined by x. Let ι : E Ñ E be the inverse map by x Þ Ñ´x, and let Ψ : EˆYˆP ιˆid YˆP ÝÑ EˆYˆP m Yˆi d P ÝÑ YˆP be the composition. We let I = [O YˆP Ñ F] be the universal stable pair on YˆP. Then Ψ˚(I) defines a family of stable pairs on Y over EˆP. By the universal property of P, this gives a map m P : EˆP Ñ P such that Ψ˚(I) -mP(I). The map m P defines a group action on P. Points x P E act on I P P by I + x := m P (x, I) = t˚x(I) = t x˚( I). Let m YˆP : EˆYˆP Ñ YˆP be the diagonal action by (e, x, I) Þ Ñ (x + e, I + e). Then Ψ˚(I) -mP(I) gives mYˆP(I) = πE(I) where π E : EˆYˆP Ñ YˆP is the projection. These data satisfy the cocycle condition which descends to the quotient: ρ : YˆP Ñ (YˆP)/E with the diagonal action m YˆP . The universal stable pair I descends along ρ and gives I on the quotient (YˆP)/E and ρ˚(I) -I. 3.4. The reduced perfect obstruction theory. For the K3 surface S, the µ r -gerbe S Ñ S is a holomorphic symplectic DM stack. We follow Oberdieck [38] and Kool-Thomas [28] to construct a reduced perfect obstruction theory on P. Let π Y : YˆP Ñ Y and π P : YˆP Ñ P be projections. We consider the Atiyah class: [18]. Since L YˆP -πY(Ω Y ) ' πP(L P ) Ñ πPL P , we use the composition maps At Y (I) P Ext 1 YˆP (I, I b L YˆP ) inExt 1 YˆP (I, I b L YˆP ) Ñ Ext 1 YˆP (RHom(I, I), πPL P ) Ñ Ext 1 YˆP (RHom(I, I) 0 , πPL P ) - ÝÑ Hom P (Rπ P˚R Hom(I, I) 0 b ω P [2], L P ) and the last isomorphism is from relative Verdier duality. Then the image At P (I) in the above image gives a perfect obstruction theory: E ‚ = Rπ P˚( RHom(I, I) 0 b ω P ) [2] Ñ L P . As in [28, §2.3], we take cup product of the obstruction sheaf of E ‚ with the Atiyah class At Y (I) and get a semi-regularity map: (3.4.1) sr : (E ‚ ) _ Ñ H 1,3 (Y) b O P [´1] by Ext 2 π P (I, I) 0 ã Ñ Ext 2 π P (I, I) At(I) ÝÑ Ext 3 π P (I, I b L YˆP ) Ñ Ext 3 π P (I, I b πYΩ Y ) tr ÝÑ R 3 π P˚πY Ω Y - ÝÑ H 1,3 (Y) b O P , which is surjective since S is a K3 gerbe. Taking the dual of the semi-regularity map sr we get: H 1,3 (Y) _ b O P [1] Ñ E ‚ and let E ‚ red := Cone(H 1,3 (Y) _ b O P [1] Ñ E ‚ ),E ‚ red Ñ L P for P with virtual dimension h 1,3 (Y) = h 0,2 (S). Proof. We generalize the proof in [28,Theorem 2.7] to the gerbe setting. Step 1: We study the twistor family for µ r -gerbes on a K3 surface S. First let us recall the twisted family for K3 surfaces. Consider H 1 (T S ) and let m be the maximal ideal of the origin 0 P H 1 (T S ). H 2 (S B , µ r ) -H 2 (S, µ r ). Then the class of the gerbe [S] P H 2 (S, µ r ) determines a µ r -gerbe on S B . We denote p : S B Ñ S B for this µ r -gerbe whose central fiber gives p : S Ñ S. This is the twistor family for the K3 gerbe S. We also let Y B := S BˆE ; Y B := S BˆE be the families of SˆE over B. Step 2: Let S j / / S B S j / / S B ; SˆE j / / Y B SˆE j / / Y B ; be the inclusions of the central fibers. We claim that the natural morphisms j˚: P n (S, β) - ÝÑ P n (S B /B, β B ) and j˚: P n (Y, (β, d)) - ÝÑ P n (Y B /B, (β B , d)) are isomorphisms, where β B = β b 1 P H 2 (S B /B) -H 2 (S, C) b O B . We follow the proof in [28, Proposition 2.2]. First P n (S B , β B ) b B t0u -P n (S, β). We need to prove that if there is an Artinian scheme A with a morphism to B, and proper flat family I A Ñ A of stable pairs over S B such that it pulls back to a stable pair I 0 over S and h 0˚[ β] = β B , then A Ñ B factors through 0 P B. But this is from [28, Lemma 2.1] since we fix β P H 2 (S, Q) such that β P H 1,1 (S) X (H 2 (S, Z)/ tor). The case of Y and Y B is similar. Step 3: We then generalize [28,Theorem 2.7] to the gerbe case and get the perfect obstruction theory. Consider the algebraic twistor family S B Ñ B and the family Y B = S BˆE Ñ B, we have the moduli stack of stable pairs P := P n (Y B /B, ι˚β B ) Ñ B on the fibers is isomorphic to the moduli stack P n (Y, β) on Y = SˆE. For the stack Y B over B, since for the stable pair I ‚ P P n (Y B /B, ι˚β B ) the deformation and obstruction are given by Ext 1 Y B (I ‚ , I ‚ ) 0 and Ext 2 Y B (I ‚ , I ‚ ) 0 , the argument in [36, §3] works for this µ r -gerbe case and gives a relative perfect obstruction theory: E ‚ rel := Rπ P˚( RHom(I, I) 0 b ω PˆBY B /P )[2] Ñ L P/B . This perfect obstruction theory fits into the following diagram: F ‚ / / E ‚ rel / / Ω B [1] = L P / / L P/B / / Ω B [1] and defines a perfect absolute obstruction theory F ‚ over P. All the stable pairs of P lie scheme theoretically on the central fiber Y, and E ‚ rel is the usual complex of stable pair theory on Y. Hence F ‚ has virtual dimension h 2,0 (S). The same proof in [28,Theorem 2.7] shows that F ‚ Ñ E ‚ Ñ E ‚ red is an isomorphism. 3.5. Symmetric obstruction theory of the quotient. Global vector field on P by the E-direction on Y. Let m : EˆP Ñ P be the action, then we have the complex T E b O EˆP ã Ñ T EˆP dm ÝÑ m˚T P where dm is the differential. We restrict it to 0 EˆP ã Ñ EˆP and get a global vector field: (3.5.1) v : H 0 (T E ) b O P = T E,0 E b O P Ñ T P -E xt 1 π P (I, I) 0 . Similar arguments in [38,Lemma 1] gives: H 0 (T E ) b O P - ÝÑ H 0 (Y, T Y ) b O P At Y (I) ÝÑ Ext 1 (I, I) b O P Ñ E xt 1 π P (I, I) which is the same as (3.5.1). Since we have H 0 (E, T E ) = H 0 (Y, T Y ), let s : H 0 (T E ) _ b O P - where G ‚ := Cone(H 1,3 (Y) _ b O P [1] Ñ I ‚ ) or G ‚ := Cone(E ‚ red B _ ÝÑ H 0 (T E ) _ b O P [´1] ). Then the above induces the following diagram: H 1,3 (Y) _ b O P [1] sr _ / / s _ [1] I ‚ / / r θ G ‚ λ H 0 (T E ) b O P [1] B / / (E ‚ red ) _ [1] / / (G ‚ ) _ [1] where λ is induced by the morphism in the left square. Then from [38, Proposition 2], λ _ [1] -λ and λ : G ‚ Ñ (G ‚ ) _ [1] is a non-degenerate symmetric bilinear form of degree one. 3.5.3. The morphism q : P Ñ E. Since we work on SˆE = Y, and P = P n (Y, (β, d)) is the moduli stack. Let β P Pic(S) be such that β P Pic(S) is a curve class. We generalize the morphism p c in [38, §2.2] (which we call it q c ) to this case. First we have for each k P Z, (3.5.2) σ k : EˆE Ñ E given by (x, e) Þ Ñ e + kx which is the action of E on itself by k-times. Also (3.5.3) m P : EˆP Ñ P is given by (x, I) Þ Ñ I + x = t x˚I . We recall the construction of the morphism (3.5.4) q c : P n (Y, (β, d)) Ñ E We let p E := Pic 0 (E) be the dual of E which is isomorphic to E. Let L Ñ Eˆp E be the Poincaré line bundle which is defined by (1) L ξ = L| Eˆξ is isomorphic to ξ for any ξ P Pic 0 (E); (2) L| 0ˆp E -O p E . So L X = L| xˆp E is isomorphic to x P E since Pic 0 ( p E) -E. Let p Y = Sˆp E and L Y Ñ Yˆp Y be the pullback of L from Yˆp Y Ñ Eˆp E. Let Φ L Y : D b (Y) Ñ D b ( p Y) be the equivalence defined by: E Þ Ñ Rp p Y˚( L Y b pY(E) ). This is the Fourier-Mukai transform. For a line bundle L P Pic(S), such that c 1 (L) = c. Let π S : p Y Ñ S be the projection. Define (¨) b πSL : D b ( p Y) Ñ D b ( p Y) to be: E Þ Ñ E b πSL. Let π p E : p Y Ñ p E be the projection to p E. We define q c in the level of geometric points: P(C) = P n (Y, (β, d))(C) ã Ñ D b (Y) Φ L Y ÝÑ D b ( p Y) (¨)bπSL ÝÑ D b ( p Y) Rπ p EÝ Ñ D b ( p E) det ÝÑ Pic m ( p E) = E for some m. The algebraic morphism q c can be constructed by giving a line bundle on P n (Y, (β, d))ˆp E. The line bundle is given by the determinant of the following Rπ p EˆP˚( Φ L YˆP (I) b πSL) where I is the universal twisted stable pair on YˆP. We have a similar result as in [38,Proposition 3]. Proposition 3.5. There exists a morphism q c : P n (Y, (β, d)) Ñ E such that for k = xc, βy + n, it is E-equivariant with respect to m P and σ k defined before. Moreover, for I P P, x P E, q c (I + x) = q c (I) + kx. Proof. Since we consider the twisted stable pairs I = (Ξ b O Y Ñ F), where our µ r -gerbe twist means the µ r -action on the generating sheaf is given by the whole primitive action of the generator of µ r . Then all the calculations in [38, Lemma 4, Lemma 5, Lemma 6] keep the same except we need to multiply by e 2πi/r on the Chern character. Therefore the calculation in [38,Lemma 6] still holds for twisted stable pairs. Also for c P Pic(S), we have xc, βy + n = xc, βy + n, where c and β are the images in H 2 (S, Q) Ñ H 2 (S, Q). Virtual class. Consider the morphism q c : P Ñ E. We will construct a perfect obstruction theory on K := q´1 c (0 E ). Let 0 EˆP ã Ñ EˆP be the inclusion, then the derivative map: d : L P Ñ Ω E,0 b O P is the second one in π˚L P/E Ñ L P Ñ L π -O P induced from π : P Ñ P/E. From the analysis of obstruction theory above we form the following diagram: G ‚ / / ϕ E ‚ red / / Ω E,0 b O P - π˚L P/E / / L P / / Ω E,0 b O P where h 0 (T E ) _ = Ω E,0 . Then it indues a morphism: ϕ : G ‚ Ñ π˚P/E As in [38,Proposition 4], let ι : L ã Ñ P be the inclusion, then Lι˚(ϕ) : Lι˚G ‚ Ñ Lι˚π˚L P/Er π˚(L P/E ) where r π : K Ñ P Ñ P/E; and r π isétale. Hence r π˚(L P/E ) -L K and the composition Lι˚G ‚ Lι˚(ϕ) ÝÑ L K defines a symmetric obstruction theory on K. Thus this symmetric obstruction theory gives a virtual fundamental class [K] vir P A 0 (K) such that 1 |G| π˚([K] vir ) = [P/E] vir P A 0 (P/E) where G is the finite group of theétale map r π. Now we show Oberdieck's theorem in [38] for µ r -gerbe S and Y = SˆE. The reduced perfect obstruction theory E ‚ red defined before gives a one-dimensional virtual fundamental cycle [P] red P A 1 (P). Let ω P H 2 (E, Z) be the class of a point and β _ P H 2 (S, Q) such that xβ, β _ y = 1. Definition 3.6. Define r N Y n,(β,d) := ż [P red ] τ 0 (πS(β _ ) Y πE(ω)) where τ 0 (¨) is the insertion operator defined in [39, §3.6], and π S : SˆE Ñ S, π E : SˆE Ñ E are projections. This is called the incidence DT-invariant. There is another invariant given by the Behrend function [3]. The elliptic curve E acts on P and we have the quotient P/E. Let ν P : P/E Ñ Z be the Behrend function on the quotient. Theorem 3.8 ([38]). We have: r N Y n,(β,d) = N Y n,(β,d) . Proof. Let π : P Ñ P/E be the projection. We first show that π˚[P/E] vir = [P red ]. We use the diagram EˆK π K / / m K r π P π / / P/E From the construction of reduced obstruction theory E ‚ red (which differs from E ‚ by O P ), we have: [P] red = ts((E ‚ ) _ ) X c F (P)u 1 where s((E ‚ ) _ ) is the Segre class and c F (P) is the Fulton class of P. Here we use the description of virtual class of Siebert [41] using Fulton class. The map m isétale, and deg( r π) = |G|, we have: [P] red = 1 |G| m˚ts((m˚E ‚ ) _ ) X m˚c F (P)u 1 . Form the following diagram: YˆP π P / / ρ P π (YˆP)/E π P/E / / P/E there exists a universal pair I P (YˆP)/E such that ρ˚I = I, and I is the universal stable pair on YˆP. Let H ‚ := Rπ P/E˚R Hom(I, I) 0 [2]. Then H ‚ is of amplitude [´1, 0] and π˚H ‚ = E ‚ = Rπ P˚R Hom(I, I) 0 [2]. From Fulton's Chern class explanation of virtual classes, if we let h : Y Ñ P/E be any properétale morphism of deg(h) with Y a scheme, we have: [P/E] vir = 1 deg(h) h˚ts((h˚H ‚ ) _ ) X c F (Y)u 0 . Thus since Fulton's Chern classes are invariant underétale pullback, and m˚E ‚ = m˚π˚H ‚ = πK r π˚H ‚ , we have [P] red = 1 |G| m˚ts((πK r π˚H ‚ ) _ ) X πKc F (K)u 1 = 1 |G| m˚πKts(( r π˚H ‚ ) _ ) X πKc F (K)u 0 = π˚[P/E] vir . From the definition of the invariants in Definition 3.6 and Definition 3.7, and look at the diagram: Y π Y / / YˆP π P / / ρ P π (YˆP)/E π P/E / / P/E let D P H 2 (S, Q) be such that xβ, Dy = 1, then we have r N Y n,(β,d) is the degree of D := (´Ch 2 (I)¨πY(πS(D) Y πE(ω)) X πP[P] red . We need to show (π P/E˝ρ )˚D = [P/E] vir . We have Ch 2 (I) = ρ˚Ch 2 (I), therefore ρ˚D = ρ˚(πE(ω) X ρ˚(α)) for α = (´Ch 2 (I) Y πS(D)) X πP /E [P/E] vir . Then the same argument in [38, §4.4] shows that ρ˚D = α, and (π P/E˝ρ )˚D = π P/E˚α = xD, βy[P/E] vir = [P/E] vir . BEHREND EQUALS TO REDUCED INVARIANTS OF OBERDIECK In this section we perform the following invariants on the Calabi-Yau threefold DM stack X := SˆC for the µ r -gerbe S Ñ S over a K3 surface S. For the Calabi-Yau DM stack X = SˆC, we choose a generating sheaf on X such that it is the pullback of the generating sheaf Ξ on S by the projection X Ñ S. We still denote this generating sheaf by Ξ. We consider the stable pairs on (O X b Ξ s Ñ F) on X. For n P Q, we let P n (X, ι˚β) = P n (SˆC, ι˚β), where ι : S ã Ñ X is the inclusion, be the moduli stack of stable pairs on X with number n given by n = ż SˆP 1 Ch(F)¨Td SˆP 1 and class [F] = ι˚β. Since X is also a Calabi-Yau DM stack, P n (X, ι˚β) admits a symmetric obstruction theory and there is virtual fundamental class [P n (X, β)] vir . There is a C˚-action on X by scaling the fiber C and induces an action on the moduli stack [P n (X, β)] vir . From [10], there is an induced virtual fundamental class on the fixed locus (P n (X, β)) C˚. Definition 4.1. P red n,β (X) := ż [(P n (X,β)) C˚]vir 1 e(N vir ) P Q where e(N vir ) is the Euler class of the virtual normal bundle of the C˚-action. On the other hand, we have the Behrend function ν P : P n (X, ι˚β) Ñ Z on the moduli stack and the invariant χ(P n (X, ι˚β), ν P ) is Behrend's weighted Euler characteristic. Similar to [35], let us form the generating function: (4.0.1) Z red P (X; q, v) := ÿ α=(β,n) P red n,β (X)q n v β which is the generating function of rational residue reduced stable pair invariants of X. We also have (4.0.2) Z χ P (X; q, v) := ÿ α=(β,n) χ(P n (X, ι˚β), ν P )q n v β whichZ χ P (Y/E; q, v) := ÿ α=(β,n) χ(P n (Y, ι˚β)/E, ν P )q n v β , where Y = SˆE and we consider the quotient of the moduli stack of stable pairs. Then we have: Z red P (Y/E; q, v) =´log(1 + Z χ P (X; q, v)). Proof. The proof is the same as in [35,Proposition 3.2], since in that proof the key points only happen at the elliptic curve E. We sketch it here. By the C˚-action on X = SˆC, the induced action on P n (X, ι˚β) preserves the Behrend function ν P and let P 0 α Ă P n (X, ι˚β) be the fixed point locus, which consists of stable pairs with set-theoretic supports on Sˆt0u Ă SˆC. Therefore the weighted Euler characteristic χ(P n (X, ι˚β), ν P ) can be localized to P α = χ(P 0 α , ν P | P 0 α ). For the Calabi-Yau DM stack Y = SˆE, the exponential map e (´) gives an isomorphism on an analytic neighborhood of any point p P E and a neighborhood of 0 P C. Then this translates stable pairs from Sˆt0u Ă X to Sˆtpu Ă Y, and P 0 αˆE is the moduli stack of stable pairs on Y supported set theoretically on a single fiber Sˆtpu. Then we stratify the moduli stack P n (Y, ι˚β) by the minimal number k of fibers S on which the stable pairs are set-theoretically supported. Let the distinct charges be α 1 ,¨¨¨, α l . Let k i be the number of fibers S such that the charge is α i = (β i , n i ), then l ÿ i=1 k i = k; l ÿ i=1 k i α i = α. Then the same argument in [35,Proposition 3.2] shows that the stratum of P n (Y, ι˚β) with this data is: (4.0.3) P 0 α 1ˆ¨¨¨ˆP 0 α 1 ˆ¨¨¨ˆ P 0 α lˆ¨¨¨ˆP 0 α l ˆ E k z∆ k /(S k 1ˆ¨¨¨ˆS k l ) where the action acts freely and ∆ k is the big diagonal. Therefore the weighted Euler characteristic of (4.0.3) is:´P k 1 α 1 k 1 !¨P k 2 α 2 k 2 !¨¨¨P k l α l k l ! χ E k z∆ k E = (´1) k 1 k k k 1 , k 2 ,¨¨¨, k l P k 1 α 1¨¨¨P k l α l . Then summing over all strata and all α, we have: 8 ÿ k=1 ÿ l,k i ,α i distinct ř l i=1 k i =k (´1) k k k k 1 , k 2 ,¨¨¨, k l (P α 1 Q α 1 ) k 1¨¨¨( P α l Q α l ) k l = 8 ÿ k=1 1 k ´ÿ α P α Q α k =´log(1 + ÿ α P α Q α ). Theorem 4.3. We have Z red P (X; q, v) =´log(1 + Z χ P (X; q, v) ). Proof. From Lemma 4.2, we need to prove: Z χ P (Y/E; q, v) = Z red P (X; q, v). We show that P red n,β (X) = χ(P/E, ν P ) = N Y n,β , where P := P n (Y, ι˚β). From Theorem 3.8, N Y n,β = r N Y n,β . Thus we need to show P red n,β (X) = r N Y n,β . Here r N Y n,β = r N Y n,(β,0) and (4.0.4) r N Y n,(β,0) = ż [P n (Y,ι˚β)] red τ 0 (ι˚β _ ) where we let β _ P H 2 (S, Q) such that ş S β Y β _ = 1. We take the degeneration of the elliptic E to a one-nodal rational elliptic curve. The degeneration formula [32], [52] expresses the right side of (4.0.4) as: (4.0.5) ż [P n (SˆP 1 /(S 0 YS 8 ),ι˚β)] red τ 0 (β _ ) We use the degeneration formula again to calculate (4.0.5), and degenerate SˆP 1 /S 8 to: (SˆP 1 )/S 8 ď S 8 "S 0 (SˆP 1 )/(S 0 Y S 8 ) and the degenerate formula gives: (4.0.6) ż [P n (SˆP 1 /S 8 ,ι˚β)] red τ 0 (ι˚β _ ) = ÿ (β i ,n i ) ż [P n 1 (SˆP 1 /S 8 ,ι˚β 1 )ˆP n 2 (SˆP 1 /(S 0 YS 8 ),ι˚β 2 )] red 1ˆτ 0 (ι˚β _ ) where the sum is over all (β 1 , n 1 ), (β 2 , n 2 ) whose sum is (β, n). Same analysis as in [35,Theorem 4.2] shows that only the case (β 2 , n 2 ) = (β, n) and (β 1 , n 1 ) = 0 contributes to (4.0.5). Therefore from (4.0.6), (4.0.7) r N Y n,(β,0) = ż [P n (SˆP 1 /S 8 ,ι˚β)] red τ 0 (ι˚β _ ) Now we use virtual localization to the C˚-action on P 1 with weight +1 on tangent space at 0. Lift ι˚β _ to HC˚(SˆP 1 ) by the map ι : Sˆt0u ã Ñ SˆP 1 . Then localization formula reduces the integral (4.0.7) to: ż [P n (X,ι˚β) C˚]red 1 e(N vir ) τ 0 (ι˚β _ ). Since over S 0 , τ 0 (ι˚β _ ) is c 1 (t) ş S β Y β _ = t, we have ż [P n (X,ι˚β) C˚]red 1 e(N vir ) = P red n,β (X). WALL CROSSING FOR CALABI-YAU GERBES AND TODA'S MULTIPLE COVER FORMULA In this section we generalize Toda's wall crossing formula for the D0-D2-D6 bound states to µ rgerbes and get the multiple cover formula for counting semistable coherent sheaves on the gerbe. 5.1. Wall-crossing in D0-D2-D6 bound states for µ r -gerbes. In this section we let X be a µ r -gerbe over a smooth projective Calabi-Yau threefold X. We study the wall crossing formula of [47,Formula (28)], and also [48] on the µ r -gerbe X. We follow the techniques in Toda's papers. The arguments should work for any Calabi-Yau threefold DM stack X. 5.1.1. Toda's weak stability conditions on the category of D0-D2-D6 bound states. Let X Ñ X be a µ r -gerbe over X, which is given by a class [X] P H 2 (X, µ r ). Let Coh ď1 (X) be the subcategory of the category of coherent sheaves Coh(X) on X consisting of coherent sheaves E on X such that supp(E) ď 1. Let A X := xO X , Coh ď1 (X)[´1]y ex . Then A X is the heart of a bounded t-structure on a derived category D X (this is proved in [48,Lemma 3.5] which works for the gerbe X): D X := xO X , Coh ď1 (X)y tr Ă D b (Coh(X)). So A X is an abelian category. From physics, D X is called the category of D0-D2-D6 bound states for the gerbe X. [48], [8] to this triangulated category D X to prove the DT/PT-correspondence for the counting invariants for X Ñ X. Since the decomposition conjecture of DT and PT invariants for the µ r -gerbe X Ñ X is proved in [11], the DT/PTcorrespondence is already known. Remark 5.1. It is possible to apply stability conditions and wall crossing in Let us recall how we construct the category A X . Let Coh ě2 (X) be the subcategory of Coh(X) given by Coh ě2 (X) := tE P Coh(X)|Hom(Coh ď1 (X), E) = 0u. Then (Coh ď1 (X), Coh ě2 (X)) is a torsion pair in the sense of [48, Definition 3.1] or [14]. There is an abelian category Coh : (X) with respect to this torsion pair and A p X := Coh : (X) = xCoh ě2 (X)[1], Coh ď1 (X)y ex . The subcategory Coh ď1 (X) Ă A p X is closed under subobjects and quotients in A p X . The category A X is the intersection A X = D X X A p X [´1] . Let us construct the weak stability conditions on D X following Toda. Definition 5.2. For the Calabi-Yau threefold DM stack X Ñ X, set Γ 0 := Q ' H 2 (X, Q) and the group homomorphism cl 0 : K(Coh ď1 (X)) Ñ Γ 0 is given by E Þ Ñ (Ch 3 (E), Ch 2 (E)). Define Γ := Q ' H 2 (X, Q) ' Z = Γ 0 ' Z. The general Chern character (not orbifold Chern character) cl : K(A X ) Ñ Γ is given by: E Þ Ñ (Ch 3 (E), Ch 2 (E) , Ch 0 (E)) and defines a group homomorphism. This is because A X is generated by O X , E P Coh ď1 (X)[´1], and by Poincare duality, Ch 3 (E) P H 6 (X, Q), Ch 2 (E) P H 4 (X, Q) -H 2 (X, Q). We recall the weak stability condition in [48] and [47]. Let A X be an abelian category and Γ be a finitely generated abelian group. Fix H Ă C such that H = tρ¨e πiφ |ρ ą 0, 0 ă φ ď 1u. Definition 5.3. A (Bridgeland) stability condition on A X is a group homomorphism Z : Γ Ñ C satisfying the following conditions: (1) For any nonzero E P A X , Z (E) = Z (cl(E)) P H and the argument: arg(Z (E)) P (0, π] is well-defined. The object E P A X is called Z-(semi)stable if for any nonzero subobject 0 ‰ F Ĺ E, arg Z (F) ă (ď) arg Z (E). (2) For any object E P A X , there is a Harder-Narasimhan filtration: 0 = E 0 Ă E 1 è¨¨Ă E m = E such that each F i = E i /E i´1 is Z-semistable and arg Z (F 1 ) ą arg Z (F 2 ) ą¨¨¨ą arg Z (F m ). Now for the group Γ, we fix a filtration: 0 = Γ´1 Ĺ Γ 0 Ĺ Γ 1 Ũ¨¨Ĺ Γ m = Γ such that the quotient Γ i /Γ i´1 is a free abelian group. Definition 5.4 ([48]). A weak stability condition on A X is given by Z = tZ i u m i=0 P m ź i=0 Hom(Γ i /Γ i´1 , C) such that the following conditions are satisfied: (1) If for any nonzero E P A X , cl(E) P Γ i /Γ i´1 , then Z (E) := Z i ([cl(E)]) P H where [cl(E)] is the class of cl(E) in Γ i /Γ i´1 . The objects E P A X is Z-(semi)stable if for any exact sequence 0 Ñ F Ñ E Ñ G Ñ 0 in A X , and arg Z (F) ă (ď) arg Z (G). (2) There is a Harder-Narasimhan filtration for any E P A X . It is not hard to see that if m = 0, any weak stability condition is a stability condition. We follow Toda to construct the following weak stability condition on A X . Recall our Γ = Γ 0 ' Z, and we take the following 2-step filtration on Γ: 0 = Γ´1 Ĺ Γ 0 Ĺ Γ 1 = Γ. The embedding Γ 0 ã Ñ Γ is given by (n, β) Þ Ñ (n, β, 0). So # Γ 0 /Γ´1 = Q ' H 2 (X, Q); Γ 1 /Γ 0 = Q. We give the following data: # ω P H 2 (X, Q) -H 2 (X, Q), ω is ample; 0 ă θ ă 1. and let Z ω,θ = (Z ω,θ,0 , Z ω,θ,1 ) P 1 ź i=0 Hom(Γ i /Γ i´1 , C) = Hom(Γ 0 /Γ´1, C)ˆHom(Γ 1 /Γ 0 , C) which is defined as: # Z ω,θ,0 (n, β) = n´(ω¨β) ?´1 ; Z ω,θ,1 (rk) = ρ¨e πiθ , where (n, β) P Γ 0 and ρ P Z, (n, β, rk) P Γ. Proof. The proof is similar to [47,Lemma 5.2]. For completeness, we provide a proof here. We check Definition 5.4. Let E P A X be such that cl(E) P Γ i /Γ i´1 . In the case i = 1, Z ω,θ (E) P R ą0 e iπθ Ă H and in the case i = 0, Z ω,θ (E) = Z ω (E[1]) P H since E P Coh ď1 (X)[´1]. Here Z ω (E) = n´(ω¨β) ?´1 . Also Z ω,θ (E) defined0 Ñ T Ñ E Ñ F Ñ 0 such that T P A p 1 and F P A p 1/2 . Let A X,1 := A p 1 [´1] X A X = xO x [´1] : x P Xy ex and A X,1/2 := A p 1/2 [´1] X A X = tE P A X |Hom(A X,1 , E) = 0u. Then (A X,1 , A X,1/2 ) is a torsion pair on A X . Also if E P A p 1/2 [´1] , and rk(E) = 0 or 1, and c 1 (E) = 0, then from [47, Lemma 5.1], E P A X,1/2 . The proof is as follows. Rank zero case is obvious. In the rank one case, H´1(E) is torsion free of rank one, so there exists a gerby curve C Ă X such that H´1(E) Ñ H´1(E) __ -O X and an exact sequence I C Ñ E Ñ F[´1]. I C , F[´1] P A X imply that E P A X . Thus for any E P A X , there exists an exact sequence: (5.1.1) 0 Ñ T Ñ E Ñ F Ñ 0 such that T P A X,1 and F P A X,1/2 . Also the categories A X,1 , A X,1/2 are of finite length with respect to strict epimorphism and strict monomorphism. This is due to [49,Lemma 2.19] and the category of twisted sheaves on X (which is the same as category of coherent sheaves on X) is equivalent to the categories of untwisted shaves as in [16]. Since Z ω,θ is defined by: # Z ω,θ,0 (n, β) = n´(ω¨β) ?´1 ; Z ω,θ,1 (rk) = rk¨e iπθ . The same proof as in [49,Lemma 2.27] provides that E P A X is Z ω,θ -semistable if and only if one of the following conditions holds: (1) E P A X,1 ; (2) E P A X,1/2 and for any exact sequence 0 Ñ F Ñ E Ñ G Ñ 0 in A X with F, G P A X,1/2 , and arg Z ω,θ (F) ď arg Z ω,θ (G). Thus the Harder-Narasimhan filtration for any E P A X is given by the exact sequence (5.1.1) for the torsion pair and the Harder-Narasimhan filtration for any F is also given by the exact sequence (5.1.1). 5.1.2. Moduli stack of semistable objects in A X . We use the following big moduli stack x M X on A X counting perfect complexes satisfying certain conditions, see [33], [47]. Let us define x M X : (Sch C ) Ñ (groupoids) to be the stack that sends S Þ Ñ tE |E P D(Coh(XˆS))|Condition*u/ - where Condition* is: $ ' & ' % ‚ E is relatively perfect; ‚ E s = Lis E P D b (Coh(X)); ‚ Ext i (E s , E s ) = 0, i ă 0 for any s P S. Then x M X is an Artin stack locally of finite type. We define a substack x M(A X ) Ă x M X to be the substack consisting of all S-valued point E P x M(S) such that E s P A X for all the points s P S. Then we may write (5.1.2) x M(A X ) = ğ vPΓ x M v (A X ) where x M v (A X ) is the substack of objects E P A X such that cl(E) = v. Note here that cl(E) = (Ch 3 (E), Ch 2 (E), Ch 0 (E)) P Γ, since we work on µ r -gerbes X Ñ X. In general the Chern character Ch should be taken as the orbifold Chern character Ă Ch : K(X) Ñ HC R (X). Since HC R (X) = H˚(X) \¨¨\ H˚(X), up to the action of µ r on E, cl(E) keeps the same on each component. Later on we only care about the first component given by X-twisted sheaves. Then x M v (A X ) Ă x M X is an open immersion for v = (n, β, rk) P Γ with rk = 0 or 1; and x M v (A X ) is an Artin stack locally of finite type. Definition 5.6. Define x M n,β (ω, θ) Ă x M v (A X ) to be the stack of Z ω,θ -semistable objects E P A X with cl(E) = v = (´n,´β, 1). Here is a similar result as in [47,Proposition 5.4]: Proposition 5.7. We have: (1) The stack x M n,β (ω, θ) is an Artin stack of finite type. (2) In the case θ Ñ 1, x M n,β (ω, θ) -[P n (X, β)/C˚] which is the trivial C˚-gerbe. (3) x M n,β (ω, θ) -x M´n ,β (ω, 1´θ) is given by E Þ Ñ RHom(E, O X ). (4) x M n,β (ω, 1 2 ) = H for |n| ąą 0. Proof. We first recall the µ-limit stability of Toda in [49] and generalize it to µ r -gerbes X Ñ X. Let B + iω P H 2 (X, C), where ω P H 2 (X, R) is an ample divisor. Then in [49] Toda defined the so called µ B+iω -limit stability on A p X . We recall it here. Let σ = B + iω P A(X) C = tB + iω|ω ample u, and let Z σ : K(X) Ñ C be given by (5.1.3) E Þ Ñ´ż X e´( B+iω) Ch(E) a Td X P C. Here we only use the original Chern character on X and therefore Toda's arguments in [49, §3], [50, §3] go through for sheaves on A p X for X. Then one can write down Z σ (E) = ´v B 3 (E) + 1 2 ω 2 v B 1 (E) + ωv B 2 (E)´1 6 ω 3 v B 0 (E) i, where v B (E) = e´B Ch(E)¨atd X = (v B 0 (E), v B 1 (E), v B 3 (E)) P H even (X, R) -H even (X, RZ σ m (E) P tρ¨e iπφ |ρ ą 0, 1 4 ă φ ă 5 4 u for m ąą 0. Therefore φ σ m (E) = 1 π Im log Z σ m (E) P ( 1 4 ,5 4 ). Then a nonzero E P A p X is called σ-limit stable (or σ-limit semistable) if for any nonzero F Ĺ E, φ σ (F) ă φ σ (E), (or φ σ (F) ď φ σ (E)). Here a lemma as in [49,Lemma 3.8]: Lemma 5.9. For σ = B + iω P A(X) C , E P A p 1/2 , such that det(E) = O X , and Ch(E) = (´1, 0, β, n). Let F P Coh ď1 (X). Then φ σ (F) ă φ σ (E), (or φ σ (F) ą φ σ (E)) if and only if one of the following conditions holds: (1) µ σ (F) ă´3 Bω 2 ω 3 , (resp. µ σ (F) ą´3 Bω 2 ω 3 ). (2) µ σ (F) =´3 Bω 2 ω 3 and ωv B 2 (E)µ σ (F) ă v B 3 (E), (resp. ωv B 2 (E)µ σ (F) ą v B 3 (E)). Proof.dimensional sheaf F ‰ 0, F ã Ñ E[1] is in A p 1/2 , we have Ch 3 (F) ω¨Ch 2 (F) ď´2k; (2) if we have E[1] ։ G for G a one-dimensional object in A p 1/2 , we have Ch 3 (G) ω¨Ch 2 (G) ě´2k. So following Toda, let k = # 1 2 tan(θ) , θ ‰ 1 2 ; 0, θ = 1 2 . Then from the above argument: For E P A p X [´1], E[1] P A p X is µ kω+iω -semistable if and only if E P A X and E is Z ω,θ -semistable such that cl(E) = v = (´n,´β, 1) P Γ. Proof of the proposition now can be done using the same method of Toda. DT n,β (ω, θ) = χ( x M n,β (ω, θ), ν p M ), is the weighted Euler characteristic, where ν p M is the Behrend function. Our goal is to count the semistable objects in A X . We use Joyce-Song method to consider the Hall algebra H(A X ) = K(St / x M X ) with ‹-product. We use the definition of Hall algebra of Joyce and Bridgeland [7]. First for a pair (n, β) P Γ 0 , let M n,β (ω) Ă x M X be the substack parametrizing Z ω -semistable E P Coh ď1 (X) such that the data is given by [E] = β and ş X Ch(E)¨Td X = n. This is an open substack of finite type. The elements in the Hall algebra H(A X ) are given by p δ n,β (ω) = [M n,β (ω) i ã Ñ x M X ] where i sends E to E[´1] P A X , and p δ n,β (ω, θ) = [ x M n,β (ω, θ) Ñ x M X ]. We have its logarithm: p ǫ n,β (ω, θ) = ÿ ℓě1,1ďeďℓ (n i ,β i )PQ'H 2 (X,Q) n 1 +¨¨¨+n ℓ =n β 1 +¨¨¨+β ℓ =β Z ω,θ (´n i ,´β i )PR ą0 e iπθ ,i‰e (´1) ℓ´1 ℓ p δ n 1 ,β 1 (ω) ‹¨¨¨‹ p δ n e´1 ,β e´1 (ω) ‹ p δ n e ,β e (ω, θ) ‹¨¨¨‹ p δ n ℓ ,β ℓ (ω) Definition 5.11. Define DT n,β (ω, θ) = lim tÑ1 (t 2´1 )P t (´ν¨p ǫ n,β (ω, θ)) Here (1) ν : H(A X ) Ñ H(A X ) is the map by inserting the Behrend function given by [Y ρ ÝÑ x M X ] Þ Ñ ÿ iPZ i¨[ρ| Y i : Y i Ñ x M X ] where Y i = (ν M˝ρ )´1(i). (2) P t : K(St / x M X ) Ñ Q(t) is the map P t [ρ : [Y/GL m (C)] Ñ x M X ] Þ Ñ P t ([Y]) P t ([GL m (C)]) and P t ([Y]) is the Poincaré polynomial of the quasi-projective variety Y. 5.1.4. Limit stable invariants L n,β . As in [47], we define Definition 5.12. Define L n,β := DT n,β (ω, θ = 1 2 ). We have the following fact: (1) When θ Ñ 1, DT n,β (ω, θ) = P n,β ; (2) L n,β = L´n ,β and they are zero when |n| ąą 0. Rank zero invariants N n,β . In the category of Coh ď1 (X), recall that we have p δ n,β (ω) = [M n,β (ω) i ã Ñ x M X ] where i sends E to E[´1] P A X , we define p ǫ n,β (ω) = ÿ ℓě1,1ďiďℓ (n i ,β i )PQ'H 2 (X,Q) n 1 +¨¨¨+n ℓ =n β 1 +¨¨¨+β ℓ =β Z ω (´n i ,´β i )=Z ω (´n,´β) (´1) ℓ´1 ℓ p δ n 1 ,β 1 (ω) ‹¨¨¨‹ p δ n ℓ ,β ℓ (ω) Definition 5.13. Define N n,β (ω) = lim tÑ1 (t 2´1 )P t (´ν¨p ǫ n,β (ω)). The invariants N n,β (ω) counts rank zero Z ω -semistable coherent sheaves in Coh ď1 (X). From [47,Lemma 4.8], it is independent to the choice of ω and we just use N n,β := N n,β (ω). Theorem 5.14. Let 0 ă φ ă 1 2 , then we have the wall crossing formula: DT(ω, φ + ) = DT(ω, φ´)¨ź ną0,βą0 n+(ω¨β)iPR ą0 e iπφ exp((´1) n´1 nN n,β q n t β ) Proof. Since we work on the category A X of objects in the derived category of coherent sheaves on the gerbe X, and from [44], the category of coherent sheaves on X is isomorphic to the category of twisted sheaves on the rigidified inertia stack X \¨¨¨\ X by some C˚-gerbe determined by the µ r -gerbe X. The category of C˚-gerbe twisted sheaves is equivalent [16] to untwisted sheaves on X \¨¨¨\ X. Therefore the method of Toda in [48, Theorem 5.8, Theorem 8.10] work in this case and we get the wall-crossing formula. Proof. Proposition 5.7 tells us that lim θÑ1 DT(ω, θ) = PT(X). The elements F P Coh ď1 (X) such that Z ω,1/2 (F[´1]) P R ą0 ?´1 will have χ(F) = 0. Then the elements with phase 1 2 are pure dimensional one sheaves. The wall crossing formula [48, Theorem 5.8, Theorem 8.10] tells us lim θÑ 1 2 DT(ω, θ) = DT(ω, θ = 1 2 ) = ř n,β L n,β q n t β . Then the result is obtained by applying θ = 1 2 to θ = 1 from the Theorem above. 5.2. Decomposition formula forétale gerbes. GW theory for µ r -gerbes. The gerbe X Ñ X is given by an element in H 2 (X, µ r ), therefore is banded. We recall some results in [1] and [2, §6]. Let K g (X, β) be the moduli stack of twisted stable maps of genus g twisted curves to X of degree β P H 2 (X, Q) = H 2 (X, Q). We borrow the following diagram from [2, Formula (41)]: K g (X, β) t / / # # ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ p % % P g q 1 / / M g (X, β) M tw g q / / M g where (1) M tw g is the moduli stack of prestable twisted curves of genus g; (2) M g is the moduli stack of prestable curves of genus g; (3) q maps a twisted curve to the underlying prestable curve; (4) M g (X, β) is the moduli space of stable genus g stable maps to X of degree β; (5) The right vertical arrow is the forgetful morphism sending a stable map f : C Ñ X to C; (6) The stack P g is defined as the fiber product; (7) The natural morphism p : K g (X, β) Ñ M g (X, β) sends a twisted stable map [ f : C Ñ X] to the underlying stable map [ f : C Ñ X] between the coarse moduli spaces; (8) The morphism K g (X, β) Ñ M tw g is defined by the forgetful morphism again; (9) The morphism t is defined by the universal property of fiber product. In [2, Proposition 5.1, Lemma 5.2], the authors show that t isétale and factors through K g (X, β) Ñ K g (X, β) µ r Ñ P g where K g (X, β) µ r is the rigidification of K g (X, β) and K g (X, β) Ñ K g (X, β) µ r is a µ r -gerbe. In [2, §6], the authors talked about the pushforward formula for virtual fundamental classes. Theorem 5.16. ([2, Theorem 6.8]) We have: p˚[K g (X, β)] vir = r 2g´1¨[ M g (X, β)] vir . We are working on the twisted stable maps to X without marked points. Therefore N GW g,β (X) := ż [K g (X,β)] vir 1 = r 2g´1¨ż [M g (X,β)] vir 1 =: r 2g´1 N GW g,β (X). We let F GW (X; u, v) = ÿ gě0 ÿ β‰0 N GW g,β (X)u 2g´2 v β and F GW (X; u, v) = ÿ gě0 ÿ β‰0 N GW g,β (X)u 2g´2 v β be the generating function of Gromov-Witten invariants for X and X. Theorem 5.17. F GW (X; u, v) = r¨F GW (X; ru, v). Proof. We calculate F GW (X; u, v) = ÿ gě0 ÿ β‰0 N GW g,β (X)u 2g´2 v β = ÿ gě0 ÿ β‰0 r 2g´1 N GW g,β (X)u 2g´2 v β = r¨ÿ gě0 ÿ β‰0 N GW g,β (X)(ru) 2g´2 v β = r¨F GW (X; ru, v). Gopakumar-Vafa invariants and the formula of Toda. The GW invariants N GW g,β (X) and N GW g,β (X) defined above are rational numbers. From Gopakumar-Vafa [12], we have the following integrality conjecture which gives the Gopakumar-Vafa invariants. This is given by string duality between type IIA string theory and M-theory. Conjecture 5.18. (Gopakumar-Vafa conjectural invariants) There exist integers n β g P Z, for g ě 0 and β P H 2 (X, Z) such that ÿ gě0,βě0 N GW g,β (X)u 2g´2 t β = ÿ gě0,βą0 kPZ ě1 n β g k 2 sin ku 2 2g´2 t kβ . We recall the GW/PT-correspondence in [39], [36]. Let Z GW (X; u, t) := exp(F GW (X; u, t)) = 1 + ÿ β‰0 Z GW (X; u) β t β . Then the GW/PT-correspondence is given by: (5.2.1) Z GW (X; u, t) = Z PT (X; q, t), for q =´e iu . Here Z PT (X; q, t) = PT(X) 1 = 1 + ÿ β‰0 PT(X) 1 β and PT(X) 1 = ř β‰0 ř nPZ P n,β (X)q n t β . Let us define PT(X) := ÿ βě0 ÿ nPZ P n,β (X)q n t β = 1 + ÿ βą0 PT β (X) for PT β (X) = ř nPZ,βą0 P n,β (X)q n t β . [47, Conjecture 6.2] made the following conjecture: Conjecture 5.19. There exist integers n β g P Z, for g ě 0 and β P H 2 (X, Z) such that (5.2.2) PT(X) = ź βą0   8 ź j=1 (1´(´q) j t β ) jn β 0¨8 ź g=1 2g´2 ź k=0 (1´(´q) g´1´k t β ) (´1) k+g¨n β g 2g´2 k   . For a µ r -gerbe X Ñ X, the GW potential of X is related to PT(X) through the GW potential of X by: Proposition 5.20. Let Z GW (X; u, t) := exp(F GW (X; u, t)). Then we have Z GW (X; u, t) = (Z GW (X; ru, t)) r . Proof. This is calculated by Z GW (X; u, t) = exp(F GW (X; u, t)) = exp(r¨F GW (X; ru, t)) = (Z GW (X; ru, t)) r . Multiple cover formula. In this section we fix X to be a smooth projective Cababi-Yau threefold; and p : X Ñ X a µ r -gerbe. Recall the wall crossing formula (5.1.4) in Corollary 5.15. If the gerbe X Ñ X is trivial, or we just count the stable pairs [O X Ñ F] for the generating sheaf Ξ = O X , then the wall crossing Formula (5.1.4) is just the same as Toda's formula in [47,Formula (66)] based on the following: Proposition 5.21. If the gerbe X Ñ X is trivial, then every coherent sheaf on X is pulled back from its coarse moduli space X. We follow [47, §6.2] to derive the multiple cover formula. First take logarithm of RHS of (5.2.2) yields: log ź βą0   8 ź j=1 (1´(´q) j t β ) jn β 0¨8 ź g=1 2g´2 ź k=0 (1´(´q) g´1´k t β ) (´1) k+g¨n β g 2g´2 k   (5.3.1) = ÿ βą0 8 ÿ j=1 j¨n g 0 ÿ kě1 (´1) jk´1 k q jk t kβ + ÿ βą0 8 ÿ g=1 ÿ aě1 n β g a 2g´2 ÿ k=0 2g´2 k (´(´q) a ) g´1´k t aβ = ÿ βą0 8 ÿ n=1 ÿ kě1 k|(β,n) (´1) n´1 n k 2 n β/a 0 q n t β + ÿ βą0 8 ÿ g=1 ÿ aě1 a|β n β/a g a f g (´(´q) a )t β , where f g (q) := 2g´2 ÿ k=0 2g´2 k q g´1´k = q 1´g (1 + q) 2g´2 . The second term is a polynomial in q˘1 and is invariant under q Ø 1 q . Now we take the logarithm of Formula N n,β = ÿ kě1 k|(β,n) 1 k 2 n β/k 0 where N 1,β = n β 0 . This conjecture is equivalent to Conjecture 5.19, and n β g for g ě 1 are written down by the invariants L n,β . Since we don't need the higher genus Gopakumar-Vafa invariants in this paper, we leave these invariants for the future research. MULTIPLE COVER FORMULA FOR TWISTED K3 SURFACES In this section we prove a multiple cover formula for counting semistable sheaves in the category of twisted sheaves on a twisted K3 surface. Let S be a smooth projective K3 surface. We let p : S α Ñ S be an optimal µ r -gerbe, which means that the order of the corresponding [α] P H 2 (S, OS ) tor in the cohomological Brauer group is r. The pair (S, α) or the gerbe S α is called a twisted K3 surface. be the reduced connected residue GW invariants of X by C˚-localization. We write its generating series in terms of "BPS" form as in Conjecture 5.18: Z red GW (X; u, t) = ÿ gě0,β‰0 N red g,β (X)u 2g´2 t β = ÿ gě0,β‰0 n β g u 2g´2 ÿ ką0 1 k sin(ku/2) u/2 2g´2 t kβ . The Gopakumar-Vafa invariants n β g are in fact integers n g,h P Z which depend only on h and ż S β 2 = 2h´2. These invariants are nonzero only for 0 ď g ď h and are determined by the KKV formula (6.1.1) ÿ gě0 ÿ hě0 (´1) g n g,h (y 1 2´y´1 2 ) 2g q h = ź ně1 1 (1´q n ) 20 (1´yq n ) 2 (1´y´1q n ) 2 . This KKV formula is proved in [40]. In [35], Maulik-Thomas prove the following result: Z red GW (X; u, t) = Z red PT (X; q, t) =´log(1 + Z χ PT (X; q, t)) =´log(1 + Z na PT (X;´q, t)) after q = e´i u and Z na PT (X;´q, t) is the generating series of the naive Euler characteristic of the stable pair moduli spaces. Here Z red PT (X; q, t), Z χ PT (X; q, t) were written as Z red P (X; q, t), Z χ P (X; q, t) in Section 4. The KKV formula of [40] implies that the PT-stable pair generating series Z χ PT (X; q, t) satisfies the "BPS rationality" condition, and Toda [47,Theorem 6.4] shows that the "BPS rationality" condition is equivalent to the multiple cover formula (5.22). Thus we obtain: Proposition 6.2 ([35]). The multiple cover formula Conjecture 5.22 holds for K3 surfaces. 6.2. Category of twisted sheaves on X = SˆP 1 . We let X := SˆC be the local K3 surface, and X = SˆP 1 . We have H 2 (SˆP 1 , µ r ) -H 2 (S, µ r ) ' H 2 (P 1 , µ r ) -H 2 (S, µ r ) ' Z r . Let S := S α Ñ S be an optimal µ r -gerbe for α P H 2 (S, µ r ). Then p : X := SˆP 1 Ñ SˆP 1 is a µ r -gerbe given by the class α P H 2 (S, µ r ) ã Ñ H 2 (SˆP 1 , µ r ), which is also optimal. Let X π / / p P 1 SˆP 1 ; ; ① ① ① ① ① ① ① ① be the projection. The gerbe twisted sheaves were introduced in [34], and was reviewed in [21, §3]. We consider X-twisted sheaves on X = SˆP 1 . Proposition 6.3. Any α-twisted coherent sheaf E on X is tensor product of a pullback α-twisted sheaf on S and a coherent sheaf on P 1 . Proof. Let π S : X Ñ S be the projection. Since α P H 2 (S, µ r ) ã Ñ H 2 (X, µ r ), from the definition of twisted sheaves in [21, §3.1], we have a commutative diagram: µ rˆE / / χ id C˚ˆE / / E where χ : µ r Ñ C˚is the inclusion. Since any coherent sheaf E = πSE 1 b π˚E 2 for E 1 and E 2 are coherent sheaves on S and P 1 , the gerbe structure implies that µ r only acts on πSE 1 . Hence E 1 defines a S-twisted sheaf on S. We define Coh tw π (X) Ă Coh(X) to be the subcategory of gerbe α-twisted sheaves on X supported on the fiber of π, and let D S 0 := D b (Coh tw π (X)) be the corresponding derived category. As in [46, Definition 2.1], we define D := xπ˚Pic(P 1 ), Coh tw π (X)y tr Ă D b (Coh tw (X)) to be the triangulated category. Twisted Chern character. We introduce the twisted Chern character on S and twisted Mukai pairs. Let us first review the Brauer-Severi variety. The gerbe α P Br 1 (S), which is the same as Br(S). Thus it determines a projective bundle P Ñ S of rank r´1. We have a vector bundle G over P is defined by the Euler sequence 0 Ñ O P Ñ G Ñ T P/S Ñ 0 of the projective bundle P Ñ S. Yoshioka [51] defined the subcategory Coh(S; P) Ă Coh(P) of coherent sheaves on P to be the subcategory of sheaves E P Coh(S; P) if and only if E| P i -p˚(E i ) b O P i (λ i ) where tU i u is an open covering of S, P i = U iˆP r´1 , E i P Coh(U i ); and there exists an equivalence: (6.2.1) Coh(S; P) - ÝÑ Coh(S, α) by E Þ Ñ p˚(E b L _ ). Here α = o([X]) P H 2 (X,L| p´1(x) = O p´1(x) (´1), and Coh(S, α) is the category of α-twisted coherent sheaves on S. We call E P Coh(S; P) a P-sheaf. A P-sheaf E is of dimension d if p˚E is of dimension d on S. Yoshioka defined the Hilbert polynomial P G E (m) = χ(p˚(G _ b E)(m)) = d ÿ i=0 α G i (E)¨ m + i i . The stability and semistability can be defined using this Hilbert polynomial. There is an integral structure on H˚(S, Q) following Huybrechts-Stellari [16]. The integer r is the minimal rank on S such that there exists a rank r S-twisted locally free sheaf E on the generic scheme S. Recall that x, y the Mukai pairing on H˚(S, Z). Definition 6.4. For a P-sheaf E, define a Mukai vector of E as : v G (E) := Ch(Rp˚(E b G _ )) a Ch(Rp˚(G b G _ )) ? td S = (rk, ζ, b) P H˚(S, Q), where p˚(ζ) = c 1 (E)´rk(E) c 1 (G) rk(G) , b P Q. One can check that xv G (E 1 ), v G (E 2 )y =´χ(E 1 , E 2 ). We define for ξ P H 2 (S, Z), an injective homomorphism: and π S˚E P K(S). The twisted Mukai vector is given by: (6.2.2) v G : K(D S 0 ) π SÝ Ñ K(S) v G ÝÑ Γ 0 . We have the general Chern character: H˚(X, Q). H˚(X, Q). Then cl = Ch : K(D) Ñ Γ = Q ' Γ 0 and every element in Γ can be written as: v = (R, rk, β, n), where R, rk are integers and β P NS(S). For any E P D, we have: Ch : K(X) ÑLet Γ := H 0 (X, Q) ' (Γ 0 b H 2 (P 1 , Q)) Ă cl(E) = (Ch 0 (E), Ch 1 (E), Ch 2 (E), Ch 3 (E)). Here Ch 3 (E) P Q since we work on H˚(X, Q). The rank of v is rk(v) = R and Ch 1 (E) = c 1 (E) = rk¨[S] and rk is an integer. (v)) of G-twisted semistable (or stable) P-sheaves E with v G (E) = v is defined. Let M P,G H,ss (v) (or M P,G H,s (v)) be its coarse moduli space. The moduli stack N P,G H,ss (v) (or N P,G H,s (v)) of G-twisted semistable (or stable) Higgs P- sheaves (E, φ) with v G (E) = v is similarly defined, where E is a G-twisted P-sheaf and φ : p˚E Ñ p˚(E) b K S is the Higgs field. This makes sense since p˚(E b L _ ) is a G-twisted sheaf on S. Let N P,G H,ss (v) (or N P,G H,s (v)) be its coarse moduli space. On the other hand, for the optimal µ r -gerbe S Ñ S, using geometric stability as in [34], reviewed in 6.3.1. The heart of a bounded t-structure on D S 0 and on D. We first recall the geometric stability of αtwisted sheaves on S Ñ S or X Ñ X = SˆP 1 . We present twisted sheaves on X, the case of S is similar. The geometric Hilbert polynomial of a twisted sheaf E is defined as: P g (E, m) = χ g (E b O X (m)) where χ g (E) := [IX : X] deg(Ch(E)¨Td X ). The geometric Hilbert polynomial can be written as: P g (E, m) = dim(E) ÿ i=0 a i (E) m i i! . The rank is defined as: rk(E) = a d (E) a d (O X ) and deg(E) := a d´1 (E)´rk(E)¨a d´1 (O X ). The slope is: µ(E) = µ ω (E) = deg(E) rk(E) . Since we are interested in twisted sheaves E, where the µ r action on E is given by λ 1 -action. By Grothendieck-Riemann-Roch theorem for DM stacks, for a twisted sheaf on S, (6.3.1) µ ω (E) = ş S c 1 (E)p˚ω rk(E) , where ω is an ample divisor on S (usually take ω = O S (1)). On the category Coh tw π (X), for an element E such that cl 0 (E) = (rk, β, n) P Z ' NS(S) ' Q the Hilbert polynomial P g (E, m) and the slope µ(E) is defined before. Then an object E P Coh tw π (X) is called Gieseker (semi)stable and µ ω -(semi)stable if for any subsheaf F Ĺ E, p g (F, m) ă (ď)p g (E, m), µ ω (F) ă (ď)µ ω (E) respectively. Here p g is the reduced Hilbert polynomial. As in [46, §2.5], for the µ ω -stability on Coh tw π (X), we have the Harder-Narasimhan filtration 0 Ñ E 0 Ă E 1 è¨¨Ă E ℓ = E for E P Coh tw π (X) which satisfies µ ω (E 1 /E 0 ) ą¨¨¨ą µ ω (E ℓ /E ℓ´1 ). Using this we construct a torsion pair. Let µ ω,+ := µ ω (E 1 /E 0 ); µ ω,´: = µ ω (E ℓ /E ℓ´1 ). Definition 6.8. Define (T ω , F ω ) to be the torsion pair of full subcategories in Coh tw π (X) as: T ω := tE P Coh tw π (X)|µ ω,´( E) ą 0u and F ω := tE P Coh tw π (X)|µ ω,+ (E) ď 0u. Then the pair (T ω , F ω ) is a torsion pair in the sense that: (1) Any T P T ω , F P F ω , we have Hom(T, F) = 0; (2) For any E P Coh tw π (X), there exists an exact sequence 0 Ñ T Ñ E Ñ F Ñ 0 such that T P T ω , and F P F ω . Let B ω := xF ω , T ω [´1]y ex Ă D S 0 be the tilting in [14]. Then B ω is the heart of a bounded t-structure on D S 0 . We remark that B tω is the same as B ω for t ą 0. Proposition 6.9. Define A ω := xπ˚Pic(P 1 ), B ω y ex Ă D then A ω is the heart of a bounded t-structure on D. Proof. The argument is the same as [46, Proposition 2.9]. 0 = Γ´1 Ă Γ 0 Ă Γ 1 = Γ be the filtration. For ω an ample divisor on S and t P R ą0 , let Z tω P 1 ź i=0 Hom(Γ i /Γ i´1 , C) be given by: # Z tω,0 (v) := ş S e´i tω v, v P Γ 0 ; Z tω,1 (R) = R ?´1 , R P Γ 1 /Γ 0 = Z. Then [46,Lemma 3.4] says that σ tω := (Z tω , A ω ) is a weak stability condition. This can be checked that [46, §8.2] works for sheaves in D b (Coh tw π (X)) and D. The wall and chamber structure and some comparison with µ iω -limit semistable objects are in [46, §3.4, §3.5]. Hall algebra H(A ω ) and counting invariants. The Hall algebra H(A ω ) for the abelian category A ω is defined in [7]. The stable pair invariants of [39] can be defined on A ω Ă D as in §3.1, and Toda proves a wall crossing formula for the stable pair invariants. Here we only care about the rank zero invariants. Let x M(A ω ) be the stack of objects in A ω , and M tω (v) Ă x M(A ω ) the substack parametrizing Z tω -semistable objects E P A ω and cl(E) = v P Γ with rk(v) ď 1. Then the moduli stack gives an element δ tω (v) := [M tω (v) ã Ñ x M(A ω )] P H(A ω ). As in Joyce [24], let (6.3.2) ǫ tω (v) := ÿ ℓě1,v 1 +¨¨¨+v ℓ =v v i PΓ arg Z tω (v i )=arg Z tω (v) (´1) ℓ´1 ℓ δ tω (v 1 ) ‹¨¨¨‹ δ tω (v ℓ ) This sum is a finite sum [46,Lemma 4.8], and δ tω (v) is well-defined. The proof is from Bogomolov inequality for semistable sheaves. We define a morphism (q´1)P q (ǫ tω (1,´v)) for (1,´v) P Γ = Z ' Γ 0 . This is the Euler characteristic version of the DT-invariants. Counting rank zero invariants. [46, §4.6] uses the DT-invariants DT χ tω (v) to recover the invariants L β,n of the limit stable objects and study the wall crossing formula for the stable pair invariants on X. Since in this paper we don't need this PT tw (X), we leave this for later study. To define the counting invariants in A ω and A ω [´1], let C(B ω ) := Im(cl 0 : B ω Ñ Γ 0 ). From [46,Definition 4.17], Definition 6.11. Define N(v) P Q for v P Γ 0 , to be (1) If v P C(B ω ), N(v) = lim q 1 2 Ñ1 (q´1)P q (ǫ ω (0, v)); (2) If´v P C(B ω ), then N(v) := N(´v); (3) If˘v R C(B ω ), N(v) = 0. This definition is the same as the invariants of Z ω,0 -semistable objects E P B ω satisfying cl 0 (E) = v. Remark 6.12. In the case of v = (0, β, n), the invariant N(0, β, n) in the above definition is the same as the one we defined in (5.1.4). Twisted Hodge isometry. Let StabΓ 0 (D S 0 ) be the component in Stab Γ 0 (D S 0 ) containing the stability conditions (Z tω,0 , B ω ). For the twisted K3 surface S α or (S, α), [17] has generalized Bridgeland [6] stability conditions on K3 surface to twisted K3 surfaces; and proved that there exists a component of maximal dimension Stab˝(S, α) Ă Stab(S, α) in Stab(S, α). Theorem 6.13. We have Ψ : StabΓ 0 (D S 0 ) " ÝÑ Stab˝(S, α). Proof. We recall the stability condition Stab˝(S, α) Ă Stab(S, α) in [17]. We have Z : Stab(S, α) Ñ NS(S, α) b C given by: σ Þ Ñ (Z, P σ ) where NS(S, α) = r H 1,1 (S, α, Z). Let P(S, α) Ă NS(S, α) b C be the open subset of vectors ϕ such that the real part and imaginary part of ϕ generate a positive plane in NS(S, α) b C. P(S, α) has two connected components and let P + (S, α) be the one containing ϕ = e B 0 +iω . This central charge Z ϕ (E) determines a torsion pair (T , F ), where T Ă Coh(S, α) is given by twisted sheaves E such that every nontrivial torsion free quotient E ։ E 1 satisfies Im(Z ϕ (E 1 )) ą 0. F Ă Coh(S, α) is given by twisted sheaves E P F if E is torsion free and every non-zero subsheaf E 1 Ă E satisfies Im(Z ϕ (E 1 )) ď 0. Then every G P Coh(S, α) can be uniquely written as Here B 0 P H 2 (S, Q) is a B-field lift of α, i.e.,0 Ñ E Ñ G Ñ F Ñ 0 for E P T , F P F . The heart of the induced t-structure is an abelian category: A(ϕ) := $ & % E P D b (Coh(S, α))ˇˇˇˇˇˇ‚ H i (E) = 0 for i R t´1, 0u ‚ H´1(E) P F ‚ H 0 (E) P T , . -Then from [17,Lemma 3.4], let ϕ = e B+iω , the induced homomorphism Z ϕ : A(ϕ) Ñ C is a stability function on A(ϕ) if and only if for any spherical twisted sheaf E P Coh(S, α), Z ϕ (E) R R ď0 . From [17,Proposition 3.6], the pair (Z ϕ , A(ϕ)) defines a stability condition on D b (Coh(S, α)). Then let Stab˝(S, α) Ă Stab(S, α) be the connected component of Stab(S, α) that contains the stability conditions described above. Then since the category Coh tw π can be taken as a family of K3 gerbes S Ñ S, the same argument as in [46,Theorem 6.5] gives Stab˝(D S 0 ) θ˝/ / Z Stab˝(S, α) Z NS(X) b C i˚/ / NS(S) b C We explain the notation: Here is a result proved in [46, Theorem 4.21]: θ : Stab(D S 0 ) Ñ Stab(S, α) Theorem 6.14. The counting invariants N σ (v) do not depends on a choice of stability condition σ P StabΓ 0 (D S 0 ) and N σ (v) is also independent of ω. Proof. The proof is the same as [46,Theorem 4.21] by using Joyce's wall crossing formula: N σ 1 (v) = N σ 0 (v) + ÿ v 1 +v 2 =v U v 1 ,v 2 χ(v 1 , v 2 )N σ 0 (v)¨N σ 1 (v) +¨¨¨¨¨f or some other terms (which involve product of χ(v i , v j )). Here σ 0 and σ 1 are two different stability conditions and σ 0 is close to σ 1 . It suffices to prove that χ(v i , v j ) = 0 for any v 1 , v 2 P Γ 0 . Let r v 1 = (0, v 1 ), r v 2 = (0, v 2 ) P Γ = Z ' Γ 0 . Then by Grothendieck-Riemann-Roch theorem for stacks χ(r v 1 , r v 2 ) = χ(v 1 , v 2 ) = 0 where v 1 , v 2 are the corresponding Mukai vectors on S. Coh tw π (X) Ă Coh tw π (X) be the subcategory consisting of sheaves supported on the fibers of π| X : X Ñ C. Still let x M(X) be the stack of coherent twisted sheaves in Coh tw π (X), which is an Artin stack locally of finite type. Let A X := Coh tw π (X) and H(A X ) the Hall algebra in [7], which consists of elements over the stack x M(X). For v P Γ 0 , we let M ω,X (v) Ă x M(X) be the substack of ω-Gieseker semistable twisted sheaves E with v G (E) = v. This gives an element: δ ω,X (v) := [M ω,X (v) ã Ñ x M(X)] P H(A X ) and define its "logarithm": ǫ ω,X (v) := ÿ ℓě1,v 1 +¨¨¨+v ℓ =v,v i PΓ 0 p ω,v i =p ω,v (m) (´1) ℓ´1 ℓ δ ω,X (v 1 ) ‹¨¨¨‹ δ ω,X (v ℓ ) where p ω,v i (m) is the reduced geometric Hilbert polynomial. The sum is a finite sum and ǫ ω,X (v) is well-defined. Let C(X) := Im(v G : Coh tw π (X) Ñ Γ 0 ) we define the invariants: (q´1)P q (ǫ ω,X (v)). (2) If´v P C(X), J(v) := J(´v). (3) If˘v R C(X), J(v); = 0. Remark 6.16. If v G (E) = v is primitive, then J(v) = χ(M tw ω,X (v)) = χ(M tw ω,S (v)ˆC) where M tw ω,S (v) is the moduli stack of S-twisted sheaves with Mukai vector v. From Yoshioka [51], M tw ω,S (v) is deformation equivalent to the Hilbert scheme Hilb 1´χ (v,v) 2 (S) . This is essential to the calculation of Vafa-Witten invariants in [21]. The comparison between N(v) and J(v). We also need to compare N(v) and J(v). Similar to Definition 6.15, we can define the counting invariants J(v) P Q to be the invariants counting geometric ω-Gieseker semistable twisted sheaves E P Coh tw π (X) with v G (E) = v P Γ 0 . We have a generalization of [46,Theorem 4.24]: Theorem 6.17. For any v P Γ 0 , we have: J v¨a td S a Ch(Rp˚(G b G _ )) = N(v) Proof. We need to generalize [45, Theorem 6.6] to P-sheaves on Coh(S, P), where P Ñ S is the Brauer-Severi variety corresponding to the optimal gerbe S Ñ S. Recall the definition of N(v), given by a weak stability condition σ ϕ = (Z ϕ , A(ϕ)) P StabΓ 0 (D S 0 ) and N(v) is independent to the stability condition we choose. Here ϕ = e B+iω . Let us take σ k = (Z kω , A ω ) and ϕ = e B+ikω , and Z ϕ (E) = xe B+ikω , v G (E)y. on the twisted K3 surface (S, α). We prove the automorphic property for J(v). For two twisted K3 surfaces (S, α) and (S 1 , α 1 ), we recall the twisted Fourier-Mukai transform in [17], [16]. For the Bfield B such that exp(B) = α; B-field B 1 such that exp(B 1 ) = α 1 , we have (´B) ' B 1 := pS(´B) + pS 1 (B 1 ) P H 2 (SˆS 1 , Q), where p S : SˆS 1 Ñ S, and p S 1 : SˆS 1 Ñ S 1 are projections and induce α´1 b α 1 P H 2 (SˆS 1 , O˚). Let E P D b (SˆS 1 , α´1 b α 1 ) be a kernel, we have: Φ : D b (S, α) Ñ D b (S 1 , α 1 ) given by E Þ Ñ p S 1˚( pS(E) b E ) . [16,Proposition 4.3] implies that if Φ is a derived equivalence, then the induced map Φ B,B 1 : r H(S, α, Z) " ÝÑ r H(S 1 , α 1 , Z) is a Hodge isometry of integral weight-2 Hodge structures, and Φ B,B 1 (´) = p S 1˚pS ´¨Ch(E ) b td SˆS 1 . We have the following commutative diagram: D b (Coh(S 1 , α 1 )) Φ / / v α 1 D b (Coh(S, α)) v α r H(S 1 , α 1 , Z) Φ˚/ / r H(S, α, Z) Therefore this equivalence Φ gives an isomorphism Φ St : Stab(S 1 , α 1 ) " ÝÑ Stab(S, α) on the stability manifolds. We have a similar proposition as in [46,Proposition 4.29]. Proposition 6.20. Let D S 0 , Γ S 0 , J S (v) be the invariants defined before for the twisted K3 surface (S, α). Assume that Φ St sends the connected component Stab˝(S 1 , α 1 ) to Stab˝(S, α). Then J S 1 (v) = J(Φ˚(v)) for any v P Γ S 1 0 . Proof. First Φ induces an equivalence r Φ : D b (X 1 , α 1 ) " ÝÑ D b (X, α) and the kernel is: So from Theorem 6.17 and the diagram above, we have E b O ∆ P 1 P D b (Coh(S 1ˆSˆP1ˆP1 , α´1 b α 1 )).J S α (Φ˚v) = N σ (Φ˚v¨btd S α´1 ) = N Φ St σ 1 (Φ˚v¨btd S α´1 ) = N σ 1 (v¨btd S α 1´1 ) = J S α 1 (v).(6.4.1) J(v) = ÿ kě1 k|v 1 k 2 χ(Hilb xv/k,v/ky+1 (S)) which is the multiple cover formula Conjecture 5.22 for v = (0, β, n). In the category Coh tw π (X) of twisted sheaves on X Ñ X = SˆP 1 , given by the cohomology class α P H 2 (X, µ r ) -H 2 (S, µ r ), if we forget about the µ r -action, this is equivalent to the category Coh(S, α) of twisted sheaves on the twisted K3 surface (S, α). Since a one dimension sheaf E with cl 0 (E) = (0, β, n) on S or on X is automatically S-twisted or X-twisted, the element ǫ ω,X (v) is the same as ǫ ω,X (v) = ÿ ℓě1,v 1 +¨¨¨+v ℓ =v,v i PΓ 0 p ω,v i =p ω,v (m) (´1) ℓ´1 ℓ δ ω,X (v 1 ) ‹¨¨¨‹ δ ω,X (v ℓ ). This is because in the category Coh(S, α), the subcategory of one dimensional sheaves is always a twisted category and keeps the same as the subcategory of one dimensional sheaves in the general untwisted category Coh(S 0 ) for the trivial gerbe S 0 . Thus the subcategory Coh(S, α) ď1 is equivalent to the subcategory Coh(S) ď1 . Another explanation can be seen from the equivalence (6.2.1) in §6.2.1 using P-sheaves, and the one dimensional pure P-sheaves corresponds to one dimensional pure twisted sheaves in Coh(S, α). The Brauer class α, when restricted to the one dimensional supported locus of the twisted sheaf, is zero. Therefore the one dimensional twisted sheaves on S is actually equivalent to the one dimensional sheaves on S. Then the geometric semistability of a one dimensional sheaf as in §6.3.1 corresponds to the general Gieseker semistability of the corresponding untwisted sheaf. Therefore the invariants J(v) defined in the category Coh(S, α) or Coh(X, α) is the same as J X (v) if v = (0, β, n). So we get: Proposition 6.22. In the category Coh tw π (X) of twisted sheaves, the multiple cover formula is still (6.4.1). Thus we prove Theorem 1.1. APPENDIX A. THE INVARIANTS FOR TWISTED SHEAVES ON TWISTED K3 SURFACES In this appendix we generalize the result in [45] for comparing the counting invariants of semistable objects and invariants of counting semistable sheaves in the derived category of coherent sheaves on K3 surfaces to twisted K3 surfaces. A.1. Review of stability conditions on twisted K3 surfaces. A.1.1. Twisted K3 surfaces. Let S be a smooth projective K3 surface. Let A be an abelian group scheme on S. An A-gerbe S Ñ S is a DM stack S over S such that for any open subset U Ă S, there exists a covering U 1 Ñ U such that S(U 1 ) ‰ H, and any sections s, s 1 P S(U) there exists U 1 Ñ U such that s| U = s 1 | U 1 . Also let (A) S be the sheaf of abelian groups A on S, we have (A) S -IS, where IS is the inertia stack of S. We mainly take A = µ r and C˚. Consider the following exact sequence: 1 Ñ µ r Ñ C˚(¨) r ÝÑ C˚Ñ 1 and taking cohomology:¨¨¨Ñ Let α := ϕ([S]) P H 2 (S, OS ) tor be the class of the gerbe S in Br 1 (S). Then α is called a Brauer class. A K3 surface (S, α) together with a Brauer class α is called a twisted K3 surface in [16]. In particular an optimal µ r -gerbe S Ñ S determines a twisted K3 surface. H 1 (S, C˚) ψ ÝÑ H 2 (S, µ r ) ϕ ÝÑ H 2 (S, C˚) Ѩ¨Ä µ r -gerbe S Ñ S is Remark A.2. We use the notation (S, α), for α P H 2 (S, OS ) tor , or S α (where α also determines one class in H 2 (S, µ r )) to represent the twisted K3 surface, where we can view p : S α Ñ S an optimal µ r -gerbe. We may exchange the notation (S, α) and S α arbitrarily in the paper. Remark A.3. Since Br 1 (S) = Br(S), any optimal µ r -gerbe S α Ñ S for α P H 2 (S, µ r ) Ñ H 2 (S, OS ) gives an element [P] P H 1 (S, PGL r ), which classifies PGL r -bundles on S. A.1.2. Bridgeland stability conditions on twisted K3 surfaces. In this section we review the Bridgeland stability conditions on twisted K3 surfaces in [17]. Huybrechts, Macri and Stellari [17] studied the stability condition on any generic K3 category and we only fix to category of twisted sheaves on K3 surfaces. For the twisted K3 surface (S, α) or S α , we let Coh(S, α) or Coh tw (S α ) be the category of twisted sheaves on S or S α . Let D b (Coh(S, α)) or D b (Coh tw (S α )) be the corresponding derived category. These two categories Coh(S, α) and Coh tw (S α ) are equivalent if we forget about the µ r -gerbe structures on the twisted sheaves. We denote by Stab(S, α) (or Stab tw (S α )) the stability manifold. From [16], there is a twisted Hodge structure r H(S, α, Z) = H˚(S, Z), x, y,´ξ r , reviewed in Definition 6.6. Here ξ P H 2 (S, Z) is a lift of α P H 2 (S, µ r ). Since our µ r -gerbe S α is optimal, there is a corresponding Brauer-Severi variety p : P Ñ S, where P is a projective P r´1 -bundle over S. Then in [51], and §6.2.1 we defined the category of P-sheaves Coh(S, P) on P which is equivalent to Coh(S, α). We also define the Mukai vector v G (E) for a twisted sheaf E, where the rank r vector bundle G on P is determined by the Euler sequence 0 Ñ O P Ñ G Ñ T P/S Ñ 0 Note that from Proposition 6.5 (also [ Then (rk(E), D, a) P H˚(S, Z) and D mod r = ω(E), and ω(E) P H 2 (S, µ r ) is c 1 (E) mod r. The Mukai vectors in [16] are of the form e Let B 0 P H 2 (S, Q) be a B-field lift of α, that is: exp(B 0 ) = α. Then for any real ample class ω P H 1,1 (S, Z) b R, we let ϕ = e b 0 +iω = 1 + (B 0 + iω) + (B 2 0´ω 2 ) 2 + i(B 0¨ω ) P NS(S, α) b C. The subset P(S, α) Ă NS(S, α) b C has two connected components and we shall denote the one that contains ϕ = exp(B 0 + iω) by P + (S, α) and P + (S, α) Ă P(S, α) Ă NS(S, α) b C. Also π 1 (P + (S, α)) -Z. If B 1 P NS(S) b Q, and B = B 1 + B 0 , we have ϕ = e B+iω = exp(B 1 )¨exp(B 0 + iω) P NS(S, α) b C and ϕ P P + (S, α). Now for ϕ P P + (S, α), recall for any twisted sheaf E on S, we can define the slope as: µ(E) = c 1 (E)¨ω rk(E) which defines the geometric slope semistability. For a torsion free twisted sheaf E, let α) to be the subcategory consisting of twisted sheaves whose torsion free part has µ´ω-semistable Harder-Narasimhan factors of slope 0 = E 0 Ă E 1 è¨¨Ă E m´1 Ă E m = E be the Harder-Narasimhan filtration, where E i /E i+1 = F i is µ ω -semistable and µ(F i ) ą µ(F i+1 ). Defne T Ă Coh(S,µ ω (F i ) ą B¨ω (or Im Z ϕ (F i ) ą 0). A nontrivial twisted sheaf E is an object in F Ă Coh(S, α) if E is torsion free and µ ω (F i ) ď B¨ω (or Im Z ϕ (F i ) ď 0). Then (T , F ) is a torsion pair on Coh(S, α) and let A(ϕ) := $ & % E P D b (Coh(S, α))ˇˇˇˇˇˇ‚ H i (E) = 0 for i R t´1, 0u ‚H´1(E) P F ‚H 0 (E) P T , . Then let Stab˝(S, α) Ă Stab(S, α) be the connected component of Stab(S, α) that contains the stability conditions described above. Also [17,Proposition 3.8] proved that if σ = (Z, P) is contained in a connected component of maximal dimension Stab˝(S, α), and for any closed point x P S, the skyscraper sheaf k(x) is σ-stable of phase one with Z (k(x)) =´1. Then there exists ϕ = exp(B + iω) P P + (S, α) such that the heart of σ is A(ϕ). For any ϕ = B + iω P P + (S, α), we can calculate (see [17,Lemma 3.4]) Z ϕ : A(ϕ) Ñ C If v G (E) = (r, l, s) for r ą 0, then (A.1.1) Z ϕ (E) = 1 2r (l 2´2 rs) + r 2 ω 2´( l´rB) 2 + (ω¨l´r(ω¨B) i Also for r = 0, (A.1.2) Z ϕ (E) = (´s + l¨B) + (l¨ω) i which is the same as in [45,Formula (31)]. Let V Ă Stab˝(S, α) be the open subset in Stab(S, α) such that it consists of all stability conditions σ ϕ for ϕ = B + iω P P + (S, α). Then V satisfies the following properties: ‚ For any σ ϕ P Stab˝(S, α), there exists Φ P Auteq(D b (S, α)) and g P Ă GL + (2, R) such that g˝Φ(σ ϕ ) is also algebraic and is contained in V, see [17,Remark 3.9]. Now we also introduce the geometric twisted Gieseker stability. We follow [45, §4.2] and let L, M P Pic(S) be two line bundles, and L is ample. Define for any E P Coh(S, α), the twisted Hilbert polynomial: χ g (E b M´1 b L n ) = d ÿ i=0 a i n i for a i P Q, a d ‰ 0. Let ω = c 1 (L), β = c 1 (M) and the twisted reduced Hilbert polynomial is: r¨ n + (s´β¨l) l¨ω These formula are the same as [45,Formula (28), (29)] up to a factor r due to the gerbe structure. The twisted geometric stability of E can be similarly defined. The following lemma is based on [45,Lemma 4.6]. (A.1.3) p(E, β, ω, n) = χ g (E b M´1 b L n ) a d If v G (E) = (rk, l, Lemma A.7. We have: (1) Let E P D b (S, α) be simple, then v G (E) 2 ě´2. (2) For any ϕ = B + iω, m P R ą0 , the number #tv α P NS(S, α)|v 2 α ě´2, |xexp(ϕ), v α ď m| is finite. (3) Let E P N (S, α), and v α (E) = (0, l, s) P NS˚(S, α), l ‰ 0, then p(E, β, ω, n) = n´R e Z ϕ (E) Im Z ϕ (E) P Q[n]. (4) Let E, E 1 P NS(S, α), p(E, β, ω, n) = p(E 1 , β, ω, n) if and only if Im A.2. The moduli stack counting semistable objects. We list some boundedness of semistable objects following [45, §4.5]. The boundedness results are used to show that the moduli stack M c σ(β,ω) (S, α) with topological invariant c is an Artin stack of finite type. Since for now, that the moduli stack of Bridgeland semistable objects is an Artin stack is a well-known fact, we only need the boundedness results to prove our later results we interested in. Z ϕ k (E 1 ) Z ϕ k (E) = 0 for infinitely many k P Q, where ϕ k = B + ikω. Proof. (1) is from [51, Let E P A(ϕ) be an object. Set H 0 (E) tor Ă H 0 (E) to be the maximal torsion subsheaf of H 0 (E), and H 0 (E) free = H 0 (E)/H 0 (E) tor . We take Toda's notations: T 1 ,¨¨¨, T a(E) P Coh(S, α) F 1 ,¨¨¨, F d(E) , F d(E)+1 ,¨¨¨, F e(E) P Coh(S, α) represent the µ ω -stable factors of H 0 (E) free and H´1(E) respectively. Let T a(E)+1 ,¨¨¨, T b(E) , T b(E)+1 ,¨¨¨, T c(E) P Coh(S, α) be the ϕ = B + iω-twisted stable factors of H 0 (E) tor . Set dim(T i ) = 2 (1 ď i ď a(E)); dim(T i ) = 1 (a(E) ă i ď b(E)); dim(T i ) = 0 (b(E) ă i ď c(E)); Im Z ϕ (F i [1]) ą 0 (1 ď i ď d(E))); Im Z ϕ (F i [1]) = 0 (d(E) ă i ď e(E))). Define for c P N (S, α), M c (ϕ = (β, ω)) = tE P A(ϕ)| Im Z ϕ (E) ď Im Z ϕ (c)u.tIm Z ϕ (T i ) P Q|1 ď i ď c(E), E P M c (ϕ = (β, ω))u and tIm Z ϕ (F i [1]) P Q|1 ď i ď e(E), E P M c (ϕ = (β, ω))u are finite sets. Lemma A.9. There exist constants C, C 1 , N (depending only on c, B, ω), such that 1 k Re Z ϕ k (T i ) ě Re Z ϕ (T i ) ě C; (1 ď i ď a(E)) and 1 k Re Z ϕ k (F i [1]) ě Re Z ϕ (F i [1]) ě C 1 ; 1 ď i ď e(E) for any E P M c (ϕ) and k ě N, where ϕ k = B + ikω. A.3. Counting twisted semistable objects and twisted semistable sheaves. Let H(A(ϕ)) be the Hall algebra of the category A(ϕ). One can define Bridgeland stability Z ϕ k and define the counting invariants N(v) which count semistable twisted objects with Mukai vector v, see Definition 6.11. We also mimic the definition of invariants J(v) for X to define the invariants counting semistable twisted sheaves. Let Λ := Q(q (q´1)P q (ǫ ω,S α (c)). If´c P C(S, α), define J(c) = J(´c). The invariant J(c) does not depend on the choice of ω. Our aim is to compare J(c) with the invariant N(c) counting semistable objects in A(ϕ). For this purpose, we let $ ' & ' % ϕ k = B + ikω; A(ϕ k ) = A(ϕ); σ k = (Z ϕ k , A(ϕ)). Let us fix a c = (rk, l, s) P H˚(S, Q). We first have a generalization of [45,Proposition 6.4]. Proposition A.10. Assume that ω¨l ą 0 or rk = l = 0. Let us choose 0 ă φ k ď 1 such that Z ϕ k (c) P R ą0 e iπφ k . Then there exists a N ą 0 such that for all k ě N, and c 1 satisfying c 1 P C σ k (φ k ); | Im Z ϕ (c 1 )| ď | Im Z ϕ (c)|, then any E P M (c 1 ,φ k ) (σ k ) (which is σ k -semistable with (c 1 , φ k )) is ω-Gieseker semistable as a coherent sheaf. Proof. In the case rk = l = 0, any object E P A ω of numerical invariant c is a zero dimensional sheaf, so must be a semistable sheaf. Now let ω¨l ą 0. From the formula of Z ϕ (E) in (A.1.1) and (A.1.2), when rk ą 0, the phase φ k Ñ 0(k Ñ 8), and when rk = 0, φ k Ñ 1 2 (k Ñ 8). Therefore there exists N ą 0 such that φ k ď 3 4 for all k ě N. Let E P M (c 1 ,φ k ) (σ k ) and c 1 satisfies the condition the the proposition, we have φ k (H´1(E)[1]) ď φ k ď 3 4 . Look at E Þ Ñ Re Z ϕ k (H´1(E) [1]) Im Z ϕ k (H´1(E)[1]) = 1 2 rk (l 2´2 rk s + rk 2 k 2 ω 2´( l´rk B) 2 ) (kω¨l)´rk(kω¨B) = Re Z ϕ k (H´1(E) [1]) k¨Im Z ϕ (H´1(E) [1]) . On the moduli spaces Ť kěN,c 1 M (c 1 ,φ k ) (σ k ) which is bounded below, we have E P M c (ϕ) since | Im Z ϕ (E)| ď | Im Z ϕ (c)|. Then the map E Þ Ñ Im Z ϕ (E) on M c (ϕ) is bounded by Lemma A.8. Therefore the map M (c 1 ,φ k ) (σ k ) , . is a finite set. If we let tv 1 ,¨¨¨, v n u be this set. Then lim kÑ8 φ k (v i ) = 1, so we can make φ k (v i ) ą 3 4 for k ě N for some N ą 0. This implies that E P M (c 1 ,φ k ) (σ k ) has that H´1(E) = 0 meaning that E is a coherent sheaf. The next step is to follow [45,Proposition 6.4] to show that E is actually a Gieseker semistable twisted sheaf. If not, then let T be the ω-Gieseker semistable factor of E of smallest reduced geometric Hilbert polynomial. Let v α (E) = ((rk) 1 , l 1 , s 1 ); v α (T) = ((rk) 2 , l 2 , s 2 ). If (rk) 1 = 0 which implies that (rk) 2 = 0, then from (3) in Lemma A.7, E is Gieseker twisted semistable. So we assume that (rk) 1 ą 0, (rk) 2 ą 0. In this case φ k Ñ 0(k Ñ 8), we have E Ñ T is surjective and since E is σ k -semistable for k ě N, φ k (E) ď φ k (T). So we have (A.3.1) Re Z ϕ k (E) Im Z ϕ k (E) ě Re Z ϕ k (T) Im Z ϕ k (T) . We calculate above as: (A.3.2) ωl 2´( rk) 2 ωB ωl 1´( rk) 1 ωB ´s 1 + 1 2 (rk) 1 k 2 ω 2 + l 1 B´1 2 (rk) 1 B 2 ě´s 2 + 1 2 (rk) 2 k 2 ω 2 + l 2 B´1 2 (rk) 2 B 2 . We also have from the Mukai vector property: , . is a finite set. Therefore the set $ & % v α (E tor ) P NS˚(S, α)|E P ď kěN,c 1 M (c 1 ,φ k ) (σ k ) , . is a finite set. Assume that this set is: (rk) 2 + ik(µ ω (E)´µ ω (T)). Therefore replacing N if necessary, we have φ k (E) ą φ k (T) for k ě N. This N is only determined by the numerical class of T, and the finiteness of v α (T) can make this N uniformly such that φ k (E) ą φ k (T) for k ě N, which contradicts E is σ k -semistable. tv 1 1 ,¨¨¨, v 1 m u. Then φ k (v 1 i ) Ñ 1 2 when k Ñ 8. So φ k (v 1 i ) ą φ Next we have a similar result as in [45,Lemma 6.5]. Lemma A.11. If ω¨l ą 0 or rk = l = 0, then there exists a N ą 0 such that for k ě N, and c 1 P C(S, α), (A.3.4) p(c 1 , ω, n) = p(c, ω, n) then any Gieseker semistable twisted sheaf E of numerical type c 1 is σ k -semistable. Proof. First the set c 1 P C(S, α) satisfying (A.3.4) is finite. So we take c = c 1 . The case of rk = l = 0 is obvious. For the case rk ą 0, ω¨l ą 0, the smooth surface case is proved in [6,Proposition 14.2]. Since the twisted stability in [17] is similar to the construction of Bridgeland, the proof of [6, Proposition 14.2] works for twisted sheaves. For the case rk = 0, l ‰ 0, E is ω-Gieseker semistable. In this case φ k Ñ 1 2 when k Ñ 8. Then Toda's proof in [45,Lemma 6.5] works in this case. We show: Theorem A.12. For c P C(S, α), we have N σ (c) = J (c) and N σ (c) does not depend on the stability condition σ. Proof. First for the category of twisted sheaves Coh(S, α), the K-theory [51] K (Coh(S, α) ) = Q[E 0 ] ' K ď1 (Coh(S, α)) where E 0 is the minimal rank rk ą 0 (in this case is r) of S α -twisted locally free sheaves. So tensoring with E 0 gives an equivalence on K (Coh(S, α)) and hence on the derived category D b (S, α), thus the derived equivalence result of [45,Corollary 5.26] implies that: N(c b E 0 ) = N(c); J(c b E 0 ) = J(c). Thus we can assume that ω¨l ą 0, or rk = l = 0. We follow Toda to show that: N σ k (c) = J ω (c) and N is chosen before as in Proposition A.10. Let c 1 ,¨¨¨, c m P C σ k (φ k ) be such that c 1 +¨¨¨+ c m = c; m ź i=1 N σ k (c i ) ‰ 0. Here we consider σ k P B˝as an open set such that B˝= B is compact in the stability manifold. There exists a wall and chamber structure tW γ u γPΓ on B with property: S := tE P D b (S, α)|E is semistable for some σ 1 = (Z 1 , P 1 ) P B; |Z 1 (E)| ď |Z 1 (c)|u. S is a bounded mass, which means there exists m ą 0 such that m σ (E) ď M for any E P S, where m σ (E) = ř n i=1 |Z (A i )| for 0 = E 0 / / E 1 / /~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ E 2 / /~⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ / /¨¨¨¨¨¨E n = E z z t t t t t t t t t t A 1 c c • • • • • • • • • A 2`❆ ❆ ❆ ❆ ❆ ❆ ❆ A ǹ`❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ So this means that there exists Γ 1 Ă Γ and a connected component C such that C Ă č γPΓ 1 (B X W γ )z ď γRΓ 1 W γ . Infinitely many σ k 1 for k 1 P Q ěN are contained in C. Then σ k P C. If c i and c j are not proportional in N (S, α), then Im Z ϕ k 1 (c j ) Z ϕ k 1 (c i ) = 0 for infinitely many k 1 P Q ěN . Therefore from Lemma A.7 (4), p(c i , ω, n) = p(c j , ω, n) = p(c, ω, n) for i, j. From Lemma A.11, (A.3.5) M (c i ,φ k ) (σ k ) = M c i (ω) and ś m i=1 N σ k (c i ) = ś m i=1 J ω (c i ). On the other hand, if c 1 ,¨¨¨, c m P C(S, α) such that ś m i=1 J ω (c i ) ‰ 0 and c 1 +¨¨¨+ c m = c, p(c i , ω, n) = p(c, ω, n), (A.3.5) still holds for k ě N by Proposition A.10 and Lemma A.11 above. Hence ś m i=1 N σ k (c i ) = ś m i=1 J ω (c i ). Also c i P C σ k (φ k ) so J ω (c) = N σ k (c). That N σ k (c) does not depend on σ k is just from a former argument or [45,Proposition 5.17]. Theorem 1. 1 . 1Let S Ñ S be an optimal µ r -gerbe over a smooth projective K3 surface S. Then for each Mukai vector v P Γ 0 , we have a formula for the Joyce invariant: J(v) = ÿ kě1;k|v 1 k 2 χ(Hilb xv/k,v/ky+1 (S)) genus zero Gopakumar-Vafa invariant. The formula (1.0.1) now is true since in The first order neighbourhood of the origin is Spec(O H 1 (T S ) /m 2 ). The cotangent sheaf restricted to0 P H 1 (T S ) gives H 1 (T S )˚. Let S Ñ Spec(O H 1 (T S ) /m 2 ) bethe tautological flat family of K3 surfaces. As in [28, §2.1], there exists a subspace V Ă H 1 (T S ) such that Y β : V -ÝÑ H 2 (O S ) is an isomorphism. Then restricting S to B := Spec(O V /m 2 ) gives the flat twistor family S B of K3 surfaces. Since the twistor family S B Ñ B is a family over an affine scheme we have: Definition 3. 7 . 7Define N Y n,(β,d) := χ(P/E, ν P ) the weighted Euler characteristic by the Behrend function. Remark 4. 4 . 4A similar formula as in Theorem 4.3 for twisted Joyce-Song stable pair invariants also holds with the same proof in[23, Proposition 2.6]. Lemma 5. 5 . 5([47, Lemma 5.2]) The homomorphisms (Z ω,θ,i ) give a weak stability condition on the category A X . (´(´q) a ).Formula (5.3.3) gives the multiple cover formula: Conjecture 5.22. 6. 1 . 1KKV formula for K3 surfaces and the theorem of Maulik-Thomas. Let us recall how Maulik-Thomas [35] prove the multiple cover formula for K3 surfaces. Let X := SˆC be the local K3 surface. Let ι : S Ñ X be the inclusion and N red g,β := ż [M g (X,ι˚β)] vir 1 e(N vir ) Theorem 6.1. ([35, Theorem 6.3]) OX) tor is the image of the µ r -gerbe [S]. The line bundle L P Pic(P) is the line bundle with the property that :, 0 ( 0H˚(S, Z) ÝÑ H˚(S, Q) by x Þ Ñ e´ξ r¨x and T´ξ r preserves the bilinear form x, y. The following result is from [51, Lemma 3.3]. Proposition 6.5. ([51, Lemma 3.3]) Let ξ P H 2 (S, Z) be a representation of ω(G) P H 2 (S, µ r ), where rk(G) = r. Set (rk(E), D, a) := e ξ r¨v G (E). Then (rk(E), D, a) P H˚(S, Z) and D mod r = ω(E). In [15], Huybrechts and Stellari defined a weight 2 Hodge structure on the lattice (H˚(S, Z), H˚(S, Z) b A 1 κ ) := T´1 ξ r (H 2,0 (S)); H 1,1 (H˚(S, Z) b A 1 κ )) := T´1 ξ r (' 2 p=0 H p,p (S)); H 0,2 (H˚(S, Z) b A 1 κ )) := T´1 ξ r (H 0,2 (S)). and this polarized Hodge structure is denoted by H˚(S, Z), x, y,´ξ r . From [51, Lemma 3.4], this Hodge structure H˚(S, Z), x, y,´ξ r depends only on the Brauer class o([ξ mod r]), where o(ξ) is the image under the map o : H 2 (S, µ r ) Ñ H 2 (S, OS ). Definition 6.6. For the projective bundle P Ñ S and G the locally free P-sheaf. Let ξ P H 2 (S, Z) be a lifting of ω(G) P H 2 (S, µ r ), where rk(G) = r. (1) Define an integral Hodge structure of H˚(S, Q) as: T´ξ r H˚(S, Z), x, y,´ξ r . ( 2 ) 2v = (rk, ζ, b) is a Mukai vector if v P T´ξ r (H˚(S, Z)) and ζ P Pic(S) b Q. Moreover, if v is primitive in T´ξ r (H˚(S, Z)), then v is primitive. Now we let cl 0 : K(D S 0 ) π SÝ Ñ K(S) Ch ÝÑ Γ 0 with Γ 0 = r H(S, Q) X r H 1,1 (S) = Q ' NS(S) ' Q [ 21 , 21§3] Lieblich defined the moduli stack of M s,tw S (v) of stable S-twisted sheaves with Mukai vector v. We have the following result: Theorem 6.7. The natural map M s,tw S (v) Ñ M P,G H,s (v) is a µ r -gerbe. 6.3. The invariants N(v G ). 6.3. 2 . 2Toda's weak stability condition on D S 0 and D. Recall we have Toda's weak stability condition as in Definition 5.4. For our Γ = Z ' Γ 0 , let ( P q : H(A ω ) q´1)P q (ǫ tω (v)) P Q exists.Definition 6.10. For v P Γ 0 , define DT χ tω (v) H 2 ( 2S, Q) Ñ H 2 (S, OS ) by B Þ Ñ α = exp(B). Let B 1 P NS(S) b R and B := B 1 + B 0 , then ϕ = e B+iω P NS(S, α) b C and Z ϕ (E) := xv G (E), ϕy. is a map defined in[46, Proposition 5.1], and θ˝is the restriction to Stab˝(D S 0 ). Leti : S ã Ñ SˆP 1 = X,then i˚: NS(X) Ñ NS(S) is the pullback. This diagram gives a homeomorphism between Stab˝(D S 0 ) and one of the connected components of Stab˝(S, α)ˆN S(S)bC NS(X). Now on the derived category D S 0 , if we have a stability condition σ = (Z, A) P StabΓ 0 (D S 0 ) we define the invariants N σ (v) P Q as the Euler characteristic version of the generalized Donaldson-Thomas invariants counting Z-semistable objects E P A or A[´1] with cl 0 (E) = v in Definition 6.11. 6.4. 1 . 1Counting invariants on X := SˆC. We are actually interested in the counting invariants on the open local K3 gerbe X := SˆC. Let Definition 6 . 15 . 615For v P Γ 0 , define the invariant J(v) P Q as follows:(1) If v P C( From Theorem A.12 in the Appendix, E P B ω is Z ϕ -semistable with cl 0 (E) = v if and only if E is Gieseker semistable with v G (E) = v¨? td S ? Ch(Rp˚(GbG _ )) . 6.4.3. Automorphic property. Let G be the group of Hodge isometries G := O Hodge ( r H(S, α, Z), (‹, ‹)) given by a class [S] P H 2 (S, µ r ), and is called "essentially trivial" if it is in the image of the map ψ. Therefore an essentially trivial µ r -gerbe S Ñ S is given by a line bundle L P Pic(S).The cohomological Brauer groupBr 1 (S) = H 2 (S, OS ) tor ,is by definition the torsion part of the cohomology H 2 (S, OS ) tor . De Jong's theorem[9] implies that the Brauer group Br(S) = Br 1 (S), and Br(S) is the group of isomorphism classes of Azumaya algebras A on S, see[21, Definition 2.10]. Here an Azumaya algebra A on S is an associative (noncommutative) O S -algebra A which is locally isomorphic to a matrix algebra M r (O S ) for some r ą 0.Definition A.1. A µ r -gerbe p : S Ñ S is called "optimal" if the period per(S), which is defined as the order of [S] in Br 1 (S) = Br(S), is equal to r. 51 , 51Lemma 3.3]), if we set (rk(E), D, a) := e ξ r¨v G (E). - Then from[17, Lemma 3.4], let ϕ = e B+iω , the induced homomorphismZ ϕ : A(ϕ) Ñ C is a stability function on A(ϕ) if and only if for any spherical twisted sheaf E P Coh(S, α), Z ϕ (E) R R ď0 . Proposition A.6. ([17, Proposition 3.6]) The pair (Z ϕ , A(ϕ)) defines a stability condition on D b (Coh(S, α)). Proposition 3.6], where Yoshioka proves it for twisted sheaves, but the same argument as in [6, Lemma 5.1] works for derived simple object by Serre duality. (2) just come from [6, Lemma 8.2], which works for twisted sheaves on K3, since there are finitely many integral points in N (S, α) b R. (3) and (4) are from the above calculations (A.1.4) and (A.1.5). 1 2 1) be a Q-algebra, and γ : K(Var) Ñ Λ be the motivic invariants defined by Poincaré polynomial. Fixing a numerical invariant c P N (S, α),let M tw ω (c) Ă x M(A(ϕ)) be the stack of ω-Gieseker semistable twisted sheaves of numerical invariant c. Then there exists an elementδ ω,S α (c) = [M tw ω (c) Ñ x M(A(ϕ))]in the Hall algebra H(A S α ) = H(Coh(S, α)). Defineǫ ω,S α (c) = ÿ ℓě1,c 1 +¨¨¨+c ℓ =c, p(c i ,ω,n)=p(c,ω,n) (´1) ℓ´1 ℓ δ ω,S α (c 1 ) ‹¨¨¨‹ δ ω,S α (c ℓ )Let C(S, α) := Im(Coh(S, α) Ñ N (S, α)). If c P C(S, α), define J(c) = lim (rk) 2 ă (rk) 1 , 0 ă ωl 2 ď ωl 1 , (l 2 ) 2´2 (rk) 2 s 2 ě´2. Then (A.3.2) and (A.3.3) imply that the set $ & % v α (T) P NS˚(S, α)|E P ď kěN,c 1 M (c 1 ,φ k ) (σ k ) k for all 1 ď i ď m and k ě N after replacing N if necessary. Hence for k ě N, E P M (c 1 ,φ k ) (σ k ) must be torsion free. By definition of T,µ ω (E) ą µ ω (T) or µ ω (E) = µ ω (T), Multiple cover formula for twisted K3 surfaces 26 6.1. KKV formula for K3 surfaces and the theorem of Maulik-Thomas 26 6.2. Category of twisted sheaves on X = SˆP 11. Introduction 1 1.1. Outline 3 1.2. Convention 3 Acknowledgments 4 2.Étale gerbes and stacks, notations 4 3. Stable pair theory on some threefold DM stacks 4 3.1. Stable pair theory 4 3.2. Stable pairs on Y = SˆE. 5 3.3. The elliptic curve E-action 6 3.4. The reduced perfect obstruction theory 7 3.5. Symmetric obstruction theory of the quotient 8 4. Behrend equals to Reduced invariants of Oberdieck 13 5. Wall crossing for Calabi-Yau gerbes and Toda's multiple cover formula 15 5.1. Wall-crossing in D0-D2-D6 bound states for µ r -gerbes 15 5.2. Decomposition formula forétale gerbes 22 5.3. Multiple cover formula. 24 6. 26 6.3. The invariants N(v G ) 29 6.4. Twisted Hodge isometry 32 Appendix A. The invariants for twisted sheaves on twisted K3 surfaces 38 A.1. Review of stability conditions on twisted K3 surfaces 38 A.2. The moduli stack counting semistable objects 41 A.3. Counting twisted semistable objects and twisted semistable sheaves 42 References 46 1. INTRODUCTION is the generating function of Behrend invariants of X.Our goal in this section is to show a similar result in [35, Theorem 1.1]. First we prove Lemma 4.2. Let us form the generating function in this way is a stability condition on Coh ď1 (X), see[47, Example 3]. Therefore condition (1) in Definition 5.4 is satisfied.Tocheck the Harder-Narasimhan property, we introduce a torsion pair (A X , see [49, Lemma 2.16], which is defined by A p 1 := xF[1], O x |F is pure 2-dimensional , x P Xy ex and A From the definition of torsion pair, for any T P A p 1 , and F P A p 1/2 , we have Hom(T, F) = 0 and any E P A p X fits into the exact sequence:p 1 , A p 1/2 ) on the category A p p 1/2 := xE P A p X |Hom(F, E) = 0 for any F P A p 1 y. ). Remark 5.8. The integration (5.1.3) may depend on the gerbe structure X. Then from [49, Lemma 2.20], for any nonzero E P A p X , and σ m = B + imω, This is from similar calculations of [49, Lemma 3.8]. Then from Lemma 5.9 we have: let B = kω, E[1] P A p X is µ B+iω -limit semistable if and only if E P Ap 1/2 and (1) for any one Counting invariants and wall crossing. We define the counting invariants in the abelian category A X . If x[50, Proposition 3.17] showed that x M n,β (ω, θ) is an open substack of the stack x M X , hence an Artin stack locally of finite type. When θ Ñ 1, the stack x M n,β (ω, θ) = [P n (X, β)/C˚] comes from [50, Theorem 3.21] and [49, Theorem 4.7]. We omit the details. All of the results in (3), (4) are from [49, Lemma 2.28] and [50, Lemma 4.4]. 5.1.3. M n,β (ω, θ) = [ x M n,β (ω, θ)/C˚] is the moduli stack of Z ω,θ -stable E P A X satisfying cl(E) = (´n,´β, 1), we define Definition 5.10. 5.1.6. Wall-crossing formula of Toda. The wall crossing formula of Toda in [47, Theorem 5.7] and [48, Theorem 5.8, Theorem 8.10] works for coherent sheaves and stable pairs [O X Ñ F] on X. §5.1]. It is defined in a subset V Ă Stab Γ ‚ (D X ) introduced in [48, §5.1].For this we let DT(ω, θ) := ÿ n,β DT n,β (ω, θ)q n t β be the generating series for 0 ă θ ă 1 2 . The wall and chamber structure for weak stability conditions are given by [48, Then we have the limit: DT(ω, φ˘) := lim θÑφ˘0 DT(ω, θ). The following result is from [48, Theorem 5.8, Theorem 8.10] and [47, Theorem 5.7]. s) for rk ą 0, we can calculate using the definition of geometric Hilbert polynomial, and when rk = 0, l ‰ 0, (A.1.3) is: (A.1.5)see [21, Proposition 3.21], (A.1.3) is: (A.1.4) r¨ n 2 + 2ω(l´rk¨β) rk¨ω 2 n´( l 2´2 rk s´(l´rk¨β) 2 ) rk¨ω 2 The following results are from [45, Lemma 4.8, Lemma 4.9] which works for twisted sheaves.Lemma A.8. The maps M c (ϕ) Ñ Z given by E Þ Ñ b(E) and E Þ Ñ d(E) are bounded, and the sets E Þ Ñ 1 k Re Z ϕ k (H´1(E)[1]) on Ť kěN,c 1 M (c 1 ,φ k ) (σ k ) is bounded below. So by Lemma A.9, E Þ Ñ Re Z ϕ (H´1(E)[1]) is bounded below on Ť kěN,c 1 M (c 1 ,φ k ) (σ k ).Then this imples ([45, Lemma 4.10]) that the set $ & % v α (H´1(E)[1]) P NS˚(S, α)|E P ď kěN,c 1 Acknowledgments. Y. J. would like to thank Kai Behrend, Amin Gholampour, Martijn Kool, andNow we use Toda's method to show thatfor g a Hodge isometry on r H(S, α, Z). First on the gerbe X = SˆP 1 , we mimic the definition of J(v) in Definition 6.15 to define J(v). This invariant is defined by:and M tw ω,X (v) is the moduli stack of ω-Gieseker semistable twisted sheaves E P Coh tw π (X) with cl 0 (E) = v.For X = SˆP 1 , and each point p P P 1 , letand define ǫ ω,Z (v) accordingly. The following lemma is[46,Lemma 4.25].Proof. Since the definition of ǫ ω,X (v), (resp. ǫ ω,U p (v), ǫ ω,X p (v)) is given by δ ω,X (resp. δ ω,U p , δ ω,X p ) by the logarithm with the same coefficients as before. The proof is the same as [46, Lemma 4.5].Lemma 6.19. We have:Proof. The proof uses the Behrend function techniques in[26]. First the invariantis the closed subscheme consisting of semistable twisted sheaves E with supp(E) Ă X p for p P P 1 , then from Lemma 6.18, the constructible function ν M is zero outside X p , and we have:for X 0 = S. Any g in the automorphism group Aut(P 1 ) sends a point p to q, and the element ǫSo we have:from Theorem 6.17 and Lemma 6.19.(4) Any E P K(S, α), we have Ch B (E) P e B¨ ' p,p H p,p (S) .This twisted Chern character Ch B has the property: if B 0 = k¨B P H 2 (S, Z) for some k P Z, thenThis implies that Then the twisted Chern character:can be extended to the derived category. Let NS(S, α) := r H 1,1 (S, α, Z) be the twisted Néron-Severi group. 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Zhou, Relative Orbifold Donaldson-Thomas Theory and the Degeneration Formula, Algebr. Geom. 5 (2018), no. 4, 464- 522, arXiv:1504.02303. . Department Of Mathematics, University, Kansas, Lawrence Blvd, Ks, 66045Email address: [email protected] DEPARTMENT OF MATHEMATICS, OHIO STATE UNIVERSITY, 100 MATH TOWER 231 WEST 18TH AVE., COLUMBUS OH 43210 USA Email address: [email protected] OF MATHEMATICS, UNIVERSITY OF KANSAS, 405 SNOW HALL 1460 JAYHAWK BLVD, LAWRENCE KS 66045 USA Email address: [email protected] DEPARTMENT OF MATHEMATICS, OHIO STATE UNIVERSITY, 100 MATH TOWER 231 WEST 18TH AVE., COLUMBUS OH 43210 USA Email address: [email protected]

In simulation-based inferences for partially observed Markov process models (POMP), the by-product of the Monte Carlo filtering is an approximation of the log likelihood function. Recently, iterated filtering[14,13]has originally been introduced and it has been shown that the gradient of the log likelihood can also be approximated. Consequently, different stochastic optimization algorithm can be applied to estimate the parameters of the underlying models. As accelerated gradient is an efficient approach in the optimization literature, we show that we can accelerate iterated filtering in the same manner and inherit that high convergence rate while relaxing the restricted conditions of unbiased gradient approximation. We show that this novel algorithm can be applied to both convex and nonconvex log likelihood functions. In addition, this approach has substantially outperformed most of other previous approaches in a toy example and in a challenging scientific problem of modeling infectious diseases.

1802.08613

44e3e6f6a5ab855ac99892edd508c8c1bdabdff9

Accelerate iterated filtering Dao Nguyen [email protected] Departments of Mathematics University of Mississippi OxfordMississippiUSA Accelerate iterated filtering arXiv: arXiv:0000.0000and phrases: accelerate iterated filteringaccelerate inexact gradient methodsequential Monte Carlostate space modelparameter estimation In simulation-based inferences for partially observed Markov process models (POMP), the by-product of the Monte Carlo filtering is an approximation of the log likelihood function. Recently, iterated filtering[14,13]has originally been introduced and it has been shown that the gradient of the log likelihood can also be approximated. Consequently, different stochastic optimization algorithm can be applied to estimate the parameters of the underlying models. As accelerated gradient is an efficient approach in the optimization literature, we show that we can accelerate iterated filtering in the same manner and inherit that high convergence rate while relaxing the restricted conditions of unbiased gradient approximation. We show that this novel algorithm can be applied to both convex and nonconvex log likelihood functions. In addition, this approach has substantially outperformed most of other previous approaches in a toy example and in a challenging scientific problem of modeling infectious diseases. Introduction The last decade has seen a great increase in the use of simulation-based inference where numerical approximations are based on either Markov chain Monte Carlo or sequential Monte Carlo sampling. These approaches have become popularized, in part, because of the increasing computational power and the emergence of efficient stochastic optimization algorithms. On the Bayesian paradigm, particle Markov chain Monte Carlo has been introduced and popularized by Doucet and collaborators [1,2,31]. Similar ideas have been developed previously [21,7,10,16] but in different contexts than simulation-based inferences. On the frequentist paradigm, [14,13] have introduced an original approach to perform simulation-based parameter inference in POMP models by combining stochastic gradient approximation and particle filtering. In this paper, we will focus on improving one of the most popular algorithm of this class, namely, iterated filtering (IF). Iterated filtering uses an approximation estimate of the gradient of the log likelihood computed from particle filters while proposing an artificial perturbation moves to update the parameters. This class of algorithm is attractive because it enables routine simulation-based parameter inferences in general POMP model, even in the cases of intractable likelihoods. Due to some interesting theoretical properties [13,15,29], its applications range in various fields such as biology, ecology, economics and engineering [24,23,20,3,5,4]. Iterated filtering was later theoretically developed by Ionides et al. [13]. Recently, Lindström et al. [24] extended it to improve on numerical performance while Doucet et al. [8] expanded it to include filtering/smoothing with quite attractive theoretical properties. Ionides et al. [15] generalized Lindström et al. [24]'s approach and combined the idea with data cloning [22], developing a Bayes map iterated filtering with an entirely different theoretical approach. Nguyen and Ionides [30] revisited the approach of Doucet et al. [8], using a different perturbation noise and computed both the gradient and the Hessian. Similar to intractable likelihood in the context of iterated filtering, Poyiadjis et al. [32], Nemeth et al. [26], Doucet et al. [8] showed that the gradient and Hessian information can also be computed from particle filter. In the same line, manifold Langevin Monte Carlo (mMALA) [12] exploits the Hessian information to simplify the tedious tuning method while improving on convergence rate. However, this relies on rather strong assumptions that the gradient, and Hessian information of transition density and observation density can be sampled from. This is quite unrealistic in many real world applications. We, therefore, followed the formal approaches, based solely on very weak assumptions of being able to sample from transition density and evaluate from observation density. Motivated from the fact that the gradient and Hessian information can be approximated using the first and the second moments [13,8], we propose to use such approximations in the context of accelerate iterated filtering. Ionides uses score vector merely while Doucet includes the Hessian information for the independent white noise, which is not quite useful in the context of iterated filtering with natural random walk noise. Nguyen and Ionides [30] proposed to approximate the gradient and Hessian using random walk noise to efficiently explore the mode of the likelihood. Other than exploiting approximations of the Hessian under weak assumption, we chose an alternative approach. That is, we apply the accelerate gradient approach to the approximation of the gradient of the log likelihood for an effective estimation approach. The key contributions of this paper are three folds. Firstly, we developed and showed that accelerate iterated filtering algorithm converges using a general non-increasing step size with bias approximation of the gradient. It is simple, elegant, and generalizable to faster algorithms. Secondly, we proved that it has a higher convergence rate in general convex and non-convex conditions of the objective log likelihood. Finally, we showed substantial improvements of the method on a toy problem and on a real world challenge problem of vivax malaria model compared to previous simulation-based inference approaches. The paper is organized as follows. In the next section we introduce some notations and we develop the framework of accelerate iterated filtering. In Sections 3, we state the convergence of this approximation method to the true maximum likelihood estimation by iterating and accelerating noisy gradient of the log likelihood. We validate the proposed methodology by a toy example and a challenging inference problem of fitting a malaria transmission model to time series data in Section 4, showing substantial gains for our methods over current alternatives. We conclude in Section 5 with the suggesting of the future works to be extended. The proofs are postponed to the Appendix. Background of simulation-based inferences We are interested in a general latent variable model since this is an ubiquitous model for applied sciences. Let X be a latent state space with a density q θ (x) parameterized by θ ∈ Θ = R d , and let Y be an observation space equipped with a conditional density f θ (y|x). The observation y ∈ Y are considered as fixed and we write the log-likelihood function of the data (θ ) = log q θ (x) f θ (y|x)dx. We work with the maximum likelihood estimator,θ = arg max (θ ) where (θ ) is intractable but f θ (y|x) can be evaluated, by using samples where f θ (y|x) is also intractable. This process often uses the first order stochastic approximation [19], which involves a Monte Carlo approximation to a difference equation, θ m = θ m−1 + γ m ∇ (θ m−1 ), where θ 0 ∈ Θ is an arbitrary initial estimate and {γ m } m≥1 is a sequence of step sizes with ∑ m≥1 γ m = ∞ and ∑ m≥1 γ 2 m < ∞. The algorithm converges to a local maximum of (θ ) under regularity conditions. The term ∇ (θ ), also called the score function, is shorthand for the R d -valued vector of partial derivatives, ∇ (θ ) = ∂ (θ ) ∂ θ . Sequential Monte Carlo (SMC) approaches have previously been developed to estimate the score function [32,25,6]. However, under the simulation-based setting, which does not require the ability to evaluate transition densities and their derivatives, these approaches are not applicable. As a result, [13], [8] used an artificial dynamics approach to estimate the derivatives. Specifically, [30] considers a parametric model consisting of a density p Y (y; θ ) with the log-likelihood of the data y * ∈ Y given by (θ ) = log p Y (y * ; θ ). A stochastically perturbed model corresponding to a pair of random variables (Θ,Y ) having a joint probability density on R d × Y can be defined as pΘ ,Y (θ , y; θ , τ) = τ −d κ τ −1 (θ − θ ) p Y (y;θ ) . Suppose the following regularity conditions, identical to the assumptions of [8]: Assumption 1. There exists C < ∞ such that for any integer k ≥ 1, 1 ≤ i 1 , . . . , i k ≤ d and β 1 , . . . , β k ≥ 1, u β 1 i 1 u β 2 i 2 · · · u β k i k κ(u) du ≤ C, where κ is a symmetric probability density on R d with respect to Lebesgue measure and Σ = (σ i, j ) d i, j=1 is the non-singular covariance matrix associated to κ. Assumption 2. There exist γ, δ , M > 0, such that for all u ∈ R d , |u| > M ⇒ κ(u) < e −γ|u| δ . Assumption 3. is four times continuously differentiable and δ defined as in Assumption 2. For all θ ∈ R d , there exists 0 < η < δ , ε, D > 0, such that for all u ∈ R d , L (θ + u) ≤ De ε|u| η , where L : R d → R is the associated likelihood function L = exp . Under these regularity assumptions, [8] show that τ −2 Σ −1 E Θ − θ Y = y * − ∇ (θ ) < Cτ 2 . (2.1) These approximations are useful for latent variable models, where the log-likelihood of the model consists of marginalizing over a latent variable, X, (θ ) = log p X,Y (x, y * ; θ ) dx. In this case, the expectations in equation 2.1 can be approximated by Monte Carlo importance sampling, as proposed by [13] and [8]. In [30], the POMP model is a specific latent variable model with X = X 0:N and Y = Y 1:N . A perturbed POMP model is defined to have a similar construction to our perturbed latent variable model with X =X 0:N ,Y =Y 1:N andΘ =Θ 0:N . [13] perturbed the parameters by settingΘ 0:N to be a random walk starting at θ , whereas [8] tookΘ 0:N to be independent additive white noise perturbations of θ . We take advantage of the asymptotic developments of [8] while maintaining some practical advantages of random walk perturbations for finite computations, so we use the constructΘ 0:N as in [30] as follows. Let Z 0 , . . . , Z N be N + 1 independent draws from a density ψ. [30] introduces N + 2 perturbation parameters, τ and τ 0 , . . . , τ N , and construct a processΘ 0:N by setting Θ n = θ + τ ∑ n i=0 τ i Z i for 0 ≤ n ≤ N. We later consider a limit where τ 0:N as fixed and the scale factor τ decreases toward zero, and subsequently another limit where τ 0 is fixed but τ 1:N decrease toward zero together with τ. Let pΘ 0:N (θ 0:N ; θ , τ, τ 0:N ) be the probability density ofΘ 0:N . We define the artificial random variablesΘ 0:N via their density, pΘ 0:N (θ 0:N ; θ , τ, τ 0:N ) = (ττ 0 ) −d ψ (ττ 0 ) −1 (θ 0 − θ ) × N ∏ n=1 (ττ n ) −d ψ (ττ n ) −1 (θ t −θ t−1 ) . We define the stochastically perturbed model with a Markov process {(X n ,Θ n ), 0 ≤ n ≤ N}, observation processY 1:N and parameter (θ , τ, τ 0:N ) by the factorization of their joint probability density This extended model can be used to define a perturbed parameter log-likelihood function, defined as˘ (θ 0:N ) = log pY 1:N |Θ 0:N (y * 1:N |θ 0:N ; θ , τ, τ 0:N ). (2.2) Here, the right hand side does not depend on θ , τ or τ 0:N . We have designed (2.2) so that, settingθ [N+1] = (θ , θ , . . . , θ ) ∈ R d(N+1) , the log-likelihood of the unperturbed model can be written as (θ ) =˘ (θ [N+1] ). For the perturbed likelihood, we need an additional assumption of the extended version. Assumption 4.˘ is four times continuously differentiable. For all θ ∈ R d , there exist ε > 0, D > 0 and δ defined as in Assumption 2, such that for all 0 < η < δ and u 0: N ∈ R d(N+1) ,L (θ [N+1] + u 0:N ) ≤ De ε ∑ N n=1 |u n | η , whereL (θ 0:N ) = exp{˘ (θ 0:N )} is the perturbed likelihood. LetȆ θ ,τ,τ 0:N ,Cov θ ,τ,τ 0:N ,Var θ ,τ,τ 0:N denote the expectation, covariance and variance with respect to the associated posterior, pΘ 0:N |Y 1:N (θ 0:N |y * 1:N ; θ , τ, τ 0:N ). By usingȆ, Cov,Var instead ofȆ θ ,τ,τ 0:N ,Cov θ ,τ,τ 0:N ,Var θ ,τ,τ 0:N respectively, a theorem similar to theorem 4 of [8] but for random walk noise instead of independent white noise is derived. Theorem 1. [Theorem 2 of [30]] Suppose Assumptions 1, 2 and 4, there exists a constant C independent of τ, τ 1 , ...τ N such that, ∇ (θ ) − τ −2 Ψ −1 τ −2 0Ȇ Θ 0 − θ |Y 1:N = y * 1:N < Cτ 2 , where Ψ is the non-singular covariance matrix associated to ψ. Theorem 1 formally allows an approximation of ∇ (θ ). [30] also presents an alternative variations on these results which lead to more stable Monte Carlo estimation. Theorem 2. [Theorem 3 of [30]] Suppose Assumption 1, 2 and 4 hold. In addition, assume that τ n = O(τ 2 ) for all n = 1 . . . N, the following holds true, ∇ (θ ) − 1 N + 1 τ −2 τ −2 0 Ψ −1 N ∑ n=0 Ȇ Θ n − θ |Y 1:N = y * 1:N = O(τ 2 ). (2.3) These theorems are useful for our approaches because we can approximate the gradient of the log-likelihood of the extended model to the second order of τ which we will later show that it fits well with our accelerate simulation based setup. Proposed accelerate iterated filtering Our motivation comes from the accelerated gradient method for smooth non-linear stochastic programming literature. By using an approximation of the score function, it is possible to use an accelerated gradient method as in Nesterov acceleration scheme in optimization literature. One issue with the accelerated gradient approach is that it is not clear how the technique can be used in situations where both the likelihood and the gradient are intractable. These sorts of examples are common in scientific applications of state space models where the state process is a diffusion process or an ordinary differential equation (ODE) with stochastic coefficients. However, in these family of iterated filtering approaches, the score function can be approximated with noise under control without affecting the convergence rate. Specifically, applying an accelerated inexact gradient algorithm in the iterated filtering approach can obtain an optimal rate of convergence. In this paper, ε k denotes the error in the approximation of the gradient. Using the same notation as [11], denote the sequences of magnitudes of the errors in the gradient approximations { ε k }. Suppose the following assumptions: Assumption 5. The function : Θ → R is differentiable, bounded from above and has a L-Lipschitz-continuous gradient, i.e. for all θ , ϑ ∈ Θ, ∇ (θ ) − ∇ (ϑ ) ≤ L θ − ϑ , where ∇ denotes the gradient of . The function attains its maximum at a certain θ * ∈ Θ. In the sequel, Θ denotes a finite-dimensional Euclidean space with norm · and inner product ·, · . It can be shown that (e.g. in [27]) Assumption 5 is equivalent to | (ϑ ) − (θ ) − ∇ (θ ), ϑ − θ | ≤ L 2 ϑ − θ 2 , ∀θ , ϑ ∈ Θ (3.1) It is well-known that the gradient descent method converges for a general non-convex optimization problem but it does not achieve the optimal rate of convergence, in terms of the functional optimality gap, when (·) is convex [11]. In contrast, the accelerated gradient method in [28] is optimal for solving convex optimization problems, but does not necessarily converge for solving nonconvex optimization problems. [11] proposed a modified accelerated gradient method which can converge in both convex and non-convex optimization problem. However, they assumed unbiased estimation of the gradient which is not satisfied for most simulation-based inferences. Below, we extend the approach of Ghadimi to an accelerated inexact gradient (AIG) method in the context of accelerate iterated filtering. That is, we allow bias in gradient approximation by properly specifying the stepsize policy. We prove that it not only achieves the same optimal rate of convergence for both convex and non-convex optimizations, but also exhibits the best-known rate of convergence for simulation-based inference problems. Algorithm 1 Accelerate Inexact Gradient (AIG) Input: θ 0 ∈ Θ. {β k > 0}, {λ k > 0} for any k ≥ 2. {α k } ∈ (0, 1) for k > 1 and α 1 = 1. 1: θ ag 0 = θ 0 . Initialize 2: for k in 1...N do 3: θ md k = (1 − α k )θ ag k−1 + α k θ k−1 (3.2) 4: θ k = θ k−1 − λ k ∇ (θ md k ) (3.3) 5: θ ag k = θ md k−1 − β k ∇ (θ md k ) (3.4) where ∇ (θ md k ) is an estimation of ∇ (θ md k ) with error ε k . 6: end for In addition to Assumption 5, we assume a noise control condition for Algorithm 1. Assumption 6. Θ is bounded. There exists an A < ∞ such that ∑ N k=1 λ k ε k < A. Given some mild conditions often satisfied by controlling the artificial noises, we have the following result. Γ k := 1 k = 1 (1 − α k )Γ k−1 k ≥ 2 , (3.5) C k := 1 − Lλ k − L(λ k − β k ) 2 2λ k α k Γ k N ∑ τ=k 1 Γ τ > 0, for 1 ≤ k ≤ N, (3.6) then for any N ≥ 1, we have for some B < ∞, min k=1,...,N ∇ (θ md k ) + ε k 2 ≤ (θ 0 ) − * + B ∑ N k=1 λ k C k . (3.7) b) Suppose that (·) is convex. If sequences {α k } , {β k },{λ k } and {Γ k } satisfy α k λ k ≤ β k < 1 L , (3.8) α 1 λ 1 Γ 1 ≥ α 2 λ 2 Γ 2 ≥ . . . , (3.9) then for any N ≥ 1, we have min k=1,...,N ∇ (θ md k ) + ε k 2 ≤ 2 θ * −θ 0 2 2λ 1 + ∑ N k=1 Γ −1 k β k ε k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ 0 ∑ N k=1 Γ −1 k β k (1 − Lβ k ) , (3.10) (θ ag N ) − (θ * ) ≤ Γ N θ 0 − θ * 2 λ 1 + N ∑ k=1 Γ −1 k β k ε k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ 0 . (3.11) There are various options for selecting {α k } , {β k },{λ k },{Γ k }. By controlling error ε k , we can provide some of these selections below which guarantee the optimal convergence rate of the AIG algorithm for both convex and nonconvex problems. λ k ∈ β k , (1 + 1 k )β k , for ∀k ≥ 1, (3.12) then for any N ≥ 1, we have min k=1,...,N ∇ (θ md k ) + ε k 2 ≤ O 1 N . (3.13) Suppose that ε k = O τ 2 ≤ O( 1 k ) , then the AIG method can find a solutionθ such that ∇ (θ ) 2 ≤ ε in at most O(1/ε 2 ) iterations. b) Suppose that (·) is convex and ε k = O τ 2 ≤ O( 1 k 2+δ +δ 1 ) for some δ 1 > 0. If {λ k } satisfies λ k = k 1+δ − (k − 1) 1+δ ∀k ≥ 1, (3.14) then for any N ≥ 1, we have min k=1,...,N ∇ (θ md k ) + ε k 2 ≤O 1 N 2+δ , (3.15) (θ ag N ) − (θ * ) ≤ O 1 N 1+δ , (3.16) then the AIG method can find a solutionθ such that ∇ (θ ) 2 ≤ ε in O 1/ε 1 2+δ at most. Algorithm 2 Accelerate Iterated Filtering (AIF) Input: Starting parameter, θ 0 = θ ag 0 , sequences, α n , β n , λ n , Γ n simulator for f X 0 (x 0 |θ ), f Xn|X n−1 (x n |x n−1 |θ ), evaluator for f Yn|Xn (y n |x n |θ ) data, y * 1:N , labels designating IVPs, I ⊂ {1, . . . , p}, initial scale multiplier, C > 0 number of particles, J, number of iterations, M, cooling rate, 0 < a < 1, perturbation scales, σ 1:p Output: Maximum likelihood estimate θ MLE 1: θ md 0 = θ 0 Initialize 2: [Θ F 0, j ] i ∼ N [θ md 0 ] i , (Ca m−1 σ i ) 2 for i in 1..p, j in 1...J. Initialize filter mean for parameters 3: simulate X F 0, j ∼ f X 0 ·; Θ F 0, j for j in 1..J. Initialize states 4: for m in 1...M do 5: θ md m = (1 − α m )θ ag m−1 + α m 1 θ m−1 . 6: for n in 1...N do 7: Θ P n, j i ∼ N Θ F n−1, j i , (c m−1 σ i ) 2 for i / ∈ I, j in 1 : J. Perturb 8: X P n, j ∼ f n x n |X F n−1, j ; Θ P n, j for j in 1 : J. Simulate prediction particles 9: w(n, j) = g n (y * n |X P n, j ; Θ P n, j ) for j in 1 : J. Evaluate weights 10:w(n, j) = w(n, j)/ ∑ J u=1 w(n, u). Normalize weights 11: k 1:J with P {k u = j} =w (n, j). Apply systematic resampling to select indices 12: X F n, j = X P n,k j and Θ F n, j = Θ P n,k j for j in 1 : J. Resample particles 13: end for 14: S m = c −2(m−1) Ψ −1 ∑ N n=1 θ n − θ md m−1 Update Parameters 15: θ m i = θ m−1 − λ m−1 S m i for i / ∈ I. 16: θ ag m i = θ md m−1 − β m−1 S m i for i / ∈ I. 17: θ m i = 1 J ∑ J j=1 Θ F L, j i for i ∈ I. 18: end for We now add a few remarks about the extension results obtained in Theorem 4. First, if the problem is convex, by choosing more aggressive stepsizes {λ k } in (3.14), the AIG method exhibits the optimal rate of convergence in (3.16). It is also worth noting that with such a selection of {λ k }, the AIG method can find a solutionθ such that ∇ (θ ) 2 ≤ ε in at most O(1/ε 1/2+δ ) iterations. The latter result has been shown by [27], [11] but only for the accelerate unbiased gradient method. Second, observe that {λ k } in (3.12) for general nonconvex problems is in the order of O(1/L), while the one in (3.14) for convex problems are more aggressive (in O(k/L)). The value δ is optimal at 1 for convergence rate. However, it may not be optimal for computation of controlling the noises. Finally, we show that we can apply the stepsize policy in (3.12) for solving general inexact gradient problems for both convex and nonconvex optimization. The sequential Monte Carlo filter can be arbitrarily approximated to the exact filter by choosing sufficiently large number of particles [13]. It can be seen that we can choose the perturbation sequence so that the gradient noise satisfies condition in Theorem 4. For completeness, we present the pseudo code of the proposed algorithm as in Algorithm 2. Numerical examples To measure the performance of the new inference algorithm, we evaluate our accelerate iterated filtering on some benchmark examples and compare it to the existing simulation-based approaches. We make use of well tested and maintained code of R [33] packages such as pomp [17]. Specifically, models are coded using C snipet declarations [17]. New algorithm is written in R package is2, which provides user friendly interfaces in R and efficient matrix operations in the highly optimized Rcpp [9]. All the simulation-based approaches mentioned above use sequential Monte Carlo algorithm (SMC), implemented using bootstrap filter. Experiments were carried out on a cluster of 32 cores Intel Xeon E5-2680 2.7 Ghz with 256 GB memory. For a fair comparison, we try to use the same setup and assessment for every inference method. A public Github repository containing scripts for reproducing our results may be found at https://github.com/nxdao2000/AIFcomparisons. Toy example: A linear, Gaussian model In this subsection, we compare our accelerate iterated filtering algorithm to the original iterated filtering algorithm IF1 [14], Bayes map iterated filtering (IF2) [15] and the second-order iterated smoothing (IS2) [30]. It has been shown in [30] and [15] that the second-order iterated smoothing with white noise (IS1) [8] and particle Markov chain Monte Carlo (PMCMC) [1] do not perform as well as Bayes map iterated filtering so we leave them out. For a computationally convenient setting, simple models provide an opportunity to test the basic features of inference algorithms. Therefore, we first consider a bivariate discrete time Gaussian autoregressive process, a relatively simple mechanistic model. This model is chosen so that the Monte Carlo calculations can be verified using a Kalman filter. For this example, there are some alternatives to iterated filtering class. For example, EM and MCMC algorithms would be practical in this case although they do not scale well to large dynamic models, so we do not include them here. The model is given by the state space forms: X n |X n−1 = x n−1 ∼ N (αx n−1 , σ σ ), Y n |X n = x n ∼ N (x n , I 2 ) where α, σ are 2 × 2 matrices and I 2 is 2 × 2 identity matrix. The data are simulated from the following parameters: α = α 1 α 2 α 3 α 4 = 0.8 −0.5 0.3 0.9 , σ = 3 0 −0.5 2 . The number of time points N is set to 100 and initial starting point X 0 = (−3, 4). For each method mentioned above, we estimate parameters α 2 and α 3 for this model using J = 1000 particles and run our estimation for M = 25 iterations. We start the initial search uniformly on a large rectangular region [−1, 1] × [−1, 1]. As can be seen from Fig. 1, all of the distributions of estimated maximized log likelihoods touch the true MLE (computed from Kalman filter) at the vertical broken line, implying that they all successfully converged. The results show that AIF is the most efficient method of all because using AIF the results have higher mean and smaller variance compared to other approaches, indicating a higher empirical convergence rate. Algorithmically, AIF has similar computational costs with the first order approaches IF1, IF2, and is cheaper than the second order approach IS2. In deed, average computational time of twenty independent runs of each approach is given in Table 1. Additional overheads for estimating score make the computation time of AIF a bit larger compared to computational time of IF2. However, with complex models and large enough number of particles, these overheads become negligible and computational time of AIF will be similar to other first order approaches. The fact that it has the convergence rate of second order with computation complexity of first-order shows that it is a very promising algorithm. In addition, the results also imply that AIF is robust to initial starting guesses. To see how the final MLEs clustered around the true MLE, we only show 40 Monte Carlo replications for this toy example. As can be observed from Fig. 2, most of the replications clustered near the true MLE for AIF approach, while none of them stays in a lower likelihood region. It can be interpreted as a statistical summary of Fig. 2, with 200 Monte Carlo replications. These results indicate that AIF is clearly the best of the investigated methods for this test compared to others. Given additional computational resources, we also checked how the results of each method compared. Specifically, we set M = 100 iterations and J = 10000 particles, with the random walk standard deviation decreasing geometrically from 0.02 down to 0.0018 for each method. In this situation, we confirm that AIF is the best among other IF1, IF2 and IS2. All methods have comparable computational demands for given M and J. Malaria benchmark Many real world dynamic systems are highly nonlinear, partially observed and even weakly identifiable. To demonstrate the capabilities of accelerate iterated filtering for such situations, we apply it to evaluate the likelihood in a stochastic differential equation for vivax malaria model of Roy et al. [34]. The reason to choose this challenging model is that it provides a rigorous performance benchmark for our verification. The model SEIH 3 QS we consider splits up the study population of size P(t) into seven classes: susceptible individuals, S(t), exposure E(t), infected individuals, I(t), dormant classes H 1 (t), H 2 (t), H 3 (t) and recovered individuals, Q(t). This strain of malaria characterized by relapse following initial recovery from symptoms [30]. Therefore the the last S in the model name indicates the possibility that a recovered person can return to the class of susceptible individuals. The data, denoted by y * 1:N , are in the form of monthly time series over a 20-year period, counting the malaria morbidity. δ denotes the mortality rate, κ(t) a delay stage, µ SE (t) the current force of infection, and τ D the mean latency time. The state process is X(t) = S(t), E(t), I(t), Q(t), H 1 (t), H 2 (t), H 3 (t), κ(t), µ SE (t) , where transition rates from stage H 1 to H 2 , H 2 to H 3 and H 3 to Q are specified to be 3µ HI while infected population to dormancy transition rate is µ IH . The model satisfies the following stochastic differential equation system dS/dt = δ P + dP/dt + µ IS I + µ QS Q +aµ IH I + bµ EI E − µ SE (t)S − δ S, dE/dt = µ SE (t)S − µ EI E − δ E, dI/dt = (1 − b)µ EI E + 3µ HI H n − (µ IH + µ IS + µ IQ )I − δ I, dH 1 /dt = (1 − a)µ IH I − nµ HI H 1 − δ H 1 , dH i /dt = 3µ HI H i−1 − 3µ HI H i − δ H i for i ∈ {2, 3}, dQ/dt = µ IQ I − µ QS Q − δ Q. In addition, the malaria pathogen reproduction within the mosquito vector is given by dκ/dt = [λ (t) − κ(t)]/τ D , dµ SE /dt = [κ(t) − µ SE (t)]/τ D , where λ (t) is the latent force of infection and λ (t), κ(t) and µ SE (t) satisfies µ SE (t) = t −∞ γ(t − s)λ (s)ds, (4.1) with γ(s) = (2/τ D ) 2 s 2−1 (2−1)! exp(−2s/τ D ) , a gamma distribution with shape parameter 2. Since the latent force of infection is constrained by rainfall covariate R(t) and some Gamma white noise, from Roy et al. [34] we have: In this equation, q denotes a reduced infection risk from humans in the Q class and {s i (t), i = 1, . . . , N s } is a periodic cubic B-spline basis, with N s = 6. The observation model for Y n is a negative binomial distribution with mean M n and variance M n + M 2 n σ 2 obs where M n = ρ t n t n−1 [µ EI E(s) + 3µ HI H 3 (s)]ds is the number of new cases observed from time t n−1 to time t n and ρ it the mean age. The coupled system of stochastic differential equations is solved using an Euler-Maruyama scheme [18] with a time step of 1/20 month in our case. Given the data obtained from National Institutes of Malaria Research [34], we carried out simulation-based inference via the original iterated filtering (IF1), the perturbed Bayes map iterated filtering (IF2), the second order iterated smoothing (IS2), and the new accelerate iterated filtering (AIF). The inference goal used to assess all of these methods is to find high likelihood parameter values starting from randomly drawn values in a large hyperrectangle. In the presence of possible multi-modality, weak identifiability, and considerable Monte Carlo error of this model, we start 200 random searches. The random walk standard deviation is initially set to 0.1 for estimated parameters while the cooling rate c is set to 0.1 0.02 ≈ 0.95. These corresponding quantities for initial value parameters are 2 and 0.1 0.02 , respectively, but they are applied only at time zero. We run our experiment on a cluster computers with M = 50 iterations and with J = 1000 particles. The reason to choose these values for this model is that increasing the iterations to 100 and the number of particles to 10000 does not improve the results much but it takes significant longer time. Figure 4 shows the distribution of the MLEs estimated by IF1, IF2, IS2 and AIF. All distributions touch the global maximum as expected and the higher mean and smaller variance of IF2, AIF estimation clearly demonstrate that they are considerably more effective than IF1. Note that the computational times for IF1, IF2, IS2 and AIF are 44.86, 43.92, 53.10 and 52.25 minutes respectively, confirming that accelerate iterated filtering has essentially the same computational cost as first order methods IF1, IF2 and is cheaper a bit than IS2, for a given Monte Carlo sample size and number of iterations. In this hard problem, while IF1 reveal their limitations, we have shown that IF2 and AIF can still offer a substantial improvement. A natural heuristic idea to further improve the method is hybridizing IF2 and AIF but we leave it for the future work. λ (t) = I + qQ P × exp N s ∑ i=1 b i s i (t) + b r R(t) × dΓ(t) dt . Conclusion In this paper, we have proposed a novel class of iterated filtering theory using an accelerated inexact gradient approach. We have shown that choosing perturbation sequence and number of particles carefully results in an algorithm which has led to many advances including the statistical and computational efficiency. This is also very fruitful as it is extendable to a more generalized class of algorithm, based on proximal theory. Previous proof of iterated filtering class require some difficult conditions, which is not easily verifiable. However, in this article, we use only general standard gradient conditions. We are going further down the road of a more systematic approach which could be easily generalized to the state of the art algorithm in the optimization literatures. The convergence rate is also explicitly stated and it is better than standard theory. From a theoretical point of view, it could be an interesting perspective and insight. In addition, from practical point of view, we have provided an efficient framework, applicable to a general class of nonlinear, non-Gaussian non-standard POMP models, especially suitable in the control feedback system. There are a lot of such systems, which are not well-treated by current available modeling framework. We simultaneously present the performance of our open source software package is2 to facilitate the needs of the community. The performance of this new approaches surpass the other frameworks by a large margin of magnitude. It may be surprising that this simple accelerated inexact gradient approach has the needed convergence properties, and can easily be generalized, at least in some asymptotic sense. It is not hard to show that the accelerated inexact proximal gradient iterated filtering theory can be adapted to apply with iterated smoothing and with either independent white noise or random walk perturbations while our empirical results still show strong evidences of the improvements. In principle, different simulation-based inference methods can readily be hybridized to build on the strongest features of multiple algorithms. Our results could also be applied to develop other simulation-based methodologies which can take advantage of proximal map. For example, it may be possible to use our approach to help design efficient proposal distributions for particle Markov chain Monte Carlo algorithms. The theoretical and algorithmic innovations of this paper will help to build a new direction for future developments on this frontier. Applying this approach to methodologies like Approximate Bayesian Computation (ABC), Liu-West Particle Filter (LW-PF), Particle Markov chain Monte Carlo (PM-CMC), with different samplers scheme, e.g. forward backward particle filter, forward smoothing or forward backward smoothing are foreseeable extensions. Appendix A: Proofs We first need a simple technical result (see Lemma 1 of [11]). The proof is the same as that of Lemma 1 of [11] but we provide it here for completeness. Lemma 1. (Lemma1 of [11]. Assume sequences {α k } ∈ (0, 1) for k > 1 and α 1 = 1 and sequences {a k }, {η k } satisfy a k ≤ (1 − α k )a k−1 + η k , k = 1, 2, . . . (A.1) If we define a positive sequence {Γ k } as in 3.5 then for any k ≥ 1, we have a k ≤ Γ k k ∑ i=1 (η i /Γ i ). Proof. Since α 1 = 1 and Γ 1 = 1, from 3.5 we have a 1 ≤ η 1 or a 1 Γ 1 ≤ η 1 Γ 1 . Since Γ k > 0 for every k > 1, dividing both sides of A.1 by Γ k , a k Γ k ≤ (1 − α k )a k−1 + η k Γ k = a k−1 Γ k−1 + η k Γ k , ∀k ≥ 2. Summing up the above inequalities and rearranging the terms, the conclusion follows. Lemma 2. k ∑ τ=1 α τ Γ τ = 1 Γ k . (A.2) Proof. We have k ∑ τ=1 α τ Γ τ = α 1 Γ 1 + k ∑ τ=2 1 Γ τ (1 − (1 − α τ )) = 1 Γ 1 + k ∑ τ=2 ( 1 Γ τ − 1 Γ τ−1 ) = 1 Γ k . A.1. Proof of Theorem 3 Proof. The proof follows closely to the proof of theorem 1 of [11] except we consider bias estimate of the gradient. We first prove part a. By 3.1 and 3.3, we have (θ k ) ≤ (θ k−1 ) + ∇ (θ k−1 ), θ k − θ k−1 + L 2 θ k − θ k−1 2 = (θ k−1 ) + ∇ (θ k−1 ) − ∇ (θ md k ) − ε k + ∇ (θ md k ) + ε k , −λ k ∇ (θ md k ) + ε k + Lλ 2 k 2 ∇ (θ md k ) + ε k 2 = (θ k−1 )−λ k (1− Lλ k 2 ) ∇ (θ md k )+ε k 2 −λ k ∇ (θ k−1 ) − ∇ (θ md k ) − ε k , ∇ (θ md k ) + ε k ≤ (θ k−1 )−λ k (1− Lλ k 2 ) ∇ (θ md k )+ε k 2 +λ k ∇ (θ k−1 ) − ∇ (θ md k ) + ε k · ∇ (θ md k )+ε k , ≤ (θ k−1 )−λ k (1− Lλ k 2 ) ∇ (θ md k )+ε k 2 +λ k L θ k−1 − θ md k + ε k · ∇ (θ md k )+ε k , = (θ k−1 )−λ k (1− Lλ k 2 ) ∇ (θ md k )+ε k 2 +λ k L(1 − α k ) θ ag k−1 − θ k−1 + ε k · ∇ (θ md k )+ε k , = (θ k−1 ) − λ k 1 − Lλ k 2 ∇ (θ md k ) + ε k 2 +L(1 − α k )λ k ∇ (θ md k ) + ε k · θ ag k−1 − θ k−1 + λ k ε k · ∇ (θ md k ) + ε k ≤ (θ k−1 ) − λ k 1 − Lλ k 2 ∇ (θ md k ) + ε k 2 + Lλ 2 k 2 ∇ (θ md k ) + ε k 2 + L(1 − α k ) 2 2 θ ag k−1 − θ k−1 2 + λ k ε k · ∇ (θ md k ) + ε k = (θ k−1 ) − λ k (1 − Lλ k ) ∇ (θ md k ) + ε k 2 + L(1 − α k ) 2 2 θ ag k−1 − θ k−1 2 + λ k ε k · ∇ (θ md k ) + ε k (A.3) The second inequality is from triangular inequality and the Cauchy-Schwarz inequality while the second inequality is due to the Lipschitz of gradient assumption (1.2) and last equality comes from 3.2. We have the last inequality follows from ab ≤ (a 2 + b 2 )/2. θ ag k −θ k = (1−α k )θ ag k−1 +α k θ k−1 −β k ∇ (θ md k ) + ε k − θ k−1 − λ k ∇ (θ md k ) + ε k = (1 − α k )(θ ag k−1 − θ k−1 ) + (λ k − β k ) ∇ (θ md k ) + ε k . Applying Lemma 1 where θ ag k − θ k := a k and η k := (λ k − β k ) ∇ (θ md k ) + ε k , we obtain θ ag k − θ k = Γ k k ∑ τ=1 ( λ τ − β τ Γ τ ) ∇ (θ md τ ) + ε τ . Since · 2 is convex, using Jensen's inequality and Lemma 2 we have θ ag k − θ k 2 = Γ k k ∑ τ=1 ( λ τ − β τ Γ τ ) ∇ (θ md τ ) + ε k 2 = Γ k k ∑ τ=1 α τ Γ τ λ τ − β τ α τ ∇ (θ md τ ) + ε k 2 ≤ Γ k k ∑ τ=1 α τ Γ τ λ τ − β τ α τ ∇ (θ md τ ) + ε k 2 = Γ k k ∑ τ=1 (λ τ − β τ ) 2 Γ τ α τ ∇ (θ md τ ) + ε τ 2 . (A.4) Replacing the above bound in A.3, and the fact that Γ k = Γ k−1 (1 − α k ) as in 3.5 and that α k ∈ (0, 1] for all k ≥ 1we obtain (θ k ) ≤ (θ k−1 ) − λ k (1 − Lλ k ) ∇ (θ md k ) + ε k 2 + LΓ k−1 (1 − α k ) 2 2 k−1 ∑ τ=1 (λ τ − β τ ) 2 Γ τ α τ ∇ (θ md τ ) + ε τ 2 + λ k ε k · ∇ (θ md k ) + ε k ≤ (θ k−1 ) − λ k (1 − Lλ k ) ∇ (θ md k ) + ε k 2 + LΓ k 2 k ∑ τ=1 (λ τ − β τ ) 2 Γ τ α τ ∇ (θ md τ ) + ε τ 2 + λ k ε k · ∇ (θ md k ) + ε k (A.5) for every k ≥ 1. Using the definition of C k in 3.6 and summing up the above inequalities, we have (θ N ) ≤ (θ 0 ) − N ∑ k=1 λ k (1 − Lλ k ) ∇ (θ md k ) + ε k 2 + L 2 N ∑ k=1 Γ k k ∑ τ=1 (λ τ − β τ ) 2 Γ τ α τ ∇ (θ m τ ) + ε k 2 + N ∑ k=1 λ k ε k · ∇ (θ md k ) + ε k = (θ 0 ) − N ∑ k=1 λ k (1 − Lλ k ) ∇ (θ md k ) + ε k 2 + L 2 N ∑ k=1 (λ k − β k ) 2 Γ k α k ( N ∑ τ=k Γ τ ) ∇ (θ md k ) + ε k 2 + N ∑ k=1 λ k ε k · ∇ (θ md k ) + ε k = (θ 0 ) − ∑ λ k C k ∇ (θ md k ) + ε k 2 + N ∑ k=1 λ k ε k · ∇ (θ md k ) + ε k (A.6) Rearranging the terms in the above inequality N ∑ k=1 λ k C k ∇ (θ md k ) + ε k 2 ≤ (θ 0 ) − (θ * ) + N ∑ k=1 λ k ε k · ∇ (θ md k ) + ε k By assumption 4 that ∇ (·) and ∑ N k=1 λ k ε k are bounded. Since (θ N ) ≥ (θ * ) and in view of the assumption that C k > 0, we obtain for some constant B, min k=1,.N ∇ (θ md k ) + ε k 2 ≤ (θ 0 ) − (θ * ) + B ∑ N k=1 λ k C k which clearly implies 3.7. We now prove part b). First, from L-Lipschitz-continuous gradient property 3.4, we have (θ ag k ) ≤ (θ md k ) + ∇ (θ md k ), θ ag k − θ md k + L 2 θ ag k − θ md k 2 ≤ (θ md k ) − β k ∇ (θ md k ) + ε k 2 + ε k β k ∇ (θ md k ) + ε k + Lβ 2 k 2 ∇ (θ md k ) + ε k 2 . (A.7) By the assumption that (·) is convex and 3.2, (θ md k ) − (1 − α k ) (θ ag k−1 ) + α k (θ ) = α k (θ md k ) − (θ ) + (1 − α k ) (θ md k ) − (θ ag k−1 ) ≤ α k ∇ (θ md k ), θ md k − θ + (1 − α k ) ∇ (θ md k ), θ md k − θ ag k−1 = ∇ (θ md k ), α k (θ md k − θ ) + (1 − α k )(θ md k − θ ag k−1 ) = α k ∇ (θ md k ), θ k−1 − θ . (A.8) From 3.3, we have θ k − θ 2 = θ k−1 − λ k ∇ (θ md k ) − θ 2 = θ k−1 − θ 2 − 2λ k ∇ (θ md k ), θ k−1 − θ + λ 2 k ∇ (θ md k ) 2 , = θ k−1 − θ 2 − 2λ k ∇ (θ md k ) + ε k , θ k−1 − θ + λ 2 k ∇ (θ md k ) + ε k 2 , which implies α k ∇ (θ md k ) + ε k , θ k−1 − θ = α k 2λ k θ k−1 − θ 2 − θ k − θ 2 + α k λ k 2 ∇ (θ md k ) + ε k 2 . Hence we obtain α k ∇ (θ md k ), θ k−1 − θ ≤ α k 2λ k θ k−1 − θ 2 − θ k − θ 2 + α k λ k 2 ∇ (θ md k ) + ε k 2 + α k ε k θ k−1 − θ (A.9) Using the results of A.7, A.8, and A.9, we get (θ ag k ) ≤ (1 − α k ) (θ ag k−1 ) + α k (θ ) + α k 2λ k θ k−1 − θ 2 − θ k − θ 2 + α k ε k θ k−1 − θ −β k (1 − Lβ k 2 − α k λ k 2β k ) ∇ (θ md k ) + ε k 2 + ε k β k ∇ (θ md k ) + ε k ≤ (1 − α k ) (θ ag k−1 ) + α k (θ ) + α k 2λ k θ k−1 − θ 2 − θ k − θ 2 − β k 2 (1 − Lβ k ) ∇ (θ md k ) + ε k 2 + ε k β k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ , (A.10) where the last inequality follows from the assumption in 3.8. Subtracting (θ ) from both sides of the above inequality and using Lemma 1, we conclude that (θ ag N ) − (θ ) ≤ Γ N N ∑ k=1 α k 2λ k Γ k θ k−1 − θ 2 − θ k − θ 2 − N ∑ k=1 β k 2Γ k (1 − Lβ k ) ∇ (θ md k ) + ε k 2 + N ∑ k=1 1 Γ k ε k β k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ ≤ Γ N θ 0 − θ 2 2λ 1 − Γ N N ∑ k=1 β k 2Γ k (1 − Lβ k ) ∇ (θ md k ) + ε k 2 + Γ N N ∑ k=1 1 Γ k ε k β k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ (A.11) for every θ ∈ R n . By our contruction 3.9 that sequence α k λ k Γ k is decreasing and the fact that α 1 = Γ 1 = 1, we have N ∑ k=1 α k λ k Γ k θ k−1 − θ 2 − θ k − θ 2 ≤ α 1 θ 0 − θ 2 λ 1 Γ 1 = θ 0 − θ 2 λ 1 (A.12) which immediately implies the last inequality of A.11. Hence, we can conclude 3.11 from the above inequality and the assumption in 3.8: (θ ag N ) − (θ * ) ≤ Γ N θ 0 − θ * 2 λ 1 + N ∑ k=1 Γ −1 k ε k β k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ Finally, noting the fact that (θ ag N ) ≥ (θ * ), substitute θ := θ * , re-arranging the terms in A.11 we obtain N ∑ k=1 β k 2Γ k (1 − Lβ k ) ∇ (θ md k ) + ε k 2 k = 1, . . . , N ≤ θ * − θ 0 2 2λ 1 + N ∑ k=1 1 Γ k ε k β k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ , or min k=1,.N ∇ (θ md k )+ε k 2 ≤ 2 θ * −θ 0 2 2λ 1 + ∑ N k=1 1 Γ k ε k β k ∇ (θ md k ) + ε k + α k ε k θ k−1 − θ ∑ N k=1 Γ −1 k β k (1 − Lβ k ) which together with 3.8, clearly imply 3.10. A.2. Proof of Theorem 4 Proof. We first prove part a). Note that by choosing We also have 1 − α k = (k − 1) 1+δ k 1+δ (A.14) β k = 1 2L Γ k = 1 k 1+δ , for every k > 1, or α k = k 1+δ −(k−1) 1+δ k 1+δ = O( (1+δ )k δ k 1+δ ) = O( 1 k ). If we choose λ k such that λ k − β k = o(k −1 ) then (λ k − β k ) 2 2α k Γ k λ k ( N ∑ τ=k Γ τ ) = o(k −2 ) k −1 k −(1+δ ) 1 k δ = o(1) so for sufficiently large k we have C k = 1 − L[λ k + (λ k − β k ) 2 2α k Γ k λ k ( N ∑ τ=k Γ τ )] > 1 4 Hence, it can also be seen from 3.7 that for some positive bounded constant B 2 , min k=1,.N ∇ (θ md k ) + ε k 2 ≤ (θ 0 ) − * + B NB 2 = O( 1 N ), which concludes the first part of the proof. Since ε k = O τ 2 ≤ O( 1 k ), we have ∇ (θ md k ) converge to 0 at the rate of min O( 1 √ N ), O ( ε k ) = O( 1 √ N ), which gives us the desired result. We now show part b). Let λ k = k 1+δ − (k − 1) 1+δ c for some constant c then α 1 λ 1 Γ 1 = α 2 λ 2 Γ 2 = · · · = α k λ k Γ k . Observe that α k λ k = c 2 k 1+δ − (k − 1) 1+δ 2 k 1+δ = c 2 (1 + δ ) 2 O(k 2δ ) k 1+δ → 0 for δ < 1 so (1+δ ) 2 k 2δ k 1+δ < β k = 1 2L for sufficient large k, which implies that conditions 3.8 and 3.9 hold. Moreover, it can also be easily seen from A.3 that min k=1,.N ∇ (θ md k )+ε k 2 ≤ θ * −θ 0 2 2λ 1 +C ∑ N k=1 Γ −1 k ε k + O( 1 k ) ε k ∑ N k=1 Γ −1 k = O(N −2−δ ). The last equality is due to the fact that ∑ N k=1 Γ −1 k = ∑ N k=1 k (1+δ ) = O(N 2+δ ). Combining the above relation with 3.7, and since ε k = O τ 2 ≤ O( 1 k 2+δ +δ 1 ) for some δ 1 > 0, we have ∇ (θ md k ) converge to 0 at the rate of O 1 N 2+δ . Since α k λ k < β k = 1 2L , we have δ ≤ 1 which implies that the best convergence rate is O(N −3 ). pX 0 : 0N ,Y 1:N ,Θ 0:N (x 0:N , y 1:N ,θ 0:N ; θ , τ, τ 0:N ) = pΘ 0:N (θ 0:N ; θ , τ, τ 0:N )pX 0:N ,Y 1:N |Θ 0:N (x 0:N , y 1:N |θ 0:N ), where pX 0:N ,Y 1:N |Θ 0:N (x 0:N , y 1:N |θ 0:N ; θ , τ, τ 0:N (y n |x n ;θ n ). Theorem 3 . 3(Extension of Theorem 1 of [11]). Suppose Assumptions 5 and 6 hold. In addition, let {θ k , θ ag k } k ≥ 1 be computed by Algorithm 1. a) If sequences {α k } , {β k }, {λ k } and {Γ k } satisfy Theorem 4 . 4Suppose Assumptions 5 and 6 hold. In addition, suppose that {β k } in the accelerated gradient method are set to β k = 1 2L . a) If sequences {α k } and {λ k } satisfy FIG 1 . 1Comparison of estimators for the linear, Gaussian toy example, showing the densities of the MLEs estimated by the IF1, IF2, AIF and IS2 methods. The parameters α 2 and α 3 were estimated, started from 200 randomly uniform initial values over a large rectangular region [−1, 1] × [−1, 1]. FIG 2 .FIG 3 . 23Comparison of different estimators. The likelihood surface for the linear model, with the location of the MLE is marked with a green cross. The crosses show final points from 40 Monte Carlo replications of the estimators: (A) Original iterated filtering method; (B) Bayes map iterated filtering method; (C) Accelerate iterated filtering method; (D) Second-order iterated filtering method; Each method, was started uniformly over the rectangle shown, with M = 25 iterations, N = 1000 particles, and a random walk standard deviation decreasing from 0.02 geometrically to 0.011 for both α 2 and α 3 . Comparison of estimators for the linear, Gaussian toy example, showing the densities of the MLEs estimated by the IF1, IF2, AIF and IS2 methods using M = 100 iterations and J = 10000 particles. The parameters α 2 and α 3 were estimated, started from 200 randomly uniform initial values over a large rectangular region [−1, 1] × [−1, 1]. FIG 4 . 4The density of the maximized log likelihood approximations estimated by IF1, IF2, IS2, RIS1 and AIF for the malaria model when using J = 1000 and M = 50. 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In this paper, we introduce a graph structure, called non-zero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite.

10.1080/00927872.2015.1065866

1506.04905

4571e2b6a3fe702019f7179604d9b41fef24762b

Non-Zero Component Graph of a Finite Dimensional Vector Space 20 Apr 2016 Angsuman Das [email protected] Department of Mathematics St.Xavier's College KolkataIndia Non-Zero Component Graph of a Finite Dimensional Vector Space 20 Apr 2016arXiv:1506.04905v2 [math.GM]basisindependent setgraph 2000 MSC: 05C2505C69 In this paper, we introduce a graph structure, called non-zero component graph on finite dimensional vector spaces. We show that the graph is connected and find its domination number and independence number. We also study the inter-relationship between vector space isomorphisms and graph isomorphisms and it is shown that two graphs are isomorphic if and only if the corresponding vector spaces are so. Finally, we determine the degree of each vertex in case the base field is finite. Introduction The study of graph theory, apart from its combinatorial implications, also lends to characterization of various algebraic structures. The benefit of studying these graphs is that one may find some results about the algebraic structures and vice versa. There are three major problems in this area: (1) characterization of the resulting graphs, (2) characterization of the algebraic structures with isomorphic graphs, and (3) realization of the connections between the structures and the corresponding graphs. The first instance of such work is due to Beck [5] who introduced the idea of zero divisor graph of a commutative ring with unity. Though his key goal was to address the issue of colouring, this initiated the formal study of exposing the relationship between algebra and graph theory and at advancing applications of one to the other. Till then, a lot of research, e.g., [11,2,3,1,8,6,7,4] has been done in connecting graph structures to various algebraic objects. Recently, intersection graphs associated with subspaces of vector spaces were studied in [10,12]. However, as those works were a follow up of intersection graphs, the main linear algebraic flavour of characterizing the graph was missing. Throughout this paper, vector spaces are finite dimensional over a field F and n = dim F (V). In this paper, we define a graph structure on a finite dimensional vector space V over a field F, called Non-Zero Component Graph of V with respect to a basis {α 1 , α 2 , . . . , α n } of V, and study the algebraic characterization of isomorphic graphs and other related concepts. Definitions and Preliminaries In this section, for convenience of the reader and also for later use, we recall some definitions, notations and results concerning elementary graph theory. For undefined terms and concepts the reader is referred to [13]. By a graph G = (V, E), we mean a non-empty set V and a symmetric binary relation (possibly empty) E on V . The set V is called the set of vertices and E is called the set of edges of G. Two element u and v in V are said to be adjacent if (u, v) ∈ E. H = (W, F ) is called a subgraph of G if H itself is a graph and φ = W ⊆ V and F ⊆ E. If V is finite, the graph G is said to be finite, otherwise it is infinite. The open neighbourhood of a vertex v, denoted by N(v) , is the set of all vertices adjacent to v. A subset I of V is said to be independent if any two vertices in that subset are pairwise non-adjacent. The independence number of a graph is the maximum size of an independent set of vertices in G. A subset D of V is said to be dominating set if any vector in V \ D is adjacent to at least one vertex in D. The dominating number of G, denoted by γ(G) is the minimum size of a dominating set in G. A subset D of V is said to be a minimal dominating set if D is a dominating set and no proper subset of D is a dominating set. Two graphs G = (V, E) and G ′ = (V ′ , E ′ ) are said to be isomorphic if ∃ a bijection φ : V → V ′ such that (u, v) ∈ E iff (φ(u), φ(v)) ∈ E ′ . A path of length k in a graph is an alternating sequence of vertices and edges, v 0 , e 0 , v 1 , e 1 , v 2 , . . . , v k−1 , e k−1 , v k , where v i 's are distinct (except possibly the first and last vertices) and e i is the edge joining v i and v i+1 . We call this a path joining v 0 and v k . A graph is connected if for any pair of vertices u, v ∈ V, there exists a path joining u and v. The distance between two vertices u, v ∈ V, d(u, v) is defined as the length of the shortest path joining u and v, if it exists. Otherwise, d(u, v) is defined as ∞. The diameter of a graph is defined as diam(G) = max u,v∈V d(u, v), the largest distance between pairs of vertices of the graph, if it exists. Otherwise, diam(G) is defined as ∞. Non-Zero Component Graph of a Vector Space Let V be a vector space over a field F with {α 1 , α 2 , . . . , α n } as a basis and θ as the null vector. Then any vector a ∈ V can be expressed uniquely as a linear combination of the form a = a 1 α 1 + a 2 α 2 + · · · + a n α n . We denote this representation of a as its basic representation w.r.t. {α 1 , α 2 , . . . , α n }. We define a graph Γ(V α ) = (V, E) (or simply Γ(V)) with respect to {α 1 , α 2 , . . . , α n } as follows: V = V \ {θ} and for a, b ∈ V , a ∼ b or (a, b) ∈ E if a and b shares at least one α i with non-zero coefficient in their basic representation, i.e., there exists at least one α i along which both a and b have nonzero components. Unless otherwise mentioned, we take the basis on which the graph is constructed as {α 1 , α 2 , . . . , α n }. Theorem 3.1. Let V be a vector space over a field F. Let Γ(V α ) and Γ(V β ) be the graphs associated with V w.r.t two bases {α 1 , α 2 , . . . , α n } and {β 1 , β 2 , . . . , β n } of V. Then Γ(V α ) and Γ(V β ) are graph isomorphic. Proof: Since, {α 1 , α 2 , . . . , α n } and {β 1 , β 2 , . . . , β n } are two bases of V, ∃ a vector space isomorphism T : V → V such that T (α i ) = β i , ∀i ∈ {1, 2, . . . , n}. We show that the restriction of T on non-null vectors of V, T : Γ(V α ) → Γ(V β ) is a graph isomorphism. Clearly, T is a bijection. Now, let a = a 1 α 1 + a 2 α 2 + · · · + a n α n ; b = b 1 α 1 + b 2 α 2 + · · · + b n α n with a ∼ b in Γ(V α ). Then, ∃ i ∈ {1, 2, . . . , n} such that a i = 0, b i = 0. Also, T(a) = a 1 β 1 + a 2 β 2 + · · · + a n β n and T(b) = b 1 β 1 + b 2 β 2 + · · · + b n β n . Since, a i = 0, b i = 0, therefore T(a) ∼ T(b) in Γ(V β ). Similarly, it can be shown that if a and b are not adjacent in Γ(V α ), then T(a) and T(b) are not adjacent in Γ(V β ). Remark 3.1. The above theorem shows that the graph properties associated of Γ does not depend on the choice of the basis {α 1 , α 2 , . . . , α n }. However, two vertices may be adjacent with respect to one basis but non-adjacent to some other basis as shown in the following example: Let V = R 2 , F = R with two bases {α 1 = (1, 0), α 2 = (0, 1)} and {β 1 = (1, 1), β 2 = (−1, 1)}. Consider a = (1, 1) and b = (−1, 1). Clearly a ∼ b in Γ(V α ), but a ∼ b in Γ(V β ). Basic Properties of Γ(V) In this section, we investigate some of the basic properties like connectedness, completeness, independence number, domination number of Γ(V). , α i = α j . Consider c = α i + α j . Then, a ∼ c and b ∼ c and hence d(a, b) = 2. Thus, Γ is connected and diam(Γ) = 2. Theorem 4.2. Γ(V) is complete if and only if V is one-dimensional. Proof: Let Γ(V) be complete. If possible, let dim(V) > 1. Therefore, ∃ α 1 , α 2 ∈ V which is a basis or can be extended to a basis of V. Then α 1 and α 2 are two non-adjacent vertices in Γ(V), a contradiction. Therefore, dim(V) = 1. Conversely, let V be one-dimensional vector space generated by α. Then any two nonnull vectors a and b can be expressed as c 1 α and c 2 α respectively for non-zero c 1 , c 2 ∈ F and hence a ∼ b, thereby rendering the graph complete. Proof: The proof follows from the simple observation that α 1 + α 2 + · · · + α n is adjacent to all the vertices of Γ(V α ). . . , β l } is a minimal dominating set of Γ(V α ), then l ≤ n, i.e., the maximum cardinality of a minimal dominating set is n. Proof: Since D is a minimal dominating set, ∀i ∈ {1, 2, . . . , l}, D i = D \ {β i } is not a dominating set. Therefore, ∀i ∈ {1, 2, . . . , l}, ∃ γ i ∈ Γ(V α ) which is not adjacent to any element of D i but adjacent to β i . Since, γ i = θ, ∃ α t i such that γ i has non-zero component along α t i . Now, as γ i is not adjacent to any element of D i , so is α t i . Thus, ∀i ∈ {1, 2, . . . , l}, ∃ α t i such that α t i ∼ β i , but α t i ∼ β k , ∀k = i. Claim: i = j ⇒ α t i = α t j . Let, if possible, i = j but α t i = α t j . As β i ∼ α t i and α t i = α t j , therefore β i ∼ α t j . However, it contradicts that α t i ∼ β k , ∀k = i. Hence, the claim. As α t 1 , α t 2 , . . . , α t l are all distinct, it follows that l ≤ n. Theorem 4.5. The independence number of Γ is dim(V). Proof: Since {α 1 , α 2 , . . . , α n } is an independent set in Γ, the independence number of Γ ≥ n = dim(V). Now, we show that any independent set can not have more than n elements. Let, if possible, {β 1 , β 2 , . . . , β l } be an independent set in Γ, where l > n. Since, β i = θ, ∀i ∈ {1, 2, . . . , l}, β i has at least one non-zero component along some α t i , where t i ∈ {1, 2, . . . , n}. Claim: i = j ⇒ t i = t j . If ∃i = j with t i = t j = t(say), then β i and β j has non-zero component along α t . This imply that β i ∼ β j , a contradiction to the independence of β i and β j . Hence, the claim is valid. However, as there are exactly n distinct α i 's, l ≤ n, which is a contradiction. Thus, Γ = n = dim(V). Lemma 4.1. Let I be an independent set in Γ(V α ), then I is a linearly independent subset of V. Proof: Let I = {β 1 , β 2 , . . . , β k } be an independent set in Γ. By Theorem 4.5, k ≤ n. If possible, let I be linearly dependent in V. Then ∃ i ∈ {1, 2, . . . , k} such that β i can be expressed as a linear combination of β 1 , β 2 , . . . , β i−1 , β i+1 , . . . , β k , i.e., β i = c 1 β 1 + c 2 β 2 + · · · + c i−1 β i−1 + c i+1 β i+1 + · · · + c k β k = k s=1,s =i c s β s(1) Now, since {α 1 , α 2 , . . . , α n } is a basis of V, let β j = n t=1 d tj α t for j = 1, 2, . . . , i − 1, i + 1, . . . k. Thus, the expression of β i becomes Since, β i = θ. Thus, β i has a non-zero component along some α t * . Also, ∃ some β j , j = i such that β j has a non-zero component along α t * . (as otherwise, if all β j , j = i has zero component along α t * , then by Equation 1, β i has zero component along α t * , which is not the case.) β i = Thus, β j ∼ β i , a contradiction to the independence of I. Thus, I is a linearly independent set in V. Remark 4.2. Converse of Lemma 4.1 is not true, in general. Consider a vector space V, its basis {α 1 , α 2 , α 3 , . . . , α n } and the set L = {α 1 + α 2 , α 2 , α 3 , . . . , α n }. Clearly L is linearly independent in V, but it is not an independent set in Γ(V α ) as α 1 + α 2 ∼ α 2 . Non-Zero Component Graph and Graph Isomorphisms In this section, we study the inter-relationship between the isomorphism of two vector spaces with the isomorphism of the two corresponding graphs. It is proved that two vector spaces are isomorphic if and only if their graphs are isomorphic. However, it is noted that a vector space isomorphism is a graph isomorphism (ignoring the null vector θ), but a graph isomorphism may not be vector space isomorphism as shown in Example 5.1. Lemma 5.1. Let V and W be two finite dimensional vector spaces over a field F. If Γ(V α ) and Γ(W β ) are isomorphic as graphs with respect to some basis {α 1 , α 2 , . . . , α n } and {β 1 , β 2 , . . . , β k } of V and W respectively, then dim(V) = dim(W), i.e., n = k. Proof: Let ϕ : Γ(V α ) → Γ(W β ) be a graph isomorphism. Since, α 1 , α 2 , . . . , α n is an independent set in Γ(V α ), therefore ϕ(α 1 ), ϕ(α 2 ), . . . , ϕ(α n ) is an independent set in Γ(W β ). Now, as in Theorem 4.5 it has been shown that cardinality of an independent set is less than or equal to the dimension of the vector space, it follows that n ≤ k. Again, ϕ −1 : Γ(W β ) → Γ(V α ) is also a graph isomorphism. Then, by similar arguments, it follows that k ≤ n. Hence the lemma. Theorem 5.1. Let V and W be two finite dimensional vector spaces over a field F. If V and W are isomorphic as vector spaces, then for any basis {α 1 , α 2 , . . . , α n } and {β 1 , β 2 , . . . , β n } of V and W respectively, Γ(V α ) and Γ(W β ) are isomorphic as graphs. Proof: Let ϕ : V → W be a vector space isomorphism. Then {ϕ(α 1 ), ϕ(α 2 ), . . . , ϕ(α n )} is a basis of W. Consider the restriction ϕ of ϕ on the non-null vectors of V, i.e., ϕ : Γ(V α ) → Γ(W ϕ(α) ) given by ϕ(a 1 α 1 + a 2 α 2 + · · · + a n α n ) = a 1 ϕ(α 1 ) + a 2 ϕ(α 2 ) + · · · + a n ϕ(α n ) where a i ∈ F and (a 1 , a 2 , . . . , a n ) = (0, 0, . . . , 0). Clearly, ϕ is a bijection. Now, a ∼ b in Γ(V α ) ⇔ ∃ i such that a i = 0, b i = 0 ⇔ ϕ(a) ∼ ϕ(b). Therefore, Γ(V α ) and Γ(W ϕ(α) ) are graph isomorphic. Now, by Theorem 3.1, Γ(W ϕ(α) ) and Γ(W β ) are isomorphic as graphs. Thus, Γ(V α ) and Γ(W β ) are isomorphic as graphs. Theorem 5.2. Let V and W be two finite dimensional vector spaces over a field F. If for any basis {α 1 , α 2 , . . . , α n } and {β 1 , β 2 , . . . , β k } of V and W respectively, Γ(V α ) and Γ(W β ) are isomorphic as graphs, then V and W are isomorphic as vector spaces. Proof: Since Γ(V α ) and Γ(W β ) are isomorphic as graphs, by Lemma 5.1, n = k. Now, as V and W are finite dimensional vector spaces having same dimension over the same field F, V and W are isomorphic as vector spaces. Example 5.1. Consider an one-dimensional vector space V over Z 5 generated by α (say). Then Γ(V α ) is a complete graph of 4 vertices with V = {α, 2α, 3α, 4α}. Consider the map T : Γ(V α ) → Γ(V α ) given by T (α) = 2α, T (2α) = α, T (3α) = 4α, T (4α) = 3α. Clearly, T is a graph isomorphism, but as T (2α) = α = 4α = 2(2α) = 2T (α), T is not linear. Automorphisms of Non-Zero Component Graph In this section, we investigate the form of automorphisms of Γ(V α ). It is shown that an automorphism maps {α 1 , α 2 , . . . , α n } to a basis of V of a special type, namely non-zero scalar multiples of a permutation of α i 's. Theorem 6.1. Let ϕ : Γ(V α ) → Γ(V α ) be a graph automorphism. Then, ϕ maps {α 1 , α 2 , . . . , α n } to another basis {β 1 , β 2 , . . . , β n } such that there exists σ ∈ S n , where each β i is of the form c i α σ(i) and each c i 's are non-zero. Proof: Let ϕ : Γ(V α ) → Γ(V α ) be a graph automorphism. Since, {α 1 , α 2 , . . . , α n } is an independent set of vertices in Γ(V α ), therefore β i = ϕ(α i ) : i = 1, 2, . . . , n is also an independent set of vertices in Γ(V α ). Let ϕ(α 1 ) = β 1 = c 11 α 1 + c 12 α 2 + · · · + c 1n α n ϕ(α 2 ) = β 2 = c 21 α 1 + c 22 α 2 + · · · + c 2n α n · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ϕ(α n ) = β n = c n1 α 1 + c n2 α 2 + · · · + c nn α n Since, β 1 = θ i.e., β 1 is not an isolated vertex, ∃ j 1 ∈ {1, 2, . . . , n} such that c 1j 1 = 0. Therefore, c ij 1 = 0, ∀i = 1 (as β i is not adjacent to β 1 , ∀i = 1.) Similarly, for β 2 , ∃ j 2 ∈ {1, 2, . . . , n} such that c 2j 2 = 0 and c ij 2 = 0, ∀i = 2. Moreover, j 1 = j 2 as β 1 and β 2 are not adjacent. Continuing in this manner, for β n , ∃ j n ∈ {1, 2, . . . , n} such that c njn = 0 and c ijn = 0, ∀i = n and j 1 , j 2 , . . . , j n are all distinct numbers from {1, 2, . . . , n}. Thus, c kj l = 0 for k = l and c kj k = 0, where k, l ∈ {1, 2, . . . , n} and j 1 , j 2 , . . . , j n is a permutation on {1, 2, . . . , n}. Set σ = 1 2 · · · n j 1 j 2 · · · j n . Therefore, β i = c ij i α j i = c ij i α σ(i) , with c ij i = 0, ∀i ∈ {1, 2, . . . , n}. As {α 1 , α 2 , . . . , α n } is a basis, {β 1 , β 2 , . . . , β n } is also a basis and hence the theorem. Remark 6.1. Although ϕ maps the basis {α 1 , α 2 , . . . , α n } to another basis {β 1 , β 2 , . . . , β n }, it may not be a vector space isomorphism. It is because linearity of ϕ is not guaranteed as shown in Example 5.1. However, the following result is true. Theorem 6.2. Let ϕ be a graph automorphism, which maps α i → c ij i α σ(i) for some σ ∈ S n . Then, if c = 0, ϕ(cα i ) = dα σ(i) for some non-zero d. More generally, ∀k ∈ {1, 2, . . . , n} if c 1 · c 2 · · · c k = 0, then ϕ(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) = d 1 α σ(i 1 ) + d 2 α σ(i 2 ) + · · · + d k α σ(i k ) for some d i 's with d 1 · d 2 · · · d k = 0. Proof: Since, cα i ∼ α i , therefore ϕ(cα i ) ∼ ϕ(α i ) i.e., ϕ(cα i ) ∼ c ij i α σ(i) . Thus, ϕ(cα i ) has α σ(i) as a non-zero component. If possible, let ϕ(cα i ) has a non-zero component along some other α σ(j) for some j = i. Then ϕ(cα i ) ∼ α σ(j) i.e., ϕ(cα i ) ∼ ϕ(α j ), which in turn implies cα i ∼ α j for j = i, a contradiction. Therefore, ϕ(cα i ) = dα σ(i) for some non-zero d. For the general case, since c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ∼ α i 1 ⇒ ϕ(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) ∼ ϕ(α i 1 ) = cα σ(i 1 ) for some non-zero c ⇒ ϕ(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) has a non-zero component along α σ(i 1 ) ⇒ ϕ(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) ∼ α σ(i 1 ) Similarly, ϕ(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) ∼ α σ(i 2 ) , . . . , ϕ(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) ∼ α σ(i k ) Therefore, ϕ(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) = d 1 α σ(i 1 ) + d 2 α σ(i 2 ) + · · · + d k α σ(i k ) for some d i 's with d 1 · d 2 · · · d k = 0. Corollary 6.1. Γ(V α ) is not vertex transitive if dim(V) > 1. Proof: Since dim(V) ≥ 2, by Theorem 6.2, there does not exist any automorphism which maps α 1 to α 1 + α 2 . Hence, the result. The Case of Finite Fields In this section, we find the degree of each vertices of Γ(V) if the base field is finite. For more results, in the case of finite fields, please refer to [9]. Remark 7.1. The set of vertices adjacent to α i 1 +α i 2 +· · ·+α i k is same as the set of vertices adjacent to c 1 α i 1 +c 2 α i 2 +· · ·+c k α i k i.e., N(α i 1 +α i 2 +· · ·+α i k ) = N(c 1 α i 1 +c 2 α i 2 +· · ·+c k α i k ) for c 1 c 2 · · · c k = 0. Theorem 7.1. Let V be a vector space over a finite field F with q elements and Γ be its associated graph with respect to a basis {α 1 , α 2 , . . . , α n }. Then, the degree of the vertex c 1 α i 1 + c 2 α i 2 + · · · + c k α i k , where c 1 c 2 · · · c k = 0, is (q k − 1)q n−k − 1. Proof: The number of vertices with α i 1 as non-zero component is (q − 1)q n−1 (including α i 1 itself). Therefore, deg(α i 1 ) = (q − 1)q n−1 − 1. The number of vertices with α i 1 or α i 2 as non-zero component is equal to number of vertices with α i 1 as non-zero component + number of vertices with α i 2 as non-zero component − number of vertices with both α i 1 and α i 2 as non-zero component = (q − 1)q n−1 + (q − 1)q n−1 − (q − 1) 2 q n−2 = (q 2 − 1)q n−2 . As this count includes the vertex α i 1 + α i 2 , deg(α i 1 + α i 2 ) = (q 2 − 1)q n−2 − 1. Similarly, for finding the degree of α i 1 + α i 2 + α i 3 , the number of vertices with α i 1 or α i 2 or α i 3 as non-zero component is equal to [(q−1)q n−1 +(q−1)q n−1 +(q−1)q n−1 ]−[(q−1) 2 q n−2 +(q−1) 2 q n−2 +(q−1) 2 q n−2 ]+(q−1) 3 q n−3 = (q 3 − 1)q n−3 , and hence deg(α i 1 + α i 2 + α i 3 ) = (q 3 − 1)q n−3 − 1. Proceeding in this way, we get deg(α i 1 + α i 2 + · · · + α i k ) = (q k − 1)q n−k − 1. Now, from Remark 7.1, it follows that deg(c 1 α i 1 + c 2 α i 2 + · · · + c k α i k ) = (q k − 1)q n−k − 1. Conclusion In this paper, we represent a finite dimensional vector space as a graph and study various inter-relationships among Γ(V) as a graph and V as a vector space. The main goal of these discussions was to study the nature of the automorphisms and establish the equivalence between the corresponding graph and vector space isomorphisms. Apart from this, we also study basic properties of completeness, connectedness, domination and independence number. As a topic of further research, one can look into the structure of maximal cliques and chromatic number of such graphs. Theorem 4.1. Γ(V α ) is connected and diam(Γ) = 2. Proof: Let a, b ∈ V . If a and b are adjacent, then d(a, b) = 1. If a and b are not adjacent, since a, b = θ, ∃α i , α j which have non-zero coefficient in the basic representation of a and b respectively. Moreover, as a and b are not adjacent Theorem 4 . 3 . 43The domination number of Γ(V α ) is 1. 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In this paper we study the final fate of the universe in modified theories of gravity. As compared with general relativistic formulations, in these scenarios the Friedmann equation has additional terms which are relevant for low density epochs. We analyze the sort of future singularities to be found under the usual assumption the expanding Universe is solely filled with a pressureless component. We report our results using two schemes: one concerned with the behavior of curvature scalars, and a more refined one linked to observers. Some examples with a very solid theoretical motivation and some others with a more phenomenological nature are used for illustration.

10.1016/j.physletb.2008.10.061

0805.2284

bae2535a9468fc813d2706baf8e8f1f26878cd5d

Singular fate of the universe in modified theories of gravity 4 Nov 2008 (Dated: November 4, 2008) L Fernández-Jambrina Matemática Aplicada, E.T.S.I. Navales Universidad Politécnica de Madrid Arco de la Victoria s/nE-28040MadridSpain Ruth Lazkoz Física Teórica Facultad de Ciencia y Tecnología Universidad del País Vasco Apdo. 644E-48080BilbaoSpain Singular fate of the universe in modified theories of gravity 4 Nov 2008 (Dated: November 4, 2008)PACS numbers: 04.20.Dw, 98.80.Jk, 95.36.+x, 04.50.+h In this paper we study the final fate of the universe in modified theories of gravity. As compared with general relativistic formulations, in these scenarios the Friedmann equation has additional terms which are relevant for low density epochs. We analyze the sort of future singularities to be found under the usual assumption the expanding Universe is solely filled with a pressureless component. We report our results using two schemes: one concerned with the behavior of curvature scalars, and a more refined one linked to observers. Some examples with a very solid theoretical motivation and some others with a more phenomenological nature are used for illustration. I. INTRODUCTION Refined astronomical observations of luminosity distances derived from Type Ia supernovae provide reliable evidence of the current cosmic speed up of the Universe (see [1] for the pioneering results and [2,3] for the latest). In fact, such measurements are the only direct indication of that phenomenon (see for e.g. [4]), but at the same time they are complementary with other key observations such as those of the CMB spectrum and the global matter distribution. Explaining this surprising behavior in the large-scale evolution of the Universe represents a major theoretical problem in cosmology, and several approaches have been coined to try and provide a compelling answer to this riddle. The main stream approach is to consider the Universe is filled with an exotic fluid, known as dark energy [6,7,8], but then one also has to demand the cosmic soup (made of dark energy and the rest of components) has some Goldilocks properties to comply with the observations. Alternatively, the idea that cosmic acceleration might be due to modifications to general relativity has received considerable attention as well (see [9,10] for reviews and for specific modifications). In such frameworks models displaying cosmic acceleration could be devised with less fine-tuning and unnaturality as compared to general relativistic dark energy scenarios [11]. Speculations in the direction of modified gravity are, in principle, legitimate as there are no cosmological tests probing scales as large as the Hubble radius. We only have reasonable evidence of the validity of the gravitational inverse square law up to 300 Mpc (through the ISW effect) [12]. However, the Hubble radius is two orders of magnitude larger, so our large-scale tests on general relativity are a [email protected]; http://debin.etsin.upm.es/ilfj.htm b [email protected]; http://tp.lc.ehu.es/rls.html not stringent enough. The additional degrees of freedom of these various settings, as compared to the standard picture of cosmology prior to the revolution ignited in 1998, have given rise to a collection of new cosmological evolutions with bold features, future singularites being the most perplexing ones. In this respect, attempts to classify somehow the sort of future singularities to be expected in new devised cosmological evolutions are of interest. A popular classification route in the literature [13] relies exclusively on properties of the curvature tensor and scalar quantities derived from it. From that perspective, a number of new terms in cosmology, such as the celebrated "big rip" [16], have been coined to designate extremality events associated with blow-ups of scalars constructed from the curvature tensor, along with less popular ones like "quiescent singularities" [17], "sudden singularities" [18], "big brake" [19] or "big freeze" [20] (the number of names is larger than the actual name of different extremality events). Now, even though treatments of singularities in the fashion of [13] are of interest, there are subtle and most relevant properties inherent to cosmic evolution which can only be unveiled through the more sophisticated consideration of observers (see [14,15] for a detailed account). Indeed, curvature is a static concept, as it is only provides information of what happens at each event. Conversely, information retrieved from tracking the observers along their trajectories is more dynamical in nature, and therefore more enlightening if carefully analyzed. Interestingly, this scheme allows discussing whether the singularities encountered are weak or strong. Thus, if one's ultimate goal is to draw rigorous conclusions about the final fate in the Universe, both approaches are, in our view, complementary. In this paper we address the problem of future singularities in modified gravity cosmologies. We examine carefully the interrelation between the modifications and the singularities to be expected, and we try and give a unified vision by reporting our results using the scheme concerned with the behaviour of curvature scalars [13] and the one grounded on observers [14,15]. Ideally, modifications of general relativity should be derivable of a parent theory allowing for a covariant formulation of full-fledged field equations, otherwise, neither density perturbations nor solar system predictions could be computed. This is, actually, an aspect of the problem which does not affect our discussion, as we only work at the level of the Friedmann and energy conservation equations. Whenever the literature offers relevant examples for which the underlying theory is known, we will use them to illustrate our findings, but, occasionally, we will also resort to phenomenological examples. The plan of the paper is as follows. We propose a perturbative formulation of the Friedmann equations, for which two cases are distinguished depending on whether there is a critical energy density (which affects the form of the formulation). Then, we calculate the corresponding asymptotic expression of the scale factor, and bulding on earlier works we present our classification. We round up the dissertation with relevant examples and summarize in the last section. II. MODIFICATIONS OF FRIEDMANN EQUATION There have been many attempts to modify Einstein's theory of gravity from different points of view in order to cope with the observed acceleration of the expansion of the universe. One possibility arises from modifications to the Einstein-Hilbert action leading to the so called f (R) gravity theories (many aspect of this theoretical setup have been recently reviewed in [21]. The equations governing the large-scale geometry of the Universe in such settings are of fourth order in the metric approach, and, on top of that, for f (R) gravity theories to evade compatibility issues with observational tests complicated models are required [22] (see however [23] for a different perspective). Mild applications of Ockham's razor principle, combined in graceful cases with physical motivations, have lead to the consideration that contending modified gravity schemes could perhaps be more advisable. This is the case of the proposals originated by assuming the Universe is a 3-brane embedded in a higher dimensional bulk. Instead of grounding our discussion in specific theoretical frameworks, we propose a perturbative expression for the Friedmann equation of an expanding universe, which intends to comprise most of the models in the literature. With this aim in mind, we write a modified Friedmann equation in the form ȧ a 2 = H 2 = h 0 (ρ − ρ * ) ξ0 + h 1 (ρ − ρ * ) ξ1 + · · · . (1) Thus, we assume the squared Hubble factor can be expressed as a power series in the density ρ of the matter content of the universe around a specific value ρ * , for which a qualitative change of behavior is expected. The exponents ξ i are real and ordered, ξ 0 < ξ 1 < · · · . The coefficient h 0 is obviously positive. η0 η1 η2 Tipler Królak N.O.T. (−∞, 0) (η0, ∞) (η1, ∞) Strong Strong I 0 (0, 1) (η1, ∞) Weak Strong III 1 (1, 2) Weak Weak II [2, ∞) Weak Weak IV (1, 2) (η1, ∞) Weak Weak II [2, ∞) (η1, ∞) Weak Weak IV (0, ∞) (η0, ∞) (η1, ∞) Strong Strong Crunch The equation system is closed by assuming, in addition, the validity of the usual energy conservation equationρ + 3H(ρ + p) = 0.(2) The perturbative formulation represented by Eq. (1) can accommodate the Friedmann equations of the existing modified gravity proposals with a known parent covariant theory, as well as others with a phenomenological origin. Note, as well, that the Λ-cold dark matter (LCDM) or cosmic concordance scenario [24] is trivially comprised within this framework: H 2 = h 0 + h 1 (ρ − ρ * ), with ξ 0 = 0, ξ 1 = 1, h 1 = 8πG/3, and h 0 − h 1 ρ * = Λ/3, so, actually, the parameter ρ * is not fixed. The main purpose of the modifications is to provide an accelerated evolution of the Universe without resorting to an exotic fluid, so it is usually assumed the Universe is simply filled with cold dark matter (p = 0), and this will be our working hypothesis as well. In this case, the energy conservation equation (2) can be straightforwardly integrated:ρ ρ = −3ȧ a ⇒ ρa 3 = K,(3) which gives a one-to-one map between the energy density and the scale factor through the integration constant K. If we perform a power expansion of the scale factor in time, a(t) = c 0 |t − t 0 | η0 + c 1 |t − t 1 | η1 + · · · , where the exponents η i are real and ordered, η 0 < η 1 < · · · , following [15] we shall be able to classify the singularities encountered at a time t 0 . It is expected that most models allow this sort of expansion. However, there are models arising in loop quantum cosmology [25] which show accelerated oscillations that fall out of this scheme, though most of our conclusions can be extended to them. The classification of singularities in weak and strong follows the ideas of Ellis and Schmidt [26]: The curvature may be finite or infinite at one event, but what is physically relevant is whether free-falling (or even accelerated) observers meet the singularity in finite proper time [27]. It is clear that if they take infinite time in reaching the curvature singularity, this would be indetectable. Furthermore, if instead of ideal unextended observers we consider finite objects, the key issue is whether tidal forces at the singularity are strong enough to destroy them or weak, so that there could be objects that would survive beyond the singularity. This would mean that the weak singularity is by no means the end of the universe. Following this ideas, Tipler [28] modelled extended objects by three perpendicular vorticity-free Jacobi fields travelling along a causal geodesic and forming and orthonormal frame with the velocity u of this. If the geodesic hits a singularity in finite time and the volume spanned by a set of three three such vectors remains finite, the singularity is considered weak, since an object is not crushed. Tipler argues that in this case the metric could be generically extended beyond the singularity. Otherwise, if the volume is not finite for every set of vectors, the singularity is considered strong. Thinking of cosmic censorship conjectures, Królak [29] suggested and alternative definition of strong curvature singularities that just required diminishing, instead of vanishing, volume of the finite object and is therefore easier to comply. Compliance with these definitions for FLRW models can be checked resorting to integrals of the Ricci tensor along causal geodesics with respect to proper time τ [30]: If this integral diverges at a value τ 0 τ 0 dτ ′ τ ′ 0 dτ ′′ R ij u i u j ,(4) the causal geodesic meets a strong curvature singularity according to Tipler's definition. And for Królak's definition divergence of this other integral at τ 0 τ 0 dτ ′ R ij u i u j ,(5) means meeting a strong curvature singularity. The application of these results to FLRW models is summarized in Table I, which is a simplified version of the one in [14]. The last columns refers to the classification of future singularities in [13]): • Type I: "Big Rip": divergent a, ρ, p. • Type II: "Sudden": finite a, ρ, divergent p. • Type III: "Big Freeze": finite a, divergent ρ, p. • Type IV: "Big Brake": finite a, ρ, p, but divergent higher derivatives. Even though the modifications will only lead to acceleration for certain values of the parameter ξ 0 , our forthcoming discussion on the asymptotic behavior of the scale factor is valid for any value of ξ 0 , and it only relies on the ordering of the exponents. For this reason, our scheme comprises as well modifications to gravity which are not able to explaining the current acceleration, such as, for instance, the non-self-accelerating branch of the DGP modification scenario. Inserting the modified Friedmann equation into the conservation equation one gets: ρ ρ = −3 h 0 (ρ − ρ * ) ξ0/2 − − 3 2 h 1 √ h 0 (ρ − ρ * ) ξ1−ξ0/2 + · · · .(6) Our purpose it to integrate the latter by considering all the possibilities which arise from different values of the parameters, and then use the aforementioned map between the energy density and the scale factor so that we can finally obtain asymptotic expressions for the expansionary behaviour of the models. Then, we will identify the specific late-time behaviour of the models, focusing on the existence of future singularities of various types. This classification resorts to earlier works by ourselves. A separate treatment of the cases ρ * = 0 and ρ * = 0 cases is required, so we split the discussion into two subsections. A. Absent critical density In the case of a theory with no critical density, i.e. density ρ * = 0, expressions get considerably simplified: ρ ρ ≃ −3 h 0 ρ ξ0/2 , ρ(t) ≃          3 2 ξ 0 h 0 (t − t 0 ) −2/ξ0 for ξ 0 = 0, e −3 √ h0(t−t0) for ξ 0 = 0. Correspondingly, in terms of the expansion factor we get a(t) ≃ 3 √ K 3 2 ξ 0 h 0 (t − t 0 ) 2/3ξ0 , which provides the following expected results: • ξ 0 < 0: As matter density decreases smoothly, an eventual blow up of the corrections to the Friedmann equation is approached. At a finite time t 0 the scale factor becomes infinite, and the Universe experiences a type of singularity which has been called "big rip" [16] (type I in the classification in [13]). • ξ 0 > 0: The matter density decreases and the scale factor increases smoothly as t grows towards infinity. This case comprises both quintessence-like behaviors for ξ 0 ∈ (0, 2/3), and non-accelerated evolutions for ξ 0 ≥ 2/3. • ξ 0 = 0: The lowest order term is that of a cosmological constant, and we have to resort to the first correction with a positive exponent ξ 1 , which leads again to an expression solvable as a Bernouilli equation: ρ ρ ≃ −3 h 0 − 3 2 h 1 √ h 0 ρ ξ1 (8) ρ(t) ≃ e 3ξ1 √ h0(t−t0) − 1 2 h 1 h 0 −1/ξ1 . In this case a(t) ≃ 3 √ K e 3ξ1 √ h0(t−t0) − 1 2 h 1 h 0 1/3ξ1 ,(9) so, this situation represents an exponential expansion of the Universe, with a corresponding exponential decrease of matter density, with no future singularity at all. Therefore, in the case ρ * = 0, the modifications considered do not produce a qualitative change of behavior towards the future, except for dramatic modifications produced by negative exponents, which lead to a "big rip" singularity in the future. We close this subsection with several examples which fit in ρ * = 0 case of the general perturbative expression of H 2 we started from, namely they satisfy H 2 = h 0 ρ ξ0 + h 1 ρ ξ1 + . . . .(10) These results are summarized in Table II. The first case we consider for illustration is that of the power-law Cardassian models [31], for which H 2 = 8 3 Gπρ 1 + ρ ρ card n−1 .(11) This expression can be accommodated into (10) with the following identifications between our parameters and those of the original reference: ξ 0 = 1, ξ 1 = n < 2/3, h 0 = 8πG/3, h 1 = (8πG/3)ρ 1−n card > 0. The constant ρ card signals the amount of matter energy density ρ below which the Cardassian corrections start to dominate (ρ card ∼ ρ) DGP cosmologies [32,33,34,35,36] provide another interesting set of examples. If the brane has no tension and the bulk is the Minkowski spacetime, one has [34,37] H H 0 = Ω rc + 8Gπρ 3H 2 0 ± Ω rc .(12) Here H 0 is the value of the Hubble factor today, and Ω rc is the present value of the fractional energy density associated with the scale at which the crossover to a corrections dominated regime occurs. In the perturbative formulation required for the discussion we get H 2 = Ω rc (1 ± 1) 2 H 2 0 + 8Gπρ 3 (1 ± 1) ∓ 16G 2 π 2 ρ 2 9H 0 2 Ω rc + . . . .(13) The self-accelerating branch [33,34] arises by taking the upper signs, and it is characterized by ξ 0 = 0 and ξ 1 = 1, whereas for the so called normal branch [34,35], which arises by taking the lowers signs, one has ξ 0 = 2. Finally, we can bring about the reinterpretation of Chaplygin-like cosmic evolutions as a modified gravity proposal [39,40]. In these frameworks one has H 2 = 8πG 3 A + ρ γ(α+1) 1 α+1 .(14) with α > 0 and A = (3H 2 0 /(8πG)) (1+α) 1 − Ω 1+α m , and γ = 1 in the models considered in [39], whereas in the more general framework of [40] γ is free. Two cases are to be distinguished. In the γ > 0 case, the correspondence up to order γ(1 + α) in ρ is given by h 0 = (8Gπ/3)A 1/(1+α) , ξ 0 = 0, h 1 = (8πG/(3(1 + α))A −α/(1+α) , ξ 1 = γ(α + 1). But in the γ < 0 the correspondence is rather different, as the identification up to order γ is given by h 0 = 8Gπ/3 and ξ 0 = γ. In all the examples but the last one, according to the discussion above, no singular fate of the universe is faced. On the contrary, in the last kind of models the singularity is of "big rip" type. B. Non-trivial critical density New features appear for general modifications endowed with a non-trivial critical density ρ * . For this case, we assume the matter density has an expansion around the critical value ρ * = ρ(t 0 ) at a time t 0 : (15) and from the latter we obtain ρ(t) = ρ * + ρ 1 (t 0 − t)η 1 + ρ 2 (t 0 − t)η 2 + · · · ,a(t) = 3 √ K ρ 1/3 * 1 − ρ 1 3ρ * (t 0 − t)η 1 + · · · ,(16) so that the first exponents are the same in both expansions, η 0 = 1, η 1 =η 1 , . . . and we may drop the tildes. At lowest order we have, ρ ρ ≃ −3 h 0 (ρ − ρ * ) ξ0/2 = −3 h 0 {ρ 1 (t 0 − t) η1 } ξ0/2 , which upon the requirement of compatibility with Eq. (6) fixes the first exponent as η 1 = 2 2 − ξ 0 .(17) The following three cases are to be distinguished: • ξ 0 < 0: Since 0 < η 1 < 1, according to [15] or Table I, these models have a singularity at t 0 with divergent H (a "big freeze" or singularity type III [13]), which is a weak curvature singularity according to Tipler [28], but strong according to Królak [29]. • ξ 0 ∈ (0, 2): In this case η 1 > 1, so these models could show a weak singularity at t 0 according to [15] or Table I (sudden singularity [18] or type II in [13], or even type IV if η 1 ≥ 2, ξ 0 ≥ 1). • ξ 0 = 0: The cosmological constant term is dominant against modifications of the Friedmann equation. At first order, we havė ρ ρ ≃ − η 1 ρ 1 ρ * (t 0 − t) η1−1 = −3 h 0 , that is, we find a linear behavior for matter density: η 1 = 1, ρ 1 = 3ρ * h 0 . This being so, it turns out we have to expand the equation a bit further in order to reveal new qualitative behavior: ρ ρ ≃ −3 h 0 − η 2 ρ 2 ρ * h 0 (t 0 − t) η2−1 + . . . = −3 h 0 − 3 2 h 1 √ h 0 3 h 0 ρ * (t 0 − t) ξ1 + · · · = −3 h 0 − 3 2 h 1 √ h 0 (ρ − ρ * ) ξ1 + · · · . Necessarily, η 2 = 1 + ξ 1 , ρ 2 = 1 2η 2 h 1 h 0 3 h 0 ρ * η2 ,(18) and therefore, according to [15] or Table I, a(t) = 3 √ K ρ 1/3 * 1 − ρ 1 3ρ * (t 0 − t) − ρ 2 3ρ * (t 0 − t) η2 + · · · } ,(19) there is a singularity at t 0 due to the lack of smoothness of the density and the scale factor. But this singularity is weak in both Tipler's [28] and Królak' [29] classification, so it does not exert any infinite distortion on finite objects going through it and cannot, therefore, be considered as a final stage of the Universe. It is a sudden singularity or type II in [13]) for which the scale factor and the density remain finite, butḢ blows up. It is worthwhile mentioning that milder singularities for which H and alsoḢ are finite (type IV in [13]) could, in principle, appear within this framework, but they would involve choosing η 2 ≥ 2, and thereby ξ 1 ≥ 1, so that the linear term in the density in Friedmann equation would be absent. Obviously models with analytical expansion, that is, natural exponents ξ 0 , ξ 1 ,. . . (such as LCDM, for instance) do not show future singularities, neither weak nor strong. These results are summarized in Table III. The normal branch of DGP cosmologies provide a relevant example for this section. If the bulk on which the brane lives is an anti-de Sitter spacetime one has H H 0 = 8Gπρ − |Λ b | 3H 2 0 + Ω rc − Ω rc .(20) The identification with our perturbed formulation is given by ρ * = (|Λ b | − 3H 2 0 Ω rc )/(8πG), h 0 = Ω rc H 2 0 , ξ 0 = 0, h 1 = −4H 0 2πGΩ rc /( √ 3) and ξ 1 = 1/2. This singularity is a sudden one, also referred to as quiescent [38], or using our terminology, it is a weak extremality event. A slight variation leading to a singularity of the same sort consists in letting the brane have a negative brane tension σ. In this case the bulk can either be the Minkowski or the anti-de Sitter spacetime. The above expression can be adapted to this variation by simply letting ρ → ρ + σ and ρ * → ρ * + σ. Finally, we may consider models arising in loop quantum cosmology as those in [41], for which H 2 = 8πG 3 ρ 1 − ρ ρ * , but for these models the critical density is relevant for the high density regime, imposing a maximum density which is reached as the energy density grows. As this is the opposite of our working hypothesis (remember we demandedρ < 0) these models do not quite fit in our description here, but could be treated in an analogous way, with the corresponding adjustements. III. DISCUSSION We here put forward a detailed classification of the future behavior of FRW cosmologies in modified gravity proposals. Departures from the standard description of the expansion of the Universe according to Einstein's theory have been considered of interest, as they could provide an explanation of what is the agent responsible for the accelerated expansion of the Universe. The main question we pose is what are the characteristics of the modifications in connection with the presence of a singular future behavior of the Universe. As we have reflected here, not all the relevant properties of cosmic evolution emerge by considering curvature scalars, and the deeper insight provided by the consideration of observers is needed. The spirit of the modified gravity proposals we consider is to assume the Universe is simply filled with cosmic dust, and no blueshifting component whatsoever is considered (unlike when one assumes the current cosmic acceleration is due to an exotic fluid or dark energy). Our starting point is a perturbative low-energy or infrared expansion of the modified Friedmann equation. Two classes emerge: those with a critical energy density and those without it. We find one has to consider at most the exponents of the first two terms of the expansion in order to differentiate the possible behaviors, and, more importantly, whether the future singularity, if it exists, is weak or strong. The scheme we propose provides an easy route to conclude the sort of singular behavior present in potential new candidates to explain the current acceleration in the universe in terms of a modification of gravity. 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The auxiliary field method is a technique to obtain approximate closed formulae for the solutions of both nonrelativistic and semirelativistic eigenequations in quantum mechanics. For a many-body Hamiltonian describing identical particles, it is shown that the approximate eigenvalues can be written as the sum of the kinetic operator evaluated at a mean momentum p 0 and of the potential energy computed at a mean distance r 0 . The quantities p 0 and r 0 are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as well as its connection with the perturbation theory. For a nonrelativistic kinematics, general results are obtained for the structure of critical coupling constants for potentials with a finite number of bound states.

10.4303/jpm/p111101

1106.6123

f00206c600a062c93de1f75422e818b95aabb875

Further developments for the auxiliary field method 7 Nov 2011 Claude Semay [email protected] Service de Physique Nucléaire et Subnucléaire Université de Mons -UMONS 20 Place du Parc7000MonsBelgium Fabien Buisseret [email protected] Service de Physique Nucléaire et Subnucléaire Université de Mons -UMONS 20 Place du Parc7000MonsBelgium HauteÉcole Louvain en Hainaut (HELHa) Chaussée de Binche 1597000MonsBelgium Bernard Silvestre-Brac LPSC Université Joseph Fourier Grenoble 1 CNRS/IN2P3 Institut Polytechnique de Grenoble Avenue des Martyrs 5338026Grenoble-CedexFrance Further developments for the auxiliary field method 7 Nov 2011PACS numbers: 0365Ge, 0365Pm The auxiliary field method is a technique to obtain approximate closed formulae for the solutions of both nonrelativistic and semirelativistic eigenequations in quantum mechanics. For a many-body Hamiltonian describing identical particles, it is shown that the approximate eigenvalues can be written as the sum of the kinetic operator evaluated at a mean momentum p 0 and of the potential energy computed at a mean distance r 0 . The quantities p 0 and r 0 are linked by a simple relation depending on the quantum numbers of the state considered and are determined by an equation which is linked to the generalized virial theorem. The (anti)variational character of the method is discussed, as well as its connection with the perturbation theory. For a nonrelativistic kinematics, general results are obtained for the structure of critical coupling constants for potentials with a finite number of bound states. Introduction The auxiliary field method (AFM) is a very powerful method to obtain approximate analytical expressions for the eigenvalues of one, two and many-body systems with both nonrelativistic or semirelativistic kinematics. It has been shown in a series of paper [1,2,3,4,5,6,7,8] that it can be applied with great success in many physical situations. The basic idea is to replace a problem which is not solvable, for example because of a complicated potential or a semirelativistic kinematics, by another one which can be treated analytically. In so doing, it is necessary to introduce auxiliary fieldsν k . The original Hamiltonian H is replaced by a new Hamiltoniañ H(ν k ), called the AFM Hamiltonian. If these auxiliary fields are chosen asν k (0) in order to extremize the AFM Hamiltonian, this one coincides with the original Hamiltonian:H(ν k (0)) = H. Thus, both formulations are completely equivalent. The approximation lies in the fact that the auxiliary fields are considered no longer as operators, but as a real constants ν k . An approximate value of the exact eigenenergy E is then given by an extremal eigenenergy E(ν k (0)) of the AFM Hamiltoniañ H(ν k ), which is in principle much simpler than H. An approximate state for the corresponding eigenvalue can also be obtained. The quality of this approximation has been studied and discussed in detail in the papers mentioned above. Among the interesting properties of the AFM, we can mention its great simplicity and its ability to treat on equal footing the ground state and the various excited states. This procedure was first introduced to get rid off the square root kinetic operator in calculations for semirelativistic eigenvalue equations [9,10]. As the AFM is an extension of these first calculations, we just keep the same name for the method. As it is shown in [11], the AFM has strong connections with the envelope theory [12,13,14,15]. Nevertheless both methods have been introduced from completely different starting points. In particular, the AFM introduces the notion of auxiliary fields which is a key ingredient to interpret the method as a mean field approximation and which can be very useful to compute mean values of observables [1,6]. In this work, we present some new and general properties of the AFM. So, only the basic ingredients necessary for the understanding of the subject treated here are recalled in Sect. 2. We refer the reader to our works mentioned above or to our review paper [16] for an exhaustive overview of the method and its applications. New results or generalizations are presented in the following sections. The connection of the AFM with the generalized virial theorem is presented in Sect. 3. The use of the perturbation theory for the AFM is explained in Sect. 4. For a nonrelativistic kinematics, the general structures of critical coupling constants for potentials with a finite number of bound states are presented in Sect. 5. A summary of the results is given in Sect. 6. The auxiliary field method Let us consider a system composed of N particles, interacting via one-body potentials U i and two-body potentials V ij , and moving with a nonrelativistic or a semirelativistic kinetic energy. In principle, the AFM can treat this problem in such a general form, but it is manageable in practice only if the particles are identical. This implies that they all have the same mass m, that the form of the one-body potentials is the same for all particles U i ≡ U , and that the form of the two-body potentials is the same for all pairs of particles V ij ≡ V . So, the most general Hamiltonian we will consider in this paper has the following form H = N i=1 p 2 i + m 2 + N i=1 U (|s i |) + N i<j=1 V (|r ij |),(1) with s i = r i − R and r ij = r i − r j . r i is the position of the particle i, p i is its conjugate momentum and R is the position of the center of mass of the N particles (N ≥ 2). It is assumed that N i=1 p i = 0. Following the AFM, this Hamiltonian is "replaced" by the auxiliary HamiltonianH, with auxiliary potentials P (x) and S(x), and which depends on 3 auxiliary fields µ, ν and ρ H(µ, ν, ρ) = B(µ, ν, ρ) + N i=1 p 2 i 2µ + ν N i=1 P (|s i |) + ρ N i<j=1 S(|r ij |),(2) provided the states considered are completely symmetrized. Most of the results presented in this section come from [6], but we use here the more convenient notations developed in [8,16]. The function B(µ, ν, ρ) is not useful to detail in this work, but it can be rebuilt from results in [6]. If this Hamiltonian is analytically solvable, an AFM analytical approximation of a mass M of the N -body Hamiltonian H is given by an eigenvalue M 0 of the HamiltonianH(µ 0 , ν 0 , ρ 0 ) where the 3 optimal auxiliary parameters µ 0 , ν 0 and ρ 0 extremize this eigenvalue. These parameters depend on the quantum numbers of the state. At this stage, it is not obvious that the solution M 0 is a good one. But, the comparison theorem of the quantum mechanics can be used to obtain significant information about the AFM eigenvalues. This theorem states that, for some eigenvalue equations, if two Hamiltonians are ordered, H (1) ≤ H (2) ( H (1) ≤ H (2) for any state), then each corresponding pair of eigenvalues is ordered E (1) {θ} ≤ E (2) {θ} , where {θ} represents a set of quantum numbers. This inequality can be obtained from the Ritz variational principle [17], but it can also be derived from the Hellmann-Feynman theorem [18]. If we can show that the auxiliary HamiltonianH(µ 0 , ν 0 , ρ 0 ) is greater or lower than the genuine Hamiltonian H, then it is possible to use the comparison theorem to locate the AFM eigenvalues with respect to the exact ones. In the case of a nonrelativistic kinematics, The AFM yield an upper (lower) bound if the potentials U (x) and V (x) could be bounded from above (below) by the auxiliary potentials P (x) and S(x) respectively [6]. For a semirelativistic kinematics, the AFM implies a replacement of the square root operators by a nonrelativistic form of the kinetic energy (see (2)) and this yields to an increase of the eigenvalues [5,18]. So, in this case, the AFM solutions are upper bounds of the exact ones if the potentials U (x) and V (x) can be bounded from above by the auxiliary potentials P (x) and S(x) respectively. In other cases, nothing can be said about the possible variational character of the solutions. Let us note that a lower bound for the ground state (and then for the whole spectrum) of the general Hamiltonian (1) for a boson-like system ‡ has been proposed ‡ A boson-like system is composed of particles whose total spatial wavefunction can be completely symmetrical. For instance, this is the case for a system of quarks inside a baryon: quarks are fermions, but the baryon is characterized by a completely antisymmetrical colour function so that the rest of the total wavefunction must be completely symmetrical. Similarly, a fermion-like system is composed of particles whose total spatial wavefunction can be completely antisymmetrical. in [19]. It takes the following general form (the one-body potential is introduced using the fact that |s 1 | = |s 2 | = |r 12 |/2 for N = 2) M ≥ N inf φ φ p 2 + m 2 + U 1 2 |r| + N − 1 2 V (|r|) φ ,(3) but also works for nonrelativistic kinematics [20]. In this latter case, using the AFM results with N = 2, a lower bound for the mean value in (3) can be computed provided the potentials U (x) and V (x) could be bounded from below by the auxiliary potentials (see Sect. 2.2). The case N ≥ 2 For arbitrary values of N , the Hamiltonian (2) is entirely analytically solvable for the unique choice P (x) and/or S(x) equal to x 2 . It can then be shown [6] that the non linear system determining the 3 variables (µ 0 , ν 0 , ρ 0 ) can be recast in the form of one transcendental equation depending on the single variable X 0 = 2µ 0 (ν 0 + N ρ 0 ). Moreover, an eigenmass can be computed from the X 0 quantity only. Thus, the eigenvalue problem for the N -body system can be determined simply by the set of the following two equations: [6] M 0 = N m 2 + Q N X 0 + N U Q N X 0 + C N V 2Q (N − 1)X 0 ,(4)X 2 0 = 2 m 2 + Q N X 0 K Q N X 0 + N L 2Q (N − 1)X 0 ,(5) where Q is a global quantum number (see below), where K(x) = U ′ (x)/P ′ (x) = U ′ (x)/(2x) and L(x) = V ′ (x)/S ′ (x) = V ′ (x)/(2x) , and where the number of pairs C N = N (N − 1) 2 (6) has been introduced for convenience. The prime denotes the derivative with respect to the argument. In this framework, an approximate AFM eigenstate is given by an eigenstate ofH(µ 0 , ν 0 , ρ 0 ). It is written in terms of Jacobi coordinates as a product of (N − 1) oscillator states with sizes depending on N and X 0 [6]. A nonrelativistic version of (4)-(5) can be obtained in the limit m → ∞ [6]. In this case, µ 0 → m. Further simplifications occur also for the ultrarelativistic limit m = 0. A state depends on (N − 1) radial quantum numbers n i and (N − 1) orbital quantum numbers l i , as well as intermediate coupling quantum numbers which are not considered here. The global quantum number resulting from the AFM treatment is then Q = N −1 i=1 (2n i + l i ) + 3 2 (N − 1).(7) All quantum numbers are not allowed, depending on the nature of the particles. In particular, the ground state for a boson-like system is just Q = 3(N − 1)/2, while the ground state of a fermion-like system is much more involved and needs the introduction of the Fermi level [6]. Since P (x) and/or S(x) equal to x 2 can only be used, an upper bound is computed for most of the relevant interactions, a fortiori for a semirelativistic kinematics. For instance, an AFM mass formula has been obtained for a system of N relativistic massless quarks interacting via a linear one-body confinement and a two-body Coulomb potential (this kind of Hamiltonian is pertinent for variant theories of the quantum chromodynamics). The accuracy of this formula has been numerically tested in [6] with N = 3: Relative errors less than 20% have been obtained for the lowest states. It has also be shown in [21] that the N -dependence of this formula is the correct one for N → ∞. When a closed formula cannot be computed, numerical solutions (generally upper bounds) can always be easily obtained. This is valuable for a N -body system. The cases N = 1 and N = 2 For N = 2, s 1 = −s 2 = r 12 /2. So, the potential U (x) becomes redundant with the potential V (x) and can be ignored. Moreover, the Hamiltonian H simplifies because p 1 = −p 2 = p. Thus, for a nonrelativistic kinematics, the case of two different particles can be considered by replacing the kinetic part 2m + p 2 /m by m 1 + m 2 + p 2 /(2m r ) where m r is the reduced mass. A priori, above calculations are only valid for N ≥ 2. But, starting from the one-body equivalent of Hamiltonian (1), it can be shown that equations (4)-(5) are also relevant for N = 1 by setting V (x) = 0 and reinterpreting p 1 and s 1 as conjugate variables. For both N = 1 and 2 systems, the more general form sgn(λ) x λ can be used for the auxiliary potential, instead of only x 2 . This leads to various expressions for Q. The complete calculation shows that the same system (4)-(5) is found and that the only trace of the auxiliary potential lies in the structure of the global quantum number Q. In practice, Q = 2n+l+3/2 with P (x) or S(x) = x 2 (see (7) with N = 2), Q = n+l+1 with P (x) or S(x) = −1/x [3] and Q = 2(−α n /3) 3/2 for S-wave states with P (x) or S(x) = x [7] , where α n is the (n + 1) th zero of the Airy function Ai. Depending on the kinematics, closed form formulae have been obtained for various potentials: sum of two power-law, logarithmic, Yukawa, exponential, square-root [1,2,3,4,5,6]. If a closed formula cannot be computed, the method is then not really interesting since a lot of numerical techniques can be harnessed to find accurate solutions for one-or two-body systems. For some nonrelativistic systems, it is possible to use two forms of the auxiliary potential to obtain both upper and lower analytical bounds of the exact solutions. The following potentials, a √ x 2 + b 2 , a ln(bx), a x − b/x, or sgn(λ) ax λ (with a > 0, b > 0, −1 ≤ λ ≤ 2) can be bounded from below (above) with the choice −1/x (x 2 ) for the auxiliary potential. For instance, let us consider the Hamiltonian H = p 2 2µ + a 2 r 2 + b 2 .(8) The eigenenergies computed with the AFM gives E AFM = 2b √ 3Y G 2 − (Y ) + 1 G − (Y ) with Y = b 2 3 32 µ a 2 Q 2 2/3 ,(9) and where G − (Y ) is the solution of the equation 4 G − (Y ) 4 − 8 G − (Y ) − 3 Y = 0. Upper (lower) bounds are obtained with Q = 2n + l + 3/2 (Q = n + l + 1). The quality of these bounds are studied in [4,5] where more details are given about this solution and the function G − (Y ). Connection with the virial theorem The general virial theorem links the mean values of the directional derivatives of the kinetics operator and the potential [22,23]. Using the Hellmann-Feynman theorem as in [24], it can be applied to the general N -body Hamiltonian (1) to yield N p k · ∇ p k T (p k ) = N s l · ∇ s l U (s l ) + C N r ij · ∇ rij V (r ij ) ,(10) with arbitrary numbers {k, l, i = j} if the mean values are taken with a completely symmetrized eigenstate of the N -body Hamiltonian. The operator T is defined by T (x) = √ x 2 + m 2 or by its nonrelativistic counterpart m+x 2 /(2m). Let us introduce the distance r 0 = N Q/X 0 and the momentum p 0 = Q/r 0 . It is a simple algebra exercise to show that formulae (4)-(5) can be written as: M 0 = N T (p 0 ) + N U r 0 N + C N V r 0 √ C N ,(11)p 0 = Q r 0 ,(12)N p 0 T ′ (p 0 ) = N r 0 N U ′ r 0 N + C N r 0 √ C N V ′ r 0 √ C N .(13) These equations have not been presented in our previous papers. Before discussing their physical meaning, let us look at the quantities r 0 and p 0 . Using formulae of the appendixes in [7] and [21], the following observables can be analytically computed: 1 N N i=1 p 2 i = p 2 0 ,(14)N N i=1 s 2 i = N i<j=1 r 2 ij = r 2 0 .(15) This shows that r 0 can be considered as a mean radius for the system and p 0 as a mean momentum per particle. Indeed, (14) and (15) imply that p 2 i = p 0 , s 2 i = r 0 N , r 2 ij = r 0 √ C N ,(16) for arbitrary i = j since the mean values are taken with completely symmetrized states. These results can also be obtained using the more general relations (66)-(68) in [6] relevant for P (x) and S(x) different from x 2 . With this new formulation, an AFM eigenvalue given by (11) is simply the kinetic operator evaluated at the mean momentum p 0 plus the potential energy computed at some mean radius depending on r 0 . As one could expect, the kinetic energy and the one-body potential energy are proportional to the number of particles and the twobody potential energy is proportional to the number of pairs. Formula (11) looks like a semiclassical approximation but this is absolutely not the case. The AFM yields an approximate N -body wavefunction [6,21], and the relation (12) between p 0 and r 0 is a full quantum link, function of the quantum numbers of the system. At last, the value of r 0 (and thus of p 0 ) is the solution of a transcendental equation (13) which is the translation into the AFM variables of the generalized virial theorem (10) which comes from very general properties of quantum mechanics. These considerations prove that the AFM really relies on very sound physical basis. Once the system (11)-(13) is written, it can appear finally quite natural to obtain such a result. The problem is to find a relevant link between the mean values r 0 and p 0 . This is solved by the AFM. It is generally possible to improve the quality of the AFM eigenvalues with a slight modification of the principal quantum number. A particularly simple form which works quite well is given by Q = N −1 i=1 (α n i + β l i ) + γ(N − 1),(17) where the values of parameters α, β and γ depend on both the interaction and the kinematics. They can be determined by an analytical procedure in some cases by using analytical results coming from WKB approximations or variational calculations [1,8]. Even if it less interesting, a fit on numerically computed exact eigenvalues can always be implemented [1,2,3,4,5,6]. With the form (17), the variational character of the AFM approximation is lost, but the relative errors can be sometimes strongly reduced. Connection with the perturbation theory It has been shown in [2] that, for one-and two-body nonrelativistic systems, the AFM and the perturbation theory give similar results when the potential is an exactly solvable one plus a small perturbation. This result is extended here for the general Hamiltonian (1), that is to say: N particles, semirelativistic kinematics and arbitrary potentials U (x) and V (x). Let us first assume that each pairwise potential V (|r ij |) is supplemented by a term ǫ v(|r ij |), with ǫ ≪ 1 in order that ǫ v(x) ≪ V (x) in the physical domain of interest. In the system (11)-(13), the potential V (x) is replaced by V (x) + ǫ v(x). In this case, new values r 1 and p 1 for the mean radius and momentum will be the solution of the new system: M 1 = N T (p 1 ) + N U r 1 N + C N V r 1 √ C N + ǫ v r 1 √ C N ,(18)p 1 r 1 = Q,(19)N p 1 T ′ (p 1 ) = r 1 U ′ r 1 N + C N r 1 V ′ r 1 √ C N + ǫ v ′ r 1 √ C N .(20) Writing r 1 = (1 + δ)r 0 , we can expect δ ≪ 1 since ǫ ≪ 1. In this case, power expansions at first order can be computed. We have p 1 ≈ (1 − δ)p 0 from (19), and we can write T (p 1 ) ≈ T (p 0 ) − δ p 0 T ′ (p 0 ), T ′ (p 1 ) ≈ T ′ (p 0 ) − δ p 0 T ′′ (p 0 ), U (r 1 /N ) ≈ U (r 0 /N ) + δ r 0 U ′ (r 0 /N )/N ,M 1 = N T (p 0 ) − N δ p 0 T ′ (p 0 ) + N U r 0 N + δ r 0 U ′ r 0 N + C N V r 0 √ C N + C N δ r 0 V ′ r 0 √ C N + C N ǫ v r 0 √ C N + O(ǫ 2 ).(21) Using (11) and (13), this equations simplifies to M 1 = M 0 + C N ǫ v r 0 √ C N + O(ǫ 2 ).(22) This result could seem quite obvious, but it demonstrates that the knowledge of r 0 is sufficient to obtain the contribution of the perturbation at the first order. Let us now look at the most general case and assume too that each [one-body potential U (|s i |) / kinetic operator T (|p i |)] is supplemented by a term [η u(|s i |) / τ t(|p i |)], with [η ≪ 1 / τ ≪ 1] in order that [η u(x) ≪ U (x) / τ t(x) ≪ T (x)] in the physical domain of interest. With similar calculations, we finally find M 1 = M 0 + N τ t (p 0 ) + N η u r 0 N + C N ǫ v r 0 √ C N + O(ǫ 2 , η 2 , τ 2 ).(23) The parameter δ is determined at the same order by the following relation N p 0 τ t ′ (p 0 ) − r 0 η u ′ r 0 N − C N r 0 ǫ v ′ r 0 √ C N = δ 2 N p 0 T ′ (p 0 ) + N p 2 0 T ′′ (p 0 ) + r 2 0 N U ′′ r 0 N + r 2 0 V ′′ r 0 √ C N .(24) Perturbed observables and wavefunctions can then be computed at first order, since r 1 = (1 + δ)r 0 and p 1 = (1 − δ)p 0 at this order. The contribution of a perturbation at the first order can thus be very easily computed within the AFM once the unperturbed problem is solved. In order to check the quality of this approximation, let us consider a case in which the unperturbed Hamiltonian H can be solved exactly by the AFM, that is M 0 is the exact solution. If the small perturbation potential is written ǫ N i<j=1 v(|r ij |), the quantum perturbation theory says that the solution M * is given by M * = M 0 + C N ǫ v(|r ij |) + O(ǫ 2 ),(25) for any pair (ij). The mean value is taken with a completely symmetrized eigenstate of the unperturbed Hamiltonian H. The comparison of (25) with (22) shows that v(|r ij |) is replaced by v r 0 / √ C N within the AFM. This is to be compared with the exact relation S(|r ij |) = S r 0 / √ C N for the auxiliary potential [1,6]. So, the AFM does not give the same result as the perturbation theory. But the agreement can be very good, as shown with several examples calculated explicitly in [2]. Similar discussions can be made for small one-body perturbation potentials or small perturbations of the kinematics. Critical coupling constants Some interactions, as the Yukawa or the exponential potentials, admit only a finite number of bound states. Let us assume that such an interaction can be written as W (x) = −κ w(x), where κ is a positive quantity which has the dimension of an energy and w(x) a "globally positive" dimensionless function such that lim x→∞ w(x) = 0. We can introduce the notion of critical coupling constant κ({θ}) where {θ} stands for a set of quantum numbers. This quantity is such that, if κ > κ({θ}), the potential admits a bound state with the quantum numbers {θ}. The interaction energy for the state with quantum numbers {θ} is then just vanishing for κ = κ({θ}). We refer the reader to [25,26,27,28] for detailed explanations about how to compute critical coupling constant in a given potential. Let us consider a nonrelativistic N -body system (no manageable calculation can be performed for a semirelativistic kinematics) with one-body potentials U (x) = −k u(x) and two-body potentials V (x) = −g v(x), both independent of the particle mass and both admitting only a finite number of bound states. The system (11)- (13) for a vanishing energy gives: N Q 2 2 m r 2 0 = N k N u r 0 N + C N g N v r 0 √ C N ,(26)N Q 2 m r 2 0 = − k N r 0 u ′ r 0 N − C N g N r 0 v ′ r 0 √ C N ,(27) where k N and g N are the critical constants for the system with N particles. The elimination of the ratio N Q 2 /(m r 2 0 ) from both equations yields the equality 2N k N u r 0 N + 2C N g N v r 0 √ C N = −k N r 0 u ′ r 0 N − C N g N r 0 v ′ r 0 √ C N .(28) When potentials u and v are both taken into account, nothing interesting can be said. So let us consider one type of potential at once. Assuming that only two-body forces are present, (28) reduces to 2 C N v r 0 √ C N + r 0 v ′ r 0 √ C N = 0,(29) where the parameter g N has disappeared. Introducing the new variable y 0 = r 0 / √ C N , we can rewrite (26) and (27) as: g N = 1 y 2 0 v(y 0 ) 2 N (N − 1) 2 Q 2 m ,(30) 2 v(y 0 ) + y 0 v ′ (y 0 ) = 0. The variable y 0 , determined by (31), is independent of N , Q and m, and depends only on the form of the function v(x). So, the general formula (30), which was not obtained in our previous works, gives precise information about the dependence of the manybody critical coupling constant g N as a function of all the characteristics of the system. With the system (30)-(31), it is easy to recover some limited previous AFM results obtained for the critical coupling constants of Yukawa and exponential interactions [3,6]. For instance, with the two-body Yukawa interaction V (x) = −g exp(−βx)/x, we have g N = 2 e β Q 2 N (N − 1) 2 m .(32) For N = 2 and Q = n + l + 1, reasonable upper bounds of the exact critical coupling constants are obtained [3]. Within the AFM approximation, the ground state (GS) of a boson-like system is characterized by Q = 3 2 (N − 1). We obtain in this case the following very general relation valid, at the AFM approximation, for all pairwise potentials with a finite number of bound states g N +1 (GS) g N (GS) = N N + 1 .(33) This ratio has previously been obtained and numerically checked for several exponential-type potentials [29,30]. Similarly, in the same general situation, g N (GS) = 2 N g 2 (GS),(34) indicating that in order to bind a N -body system, a coupling N/2 times smaller than the coupling for a two-body problem is sufficient [29,30]. Assuming that only one-body forces are present, a similar calculation gives: k N = 1 y 2 0 u(y 0 ) 1 2N 2 Q 2 m ,(35)2 u(y 0 ) + y 0 u ′ (y 0 ) = 0,(36) where the change of variable y 0 = r 0 /N has been used. Again, the general formula (35), which was not obtained in our previous works, gives precise information about the dependence of the one-body critical coupling constant k N as a function of all the characteristics of the system. For the ground state of a boson-like system, we obtain: k N +1 (GS) k N (GS) = N 2 N 2 − 1 2 ,(37)k N (GS) = 4 N − 1 N 2 k 2 (GS).(38) These results are strongly different from those for pairwise forces. If the AFM gives upper (lower) bounds for the exact eigenvalues, the critical coupling constants predicted by formulae above are upper (lower) bounds for the exact critical coupling constants. Summary The main interest of the auxiliary field method is to obtain approximate closed formulae for the solutions of nonrelativistic and semirelativistic eigenequations in quantum mechanics. The idea, strongly connected with the envelope theory, is to replace a Hamiltonian H for which analytical solutions are not known by another onẽ H which is solvable and which includes one or more auxiliary real parameters. The approximant solutions for H, eigenvalues and eigenfunctions, are then obtained by the solutions ofH in which the auxiliary parameters are eliminated by an extremization procedure for the eigenenergies. The AFM can yield upper or lower bounds (both in some favorable situations) on the exact eigenvalues. The nature of the bound depends on the fact thatH ≥ H orH ≤ H. With a semirelativistic kinematics, only upper bounds can be obtained because of the replacement of the kinetic operator by a nonrelativistic one. For many-body systems, only one type of HamiltonianH can be used. So, it is not possible to obtain both upper and lower bounds for the whole spectrum in this case. Nevertheless, for a nonrelativistic kinematics, a lower bound for the ground state can be sometimes computed. Provided the structure of the HamiltonianH is well chosen (nonrelativistic kinematics plus power-law potentials), an eigenvalue computed by the AFM is simply the kinetic operator evaluated at a mean momentum p 0 plus the potential energy computed at some functions of the mean radius r 0 . The product r 0 p 0 is equal to a global quantum number characterizing the state considered, and the value of r 0 (and then of p 0 ) is the solution of a transcendental equation which is the translation into the AFM variables of the generalized virial theorem. This new result gives sound physical basis to the method. Once a problem is solved within the AFM, it is very easy to compute the contribution of a small perturbation at the first order. It is given by the perturbation Hamiltonian evaluated at the mean momentum p 0 for a kinetic energy or at a function of the mean radius r 0 for a potential. The result does not coincide with the one obtained by the quantum perturbation theory, but the agreement can be very good. The AFM gives a very general formula for the critical coupling constants of nonrelativistic Hamiltonians with a finite number of bound states. The dependence on the quantum numbers, the mass m of the particles, the number N of particles, and the structure of the potential are predicted. Different N behaviours are obtained depending on the one-body or pairwise character of the interaction. If the AFM gives upper (lower) bounds for the exact eigenvalues, the critical coupling constants predicted are upper (lower) bounds for the exact critical coupling constants. etc. Equation(20) reduces to an expression of the form δ ≈ ǫ h(r 0 ) where h is a quite complicated function of T ′ , U ′ , V ′ and their derivatives. It confirms that δ ∼ O(ǫ). The precise form of h is given below in the most general case. It is then possible to perform an expansion of M 1 to obtain . B Silvestre-Brac, C Semay, F Buisseret, J. Phys. A: Math. Theor. 41275301Silvestre-Brac B, Semay C and Buisseret F 2008 J. Phys. A: Math. Theor. 41 275301 . B Silvestre-Brac, C Semay, F Buisseret, J. Phys. A: Math. Theor. 41425301Silvestre-Brac B, Semay C and Buisseret F 2008 J. Phys. A: Math. Theor. 41 425301 . B Silvestre-Brac, C Semay, F Buisseret, J. Phys. A: Math. Theor. 42245301Silvestre-Brac B, Semay C and Buisseret F 2009 J. Phys. A: Math. Theor. 42 245301 . C Semay, F Buisseret, B Silvestre-Brac, Phys. Rev. D. 7994020Semay C, Buisseret F and Silvestre-Brac B 2009 Phys. Rev. D 79 094020 . B Silvestre-Brac, C Semay, F Buisseret, Int. J. Mod. Phys. A. 244695Silvestre-Brac B, Semay C and Buisseret F 2009 Int. J. Mod. Phys. A 24 4695 . B Silvestre-Brac, C Semay, F Buisseret, F Brau, J. Math. Phys. 5132104Silvestre-Brac B, Semay C, Buisseret F and Brau F 2010 J. Math. Phys. 51 032104 . C Semay, B Silvestre-Brac, J. Phys. A: Math. Theor. 43265302Semay C and Silvestre-Brac B 2010 J. Phys. A: Math. Theor. 43 265302 . B Silvestre-Brac, C Semay, J. Math. Phys. 5252107Silvestre-Brac B and Semay C 2011 J. Math. Phys. 52 052107 . V L Morgunov, A Nefediev, Simonov Yu, A , Phys. Lett. B. 459653Morgunov V L, Nefediev A V and Simonov Yu A 1999 Phys. Lett. B 459 653 . Kalashnikova Yu, S Nefediev, A V , Phys. Lett. B. 49291Kalashnikova Yu S and Nefediev A V 2000 Phys. Lett. B 492 91 . B Silvestre-Brac, C Semay, F Buisseret, J. Math. Phys. 5032102Silvestre-Brac B, Semay C and Buisseret F 2009 J. Math. Phys. 50 032102 . R Hall, Phys. Rev. D. 272379Hall R L 1983 Phys. Rev. D 27 2379 . R Hall, J. Math. Phys. 24324Hall R L 1983 J. Math. Phys. 24 324 . R Hall, J. Math. Phys. 252708Hall R L 1984 J. Math. Phys. 25 2708 . R Hall, Phys. Rev. A. 395500Hall R L 1989 Phys. Rev. A 39 5500 . B Silvestre-Brac, C Semay, F Buisseret, arXiv:1101.5222Silvestre-Brac B, Semay C and Buisseret F arXiv:1101.5222 M Reed, B Simon, Methods of Modern Mathematical Physics IV: Analysis of Operators. New YorkAcademicReed M and Simon B 1978 Methods of Modern Mathematical Physics IV: Analysis of Operators (New York: Academic) . C Semay, Phys. Rev. A. 8324101Semay C 2011 Phys. Rev. A 83 024101 . R L Hall, W Lucha, J. Phys. A. 406183Hall R L and Lucha W 2007 J. Phys. A 40 6183 . R Hall, Phys. Rev. C. 201155Hall R L 1974 Phys. Rev. C 20 1155 . F Buisseret, C Semay, Phys. Rev. D. 8256008Buisseret F and Semay C 2010 Phys. Rev. D 82 056008 . W Lucha, Mod. Phys. Lett. A. 52473Lucha W 1990 Mod. Phys. Lett. A 5 2473 . W Namgung, J. Korean Phys. Soc. 32647Namgung W 1998 J. Korean Phys. Soc. 32 647 . A Frost, P G Lykos, J. Chem. Phys. 251299Frost A A and Lykos P G 1956 J. Chem. Phys. 25 1299 . F Brau, F Calogero, J. Math. Phys. 441554Brau F and Calogero F 2003 J. Math. Phys. 44 1554 . F Brau, F Calogero, J. Phys. A: Math. Gen. 3612021Brau F and Calogero F 2003 J. Phys. A: Math. Gen. 36 12021 . F Brau, J. Phys. A: Math. Gen. 369907Brau F 2003 J. Phys. A: Math. Gen. 36 9907 . F Brau, M Lassaut, J. Phys. A: Math. Gen. 3711243Brau F and Lassaut M 2004 J. Phys. A: Math. Gen. 37 11243 . J-M Richard, S Fleck, Phys. Rev. Lett. 731464Richard J-M and Fleck S 1994 Phys. Rev. Lett. 73 1464 . S Moszkowski, S Fleck, A Krikeb, L Theußl, J-M Richard, K Varga, Phys. Rev. A. 6232504Moszkowski S, Fleck S, Krikeb A, Theußl L, Richard J-M and Varga K 2000 Phys. Rev. A 62 032504

Emerging photo-induced excitonic processes in transition metal dichalcogenide (TMD) heterobilayers, e.g., coupling, dephasing, and energy transfer of intra-and inter-layer excitons, allow new opportunities for ultrathin photonic devices. Yet, with the associated large degree of spatial heterogeneity, understanding and controlling their complex competing interactions at the nanoscale remains a challenge. Here, we present an all-round dynamic control of intraand inter-layer excitonic processes in a WSe2/Mo0.5W0.5Se2 heterobilayer using multifunctional tip-enhanced photoluminescence (TEPL) spectroscopy. Specifically, we control the radiative recombination path and emission rate, electronic bandgap energy, and neutral to charged exciton conversion with <20 nm spatial resolution in a reversible manner. It is achieved through the tip-induced engineering of Au tip-heterobilayer distance and interlayer distance, GPa scale local pressure, and plasmonic hot-electron injection respectively, with simultaneous spectroscopic TEPL measurements. This unique nano-opto-electro-mechanical control approach provides new strategies for developing versatile nano-excitonic devices based on TMD heterobilayers.

2203.02136

662e1d667f4fe565fdfd932909536fedba6bab1d

Dynamical Control of Interlayer Excitons and Trions in WSe2/Mo0.5W0.5Se2 Heterobilayer via Tunable Near-Field Cavity Koo Yeonjeong Hyeongwoo 1# Department of Physics Pohang University of Science and Technology (POSTECH) 37673PohangKorea Lee Department of Physics Pohang University of Science and Technology (POSTECH) 37673PohangKorea Tatiana Ivanova School of Physics and Engineering ITMO University 197101Saint PetersburgRussia Ali Kefayati Department of Electrical Engineering University at Buffalo The State University of New York 14260BuffaloNew YorkUnited States Vasili Perebeinos Department of Electrical Engineering University at Buffalo The State University of New York 14260BuffaloNew YorkUnited States Ekaterina Khestanova School of Physics and Engineering ITMO University 197101Saint PetersburgRussia Vasily Kravtsov [email protected] School of Physics and Engineering ITMO University 197101Saint PetersburgRussia Kyoung-Duck Park Department of Physics Pohang University of Science and Technology (POSTECH) 37673PohangKorea Dynamical Control of Interlayer Excitons and Trions in WSe2/Mo0.5W0.5Se2 Heterobilayer via Tunable Near-Field Cavity Emerging photo-induced excitonic processes in transition metal dichalcogenide (TMD) heterobilayers, e.g., coupling, dephasing, and energy transfer of intra-and inter-layer excitons, allow new opportunities for ultrathin photonic devices. Yet, with the associated large degree of spatial heterogeneity, understanding and controlling their complex competing interactions at the nanoscale remains a challenge. Here, we present an all-round dynamic control of intraand inter-layer excitonic processes in a WSe2/Mo0.5W0.5Se2 heterobilayer using multifunctional tip-enhanced photoluminescence (TEPL) spectroscopy. Specifically, we control the radiative recombination path and emission rate, electronic bandgap energy, and neutral to charged exciton conversion with <20 nm spatial resolution in a reversible manner. It is achieved through the tip-induced engineering of Au tip-heterobilayer distance and interlayer distance, GPa scale local pressure, and plasmonic hot-electron injection respectively, with simultaneous spectroscopic TEPL measurements. This unique nano-opto-electro-mechanical control approach provides new strategies for developing versatile nano-excitonic devices based on TMD heterobilayers. Introduction Stacking atomically thin layers of van der Waals (vdW) materials into bilayer heterostructures provides innovative strategies for the development of next-generation optoelectronic devices [1,2,3] and substantially broadens the scope of material physics [4]. A plethora of intriguing phenomena has been already unveiled in vdW bilayers, and they are likely just the tip of the iceberg because many more structures remain unexplored with different chemical composition, stacking sequence and angle, interlayer distance, and other parameters. Hence, considerable efforts are currently focused on uncovering and controlling the inherent physical properties in vdW heterostructures. In particular, interlayer excitons (IXs), formed by electrons and holes spatially separated in the top and bottom layers of transition metal dichalcogenide (TMD) heterobilayers [5], show a range of distinct properties, which are promising for various optoelectronic applications. The reduced spatial overlap of the electron and hole wavefunctions in IXs brings about reduced radiative decay rates, with corresponding lifetimes up to μs [6], while the interlayer distance and twist angle between the constituent monolayers provide knobs for tuning the IX quantum yield [7]. In addition, the out-of-plane component of the IX dipole moment enables straightforward electric field control. IXs in TMD heterobilayers also provide long-lived valley polarization and coherence [8], circumventing the limits of TMD MLs and enabling practical valleytronic applications. Additionally, the slight lattice mismatch and twist angles in heterobilayers give rise to moiré supperlattices and corresponding confinement potentials that can effectively trap IXs [9]. Therefore, IXs in TMD heterobilayers provide much promise for realizing excitonic integrated circuits [10,11] and possibly demonstrating high temperature many-body effects, such as Bose-Einstein condensates (BEC) and superfluidity [12]. However, in order to enable practical applications of TMD heterostructures, several major challenges must be overcome, one of which is the large degree of spatial heterogeneity. The underlying processes, e.g., competing interactions of coupling, dephasing, and energy transfer of intra-and inter-layer excitons, arise at the nanoscale and cannot be understood by diffraction-limited optical approaches, calling for near-field optical probing [13,14,15,16]. Furthermore, beyond probing, it is highly important to achieve nanoscale control of local IX properties in TMD heterostructures. Yet dynamic control study of nanoscale properties of IXs with simultaneous nano-spectroscopic measurements has rarely been reported [17,18]. Here, we demonstrate an all-round dynamic control of intra-and inter-layer excitonic processes in a WSe2/Mo0.5W0.5Se2 heterobilayer with <20 nm spatial resolution using multifunctional tip-enhanced photoluminescence (TEPL) spectroscopy and imaging. The use of the alloy-based heterobilayer allows us to achieve efficient near-field enhancement for both intra-and inter-layer excitons as their spectral peaks all strongly overlap with the tip-plasmon band while remaining well separated [19]. Through hyperspectral TEPL nano-imaging, we reveal nanoscale inhomogeneities of the IX emission and identify regions of different interlayer coupling strength. At the weak interlayer coupling region, we dynamically control the radiative recombination path and competing emission rates of intra-and inter-layer excitons through the engineering of Au tip-heterobilayer distance and interlayer distance, achieving increase of the IX quantum yield compared to that of Xs. In addition, by applying GPa scale tip-pressure to the heterobilayer, we directly modify its electronic bandstructure, which is demonstrated via IX TEPL energy blueshift and supported by theoretical calculations. Furthermore, through the control of plasmonic hot-electron injection from the Au tip, we convert neutral IXs into charged IX states (trions) in a reversible manner. Our results demonstrate that interlayer excitons in TMD heterobilayers can be accurately controlled in nanoscopic volumes via a near-field approach, which opens up new avenues for the development of compact TMD-based optoelectronic devices and provides insights for studying novel many-body phenomena. Red, blue, and green colored regions are spectral regions used for hyperspectral images in a respectively. c, Schematic diagram describing the multifunction of TEPL spectroscopy to dynamically control the intralayer-and interlayer-excitonic properties, such as GPa scale tip-pressure and hot electron injection. Experimental configuration for TEPL spectroscopy We fabricate a WSe2/Mo0.5W0.5Se2 heterobilayer on a Au film by stacking exfoliated ML flakes with their crystal axes aligned for optimized IX emission. The twist angle is measured via polarization-resolved second-harmonic generation (SHG) spectroscopy to be ~1.1 o (see inset of Fig. 1a and Methods). In the assembled heterobilayer, IXs are formed by the spatially separated holes (h + ) and electrons (e -) in constituent layers in addition to intralayer excitons XWSe2 and XMo0.5W0.5Se2. Hyperspectral far-field PL imaging shows that the spatial distributions of intra-and inter-layer excitons are considerably inhomogeneous at the microscale, as shown in Fig. 1a. Such spatial heterogeneity in van der Waals heterostructures is attributed to the non-uniform interlayer coupling strength, which depends sensitively on local strain-induced deformation and interfacial contamination. Furthermore, on the smaller spatial scales below the diffraction-limit, nanoscale structural deformations such as wrinkles, bubbles, and grain boundaries [20,21,22], give rise to complex charge dynamics and interactions with competing recombination processes of intra-and inter-layer excitons. To develop comprehensive understanding of the nanoscale heterogeneity in the WSe2/Mo0.5W0.5Se2 heterobilayer and demonstrate its precise control, we develop multifunctional TEPL spectroscopy. We use a radially polarized excitation beam in the bottomillumination geometry to induce strong out-of-plane optical fields and plasmons at the Au tip-Au film junction (see Methods for more details). The plasmons then couple with the Xs and IXs in the heterobilayer and enhance their PL responses via the Purcell effect [23]. Spectral overlap between the plasmon response and the PL response of intra-and inter-layer excitons in Fig. 1b also give a clear picture of their effective resonant coupling. The tip-sample distance is regulated with a precision of ~0.2 nm using a shear-force feedback loop, with corresponding control on the plasmon enhancement and optical field strength. This allows us to dynamically manipulate the light-matter interactions at the nanoscale with simultaneous spectroscopic TEPL measurements. Fig. 1c shows a schematic of the TEPL spectroscopy experimental configuration, including different multifunctional control modalities, i.e., GPa scale tip-pressure and plasmonic hot carrier injection, as well as tip-induced engineering of the interlayer distance (dI) in a TMD heterobilayer. properties [24]. Similarly, the observed blueshift of the high density IXs is originated from the static electric dipole of IX because the repulsive interactions between the well-oriented IXs cause a mean-field shift, as revealed in previous far-field studies [11]. Note that the height h generally shows an uncorrelated behavior with the spectroscopic line profiles, which means the interlayer coupling strength is not simply characterized by the surface profiling. It should be noted that the whole region of Fig. 2a is measured with ~20 nm spatial resolution by TEPL imaging, which is much smaller than the diffraction-limited beam spot size. Hence, the observed spatio-spectral heterogeneity cannot be investigated using a conventional far-field imaging methods, such as confocal microscopy (See Fig. S1 for the confocal PL image of the same measured area). Near-field probing of the nanoscale heterogeneity in a TMD heterobilayer We then position the tip in the weak interlayer coupling region and acquire PL spectra of IX and XWSe2 as functions of the tip-sample distance d, with experimental data for selected distances shown in Fig. 2d. At d = 20 nm, we observe far-field PL spectrum exhibiting IX and XWSe2 peaks at E= 1.52 eV and 1.63 eV, respectively. The PL peak of XMo0.5W0.5Se2 is not clearly observed due to its low quantum yield. At this relatively large tip-sample distance, the XWSe2 peak shows higher PL intensity than the IX peak due to the low interlayer coupling strength. In comparison, at d = 5 nm the intensities of the X and IX PL become similar. Here, the plasmon-exciton coupling in the Au tip-Au film nanocavity is significantly stronger, and, since the plasmonic resonance predominantly enhances out-of-plane optical fields, the PL of the vertically oriented IX dipoles is increased [23]. At the same time, the intralayer excitons, while efficiently excited in the far-field via in-plane polarized fields, show increasingly inhibited PL emission inside the plasmonic cavity at smaller distances d. When the tip approaches closely to the heterobilayer with d = 3 nm, the PL intensities of the X and IX peaks are switched, and the IX emission dominates, while its spectral position and shape remain unchanged (See Fig. S2 for more details). The TEPL enhancement of IXs is attributed to the increased excitation rate and Purcell effect (See SI section 3 for the calculated enhancement factor ~1.6×10 3 ) [23]. In addition, the tipinduced charge tunneling effect further influences the observed TEPL responses of IXs and Xs [49,25]. In the near-field regime approaching tip-sample contact, the effective overlap between electron wavefunctions of the Au tip and the heterobilayer can facilitate charge tunneling processes [26] that cause the perturbation of the excitonic system. Fig. 2f illustrates the charge transport mechanism of the type-II band alignment when the tip approaches the 2D crystal surface. Since the Fermi level of Au lies lower than the conduction band minimum energy in WSe2, the electrons at the adjacent WSe2 tunnel into the Au tip. Additionally, the electrons and holes in the heterobilayer are redistributed via interlayer charge transfer. Consequently, the p-doped top layer and the n-doped bottom layer effectively facilitate the IX recombination at the local region with decreasing recombination rate of intralayer excitons, as experimentally confirmed in the result of Fig. 2d. In order to move towards practical opto-electronic device applications of vdW heterobilayers, the nanoscale heterogeneity of IX and X emission should be not only resolved, but also actively controlled. Recently, a few approaches for engineering local exciton properties in 2D heterostructures were demonstrated, for example, via electrostatic field [10,27] or high magnetic field [28]. Yet, precise nanoscale control of emission beyond the tip-sample distance modulation is a significant challenge [17,29]. To further extend our tip-induced IX emission control, we present a nano-opto-mechanical tip-pressure engineering approach through the atomic force tip control combined with in-situ TEPL spectroscopy. As schematically illustrated in Fig. 3a, the tip exerts local pressure within a ~ 25 nm 2 sample area, which is precisely regulated through changing the set-point in a shear-force feedback loop (see Methods). This pressure is expected to cause a local decrease in the interlayer distance and corresponding increase in the interlayer coupling strength. We experimentally verify this behavior by measuring TEPL spectra evolution in a reversible tip-press and -release process. As we demonstrate in Fig. S4 in the Supplementary Information, tip pressure applied to a sample region with initially weak interlayer coupling results in stronger IX emission with simultaneously decreased PL intensities of the peaks corresponding to intralayer excitons in WSe2 and Tip-induced nano-engineering of heterobilayer Mo0.5W0.5Se2, which is attributed to the improved interlayer coupling strength [30,31]. In our previous study, we demonstrated that tip-induced local pressure can exceed 10 GPa owing to its nanoscale tip-sample contact area even though the tip-force is only on the order of 0.1 pN [20,32]. Here, in the same fashion we induce ~GPa scale tip-pressure in a TMD heterobilayer (see SI section 5 and 6 for the estimation of pressure and compressive strain), which directly modifies its crystal structure and electronic bandstructure, resulting in the modified IX emission properties. Fig. 3b shows the modified TEPL spectra before (top panel) and after (bottom panel) inducing ~GPa scale tip-pressure in the WSe2/Mo0.5W0.5Se2 heterobilayer. The spectra are decomposed into 3 peaks corresponding to IX, XWSe2, and XMo0.5W0.5Se2 via fitting by Lorentzian functions. In addition to the increase in the IX/X PL ratio discussed earlier, the IX TEPL peak exhibits a clearly discernible blueshift of ~7 meV. In order to clarify the physical origin of the observed spectral changes, we simulate the associated electronic bandstructure modification with decreasing interlayer distance using density functional theory (DFT) calculations as described in Methods. In the calculations, we consider two limiting cases of a general alloy-based bilayer WSe2/Mo1-xWxSe2 with x = 0 and x = 1, where x represents the relative concentration of W atoms in the alloy layer. The calculated electronic bandstructures for x = 0 (heterobilayer WSe2/MoSe2) and x = 1 (homobilayer WSe2/WSe2) with equilibrium interlayer distance dI = 8.85 Å , corresponding to unstrained structures, are shown in gray color in Fig. 3c and 3d TEPL intensity ratio of IX-(red) and IX (light red) is derived for the selected spectra via curve fittings with a Voigt function. Subtracted TEPL spectra exhibiting apparent emerging of IX-peak (contour plot). The background contour image exhibits the evolution of the IX-peak obtained by subtracting the TEPL spectrum without hot e-injection (at d > 10 nm) from the distance-dependent TEPL spectra. b, Illustration for the tip-induced hot-electron injection process, which stimulates the IX-generation in the heterobilayer. c, Changes of the IX-/IX ratio (red) and the IX intensity (blue) as a function of tip-sample distance d. d, TEPL spectra when the Au tip presses (red) and releases (black) the heterobilayer. e, Changes of TEPL intensity ratios for IX-/IX (red) and XWSe2/IX (blue) when the Au tip presses the crystal. The experimental results reported so far have been measured at relatively low values of excitation power (≈ 10 8 W/m 2 ). By significantly increasing the excitation power, we can explore a different regime, characterized by electron transport from the Au tip to the heterobilayer, which is due to the hot electron (e -) generation at the plasmonic tip and subsequent injection into the Mo0.5W0.5Se2 (conduction band of IX) [34,35,36]. Our measurements of the excitation power dependent IX PL confirm the increased charged interlayer exciton (IX-) density in contrast to the saturating neutral IX density at the high-power regime attributed to the hot carrier injection [37,38] (see SI section 7 for more details). By approaching the plasmonic hot tip with a strongly localized field close to the heterobilayer, we achieve the dynamic local control of the interlayer trion formation and recombination rate in the near-field regime. To demonstrate such control, we locate the Au tip in the high-quality crystal region with a high interlayer coupling strength exhibiting the dominant IX emission and investigate the tipinduced hot einjection effect at the high excitation power (≈ 10 9 W/m 2 ). Fig. 4a shows the selected TEPL spectra as a function of the tip-sample distance d. At d = 1 nm, in addition to the neutral IX peak at E = 1.51 eV, a pronounced spectral shoulder emerges at the low energy, which is attributed to the charged interlayer exciton or interlayer trion (IX-) peak induced by the hot einjection. To quantify the IX-/IX PL intensity ratio as a function of d, we deconvolute the TEPL spectra into the IX and IX-peaks by the Lorentzian and calculate the integrated intensity of each spectrum. As indicated in Fig. 4a, the IX-/IX ratio increases up to ~45.3 % as the hot tip approaches the crystal. The evolution of the IX-peak is also clearly observed in the overlaid false color image, which is obtained by subtracting the TEPL spectrum at a large distance d > 10 nm, where the effect of hot electron transfer vanishes, from the distancedependent TEPL spectra. The mechanism of the tip-induced IX-generation is schematically illustrated in Fig. 4b. The hot electrons injected into the heterobilayer within the nanoscale region under the plasmonic tip bind with the neutral IXs and form the IX-. Since this process becomes increasingly efficient as the tip approaches the heterobilayer, the local density of the neutral IX is not significantly increased (see Fig. S7 in Supplementary Information for more details). To demonstrate this behavior from the result of Fig. 4a, we plot the integrated IX intensity (blue) and the IX-/IX intensity ratio (red) as a function of distance d in Fig. 4c. The hot e --induced IX-generation starts to appear at distances d < 10 nm. This experimentally observed threshold distance (d ≈ 10 nm) for the plasmonic hot einjection is in good agreement with the results of previous surface-enhanced Raman spectroscopy studies for molecular samples [39]. In addition, the IX-/IX ratio dramatically increases as d decreases while the neutral IX intensity shows no particular change indicating a highly efficient conversion from the neutral IX to IX-under the plasmonic tip as expected. We further enhance and control the charged IX emission by applying GPa-scale pressure with the tip under high-power excitation. As shown in Fig. 4d, under the tip-induced pressure the contributions to the total TEPL intensity from both IX-and IX peaks are increased, which we attribute to the higher interlayer coupling strength and correspondingly increased recombination rate for both neutral and charged IX species. This is accompanied by redshift of the IX TEPL spectrum and increased linewidth, which is in contrast to the observed blueshift of the TEPL spectrum at low excitation powers presented in Fig. 3b. Additionally, we observe that the TEPL intensity of intralayer excitons (XWSe2) is decreased, which is naturally understood from the competing recombination process between the intra-and inter-layer excitons, as discussed earlier with regard to the data presented in Fig. 2. Furthermore, the precise modification of IX, IX-and X emissions is clearly demonstrated in Fig. 4e. When we press the sample with GPa-scale tip pressure, the TEPL intensity ratios for IX-/IX (red) and XWSe2/IX (blue) show opposite behaviors with pressure. This result shows a distinct advantage of our work compared to the previous hot einjection studies, i.e., the ability to dynamically control the hot edensity and the corresponding IX-conversion rate. By regulating the tipsample distance precisely (~0.2 nm [23]) using the scanning probe tip, we can control the hot einjection at the nanoscale in a fully reversible manner, which was not possible in the previous studies [40,41] (See Fig. S8 for demonstration of reversible control). Conclusion In summary, we have investigated the nanoscale heterogeneity of the interlayer coupling strength in an aligned WSe2/Mo0.5W0.5Se2 heterobilayer and demonstrated active control of its emission via multifunctional TEPL spectroscopy inside a plasmonic tip-substrate cavity in two distinct power regimes. At low excitation powers, we control the interplay between the intralayer and neutral interlayer exciton PL via distance-tunable Purcell enhancement, where IX emission becomes dominant at small tip-sample distances. At high excitation powers, the plasmonic tip acts as a source of hot electrons, which are injected into the heterobilayer and facilitate formation of interlayer trions with distance-tunable efficiency. Beyond the simple control of interlayer excitons via tip-sample distance modulation, we reversibly modify their spectral response via applying nano-localized tip-induced GPa-scale pressure. We support the observed local pressure-dependent IX spectral evolution with DFT simulations, which provide insights into interlayer distance dependent band structure in aligned TMD bilayers. The presented results demonstrate new approaches to study the nanoscale heterogeneity of the interlayer exciton response in TMD heterobilayers and suggest ways to control that response within nanoscopic sample areas. This manifests an important step towards the development of next-generation optoelectronic devices and investigation of novel many-body effects with TMD-based heterobilayers. Acknowledgments The reported study was funded by RFBR and National Research Foundation of Korea according to the research project 19-52-51010. Methods Sample preparation Cover glass (170 um thickness) was ultrasonicated in acetone and isopropanol for 10 mins each and cleaned again by O2 plasma treatment for 10 mins. Then, a Cr adhesion layer (2 nm thickness) and an Au film (9 nm thickness) were deposited subsequently on the glass with a rate of 0.1 Å /s each at the base pressure of ~10 -6 torr using a conventional thermal evaporator. The prepared substrate was covered with a 0.5 nm thick layer of Al2O3 via atomic layer deposition. TMD monolayers (Mo0.5W0.5Se2, WSe2) were mechanically exfoliated from corresponding bulk crystals (HQ Graphene) onto polydimethylsiloxane (PDMS) stamps. For better homogeneity of the target heterobilayer, the monolayers were exposed to UV light [42] for 10 minutes. To achieve accurate layer alignment in the heterobilayer, the directions of crystallographic axes for the monolayers were determined from polarization-resolved second-harmonic generation (SHG) measurements under excitation with laser pulses of 1200 nm center wavelength and 100 fs duration. The WSe2 and Mo0.5W0.5Se2 monolayers were then stacked together on a PDMS substrate with their crystallographic axes aligned via dry transfer at a temperature of 60 °C. The twist angle between the monolayers in the resulting heterobilayer was measured again with polarization-resolved SHG. Finally, the heterobilayer was placed onto the Au-covered substrate for near-field measurements via dry transfer. Multi-functional TEPL spectroscopy and imaging setup Multi-functional TEPL spectroscopy is based on the bottom-illumination mode confocal optics setup combined with shear-force AFM using the Au tip. For the excitation beam, He-Ne laser ( = 632.8 nm, optical power of ≤ 0.5 mW) was was passed through a radial polarizer and then focused at the Au tip-Au film junction by an oil immersion objective lens (PLN100x, 1.25 NA, Olympus). The radial polarizer was used to make vertically polarized beam component as large as possible at the tip apex which leads to effective coupling of exciton and cavity plasmon inducing highly enhanced TEPL signals. The backscattered TEPL signals from a sample were collected by the same objective lens. Note that we use high NA objective lens for efficient collection of the interlayer exciton emissions which has outof-plane dipole moment. In addition, undesirable far-field background noise was reduced by using a pinhole in the detection scheme. TEPL signals (633 nm cut-off) were then sent to a spectrometer (f = 320 mm, 150 g/mm, ~1.6 nm spectral resolution, Monora320i, Dongwoo Optron) and finally imaged onto a thermoelectrically cooled charge-coupled device (CCD, DU971-BV, Andor) to obtain TEPL spectra. For hyperspectral nano-imaging, TEPL spectra at each pixel were recorded during an AFM scanning by a digital controller (Solver next SPM controller, NT-MDT) based on the Au tip attached on a quartz tuning fork. The Au tip (apex radius of ~10 nm) was prepared by the refined electrochemical etching protocol [43] and attached to a tuning fork with a super glue. The tip-sample distance was regulated by the shear-force feedback through monitoring the changing dithering amplitude of the tuning fork/tip assembly. Tip-induced pressure-engineering in heterobilayer To perform the nanoscale pressure-engineering of the heterobilayer using the Au tip, we gradually changed the setpoint of the shear-force feedback. To modify the electronic bandstructure (Fig. 3), we gradually lowered the setpoint to ~ 75 % of the initial oscillating amplitude to induce ~GPa pressure to the crystal structures. Simulations of electronic bandstructures in TMD hetero-and homo-bilayers To calculate electronic band structures, we performed plane-wave density functional theory (DFT) employing projector augmented-wave (PAW) potentials [44,45] implemented in the in the VASP package [46,47]. The electronic exchange-correlation interactions were treated using generalized gradient approximation (GGA) with the method of Perdew-Burke-Ernzerhof (PBE) [48]. In the first step, we performed a full geometry optimization for the bulk structure of the heterostructure with a planewave cutoff energy of 400 eV and a 10 × 10 × 10 grid of Monkhorst-Pack points until the change of the total energy between two relaxation steps was smaller than 10 -4 eV. Then we used the optimized geometry to construct the unit cell of the bilayer heterostructure. A distance of 20 Å was considered between the bilayers to avoid any interaction between them. To simulate the hydrostatic pressure, we added a small change to the interlayer distance followed by a self-consistent field calculation to find the electronic ground state with a plane-wave cutoff energy of 400 eV on a 15 × 15 × 1 k-point grid. The tolerance of self-consistent field calculation was set to 10 −5 eV for both the total energy change and the band-structure-energy change between two steps. Spin-orbit coupling was not included in our calculation. Fig. 1 1Schematic of multifunctional tip-enhanced photoluminescence spectroscopy to dynamically control excitonic processes in TMD heterobilayer. a, (Top) Illustration of the WSe2/Mo0.5W0.5Se2 heterobilayer with SHG polarization dependence and the Au tip to probe and control the crystal. (Bottom) Hyperspectral confocal PL images of the heterobilayer for the integrated intensities 740-760 nm (Green, XWSe2), 784-800 nm (Blue, XMo0.5W0.5Se2), and 830-900 nm (Red, IX). Scale bars are 5 μm. b, Normalized far-field PL spectra of IX, XMo0.5W0.5Se2, and XWSe2. Fig. 2 2Heterogeneous interlayer coupling strength and tip-induced charge transport control. a, Hyperspectral TEPL image of the heterobilayer exhibiting inhomogeneous IX emission at the nanoscale. b, c, Spectroscopic and topographic line profiles for the dashed line L1 in (a). Nanoscale spatial heterogeneities in TEPL peak intensity IIX, linewidth Γ, peak energy shift ∆E, and topographic height h are revealed far beyond the diffraction limit. d, Evolving TEPL spectra of the heterobilayer as a function of the tip-sample distance d. The PL responses of IX (E = 1.52 eV) and XWSe2 (E = 1.63 eV) are acquired with the tip located in the weak interlayer coupling region. e, Illustration for the more efficient plasmon-IX (out-of-plane dipole) coupling compared to the plasmon-X (in-plane dipole) coupling when the Au tip closely approaches to the crystal. f, Illustration for the type-II band alignment of a WSe2/Mo0.5W0.5Se2 heterobilayer and the work function of Au tip describing the detailed charge transport mechanisms. This energy transfer mechanism explains our experimental results of increased (decreased) TEPL intensity of interlayer (intralayer) excitons when the tip approaches to the heterobilayer. To investigate the nanoscale heterogeneity of IXs originated from the non-uniform interlayer coupling strength, we perform hyperspectral TEPL imaging of the heterobilayer, with the experimentally observed spatial distribution of the tip-enhanced IX PL shown in Fig. 2a. In our TEPL scanning, the tip-sample distance d is kept at ~5 nm to minimize tip-induced sample surface modification. To better visualize the spectroscopic information of the inhomogeneous IX distribution and corresponding topography, in Figs. 2b and 2c we present the TEPL intensity IIX, peak energy shift ∆E, linewidth , and height h along the line L1 (indicated in Fig. 2a). The variations in IIX indicate the non-uniform interlayer coupling strength and associated possible changes in the density, oscillator strength, and emission lifetimes of IXs. The regions with higher IIX generally show lower and peak energy blueshift. The higher in the low-density IX regions is possibly due to the slight deviation of the IX dipole orientation, since it can cause PL energy variation due to the quantum confinement effect on the interlayer excitonic Fig. 3 3Tip-induced control of interlayer coupling strength and electronic bandgap. a, Schematic illustration of the local tip control of the interlayer distance in a WSe2/Mo0.5W0.5Se2 heterobilayer. b, TEPL spectra before (top) and after (bottom) pressing the heterobilayer with GPa scale tip-pressure, which causes significant modifications in electronic bandstructure. c, Electronic bandstructure calculated via DFT for a WSe2/MoSe2 heterobilayer (gray), with the conduction and valence bands for different interlayer distances dI (from blue for dI = 8.85 A to brown for dI = 6.25 A); the direct K-K and indirect Y-K transitions are indicated with arrows. d, DFT-calculated bandstructure for a WSe2/WSe2 homobilayer. e, Calculated transition energies vs. interlayer distance for the two lowest transitions: direct K-K (solid curves) and indirect Y-K (dashed) in WSe2/MoSe2 (green) andWSe2/WSe2 (red) bilayers. f, Calculated energy shifts as functions of interlayer distance for WSe2/MoSe2 (green) and WSe2/WSe2 (red) bilayers; the gray shaded area indicates the experimentally measured tip-induced energy shifts for the interlayer exciton in a WSe2/Mo0.5W0.5Se2 heterobilayer. , respectively. The lower conduction band and upper valence band are highlighted with the blue curves and reveal that the two lowest-energy optical excitations correspond to the momentum-direct K-K (solid arrows) and momentum-indirect Y-K (dashed arrows) transitions. The calculated transformation of the conduction and valence bands with decreasing interlayer distance dI is shown inFig. 3c, d with curves changing color from blue (dI = 8.85 Å ) to brown (dI = 6.25 Å ).The corresponding extracted interlayer distance dependencies of the K-K and Y-K transition energies are plotted inFig. 3e. As observed in the figure, the energies of the lowest optical transitions in WSe2/MoSe2 and WSe2/WSe2 exhibit opposite trends with the decreasing interlayer distance, which is highlighted further inFig. 3f, where the distance-dependent energy shifts are plotted instead of absolute transition energies. While for WSe2/MoSe2 our DFT calculations predict blueshift of the transition energy, redshifts on a similar scale are predicted for for WSe2/WSe2. We note that due to the crossover from the direct K-K to indirect Y-K transition in WSe2/MoSe2(Fig. 3e, red curves)at dI ~ 7 Å , the highest predicted blueshifts are limited to 15-20 meV. Considering that for the studied bilayer WSe2/Mo0.5W0.5Se2 (x = 0.5) the expected energy shifts are smaller as they lie in between those for the x = 0 and x = 1 structures, our experimentally observed energy shift of 7 meV (indicated with a dashed line in Fig. 3f) is at the higher end of the calculated range of values. This can be associated with several factors, including the unknown local stoichiometry of the alloy layer, initial inhomogeneous local strain, and the strain-dependent binding energies [33] of interlayer excitons, which are not accounted for in DFT calculations. 4. 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We study the impact of LEP2 constraints on the dimensionless coefficients of the electroweak chiral Lagrangian on the precision observables using the improved renormalization group equations. We find that the current uncertainty in the triple and quartic gauge boson couplings can accommodate electroweak symmetry breaking models with S(Λ = 1 TeV) > 0.There have been recent attempts to constrain the electroweak symmetry breaking (EWSB) models by analyzing the electroweak S, T , W and Y parameters [1]. However, these studies have not considered the uncertainty in the triple gauge couplings (TGC) from LEP2 and Tevatron. In this letter, we study implications of the TGC constraints from LEP2 and Tevatron on the parameter space of the nonlinearly realized electroweak chiral Lagrangian (EWCL) by taking into account the logarithmic scale dependence of the chiral coefficients.Restricting our study on the bosonic sector of the electroweak effective field theory we include all the operators upto mass dimension four in EWCL [2] which contribute to the two, three and four point functions. We confine to consider the set of operators consistent with discrete symmetries, P , T , and C. In a similar study, Bagger et. al.[3] considered operators contributing to two point functions only.In the framework of effective field theory, the dimensionless chiral coefficients of the EWCL, such as the precision parameter S and T , depend on the renormalization scale as) We evaluate β O by including all dimensionless chiral coefficients corresponding to O(p 4 ) operators in our renormalization group equation (RGE) analysis and take into account the bounds of the TGC from the LEP2 measurements as our input. We have extended our earlier study on computation of one loop RGE using background field technique for SU(2) case [4] to improve upon the existing RGE's [2, 5] for EWCL and are presented in reference [6]. Before presenting the β functions of two point chiral coefficients, we describe the experimental or theoretical bounds of all dimensionless chiral coefficients.Two point function chiral coefficients are extracted from data collected in Z factories. We perform the analysis with the three best measured quantities m W = 80.425 ± 0.038 GeV, sin 2 θ eff W = 0.23147 ± 0.00017 and the leptonic decay width of Z, Γ l = 83.984 ± 0.086 MeV for the S, T and U fitting. The other inputs used are 1/α em (m Z ) = 128.74, m Z = 91.18 GeV, and m t = 175 GeV.The central values with 1σ errors of the S, T, U parameters are found as S = (−0.06 ± 0.11) T = (−0.08 ± 0.14) U = (+0.17 ± 0.15)which roughly agrees with[3].The fit is based on one loop calculation and performed using the procedure in reference[7]. In order to make a correspondence with the definition of S − T in EWCL, we subtract the contribution of Higgs boson from the Standard Model (SM )at a reference value m ref H as given in[3]. The validity of the subtraction method is checked by observing the independence of Higgs mass in the fit.The relations among the S − T − U parameters with the chiral coefficients α 1 , α 0 , and α 8 of EWCL are found to be α 1 (µ) = − S(µ) 16π , α 0 (µ) = − α EM T (µ) 2 ,which are in complete agreement with those in [2] provided we take into account the sign difference for β parameter accounted for the calculation performed in Euclidean space[6].

hep-ph/0603038

6a9f3a7a2a9730da7833feddf0d35a39cb44adec

Electroweak symmetry breaking and precision data 5 Mar 2006 Sukanta Dutta SGTB Khalsa College University of Delhi Delhi-110007India Theory Group KEK 305-0801TsukubaJapan. Kaoru Hagiwara Theory Group KEK 305-0801TsukubaJapan. The Graduate University for Advanced Studies 305-0801TsukubaJapan Qi-Shu Yan Theory Group KEK 305-0801TsukubaJapan. Electroweak symmetry breaking and precision data 5 Mar 2006numbers: 1110Gh1110Hi1215Ji1215Lk We study the impact of LEP2 constraints on the dimensionless coefficients of the electroweak chiral Lagrangian on the precision observables using the improved renormalization group equations. We find that the current uncertainty in the triple and quartic gauge boson couplings can accommodate electroweak symmetry breaking models with S(Λ = 1 TeV) > 0.There have been recent attempts to constrain the electroweak symmetry breaking (EWSB) models by analyzing the electroweak S, T , W and Y parameters [1]. However, these studies have not considered the uncertainty in the triple gauge couplings (TGC) from LEP2 and Tevatron. In this letter, we study implications of the TGC constraints from LEP2 and Tevatron on the parameter space of the nonlinearly realized electroweak chiral Lagrangian (EWCL) by taking into account the logarithmic scale dependence of the chiral coefficients.Restricting our study on the bosonic sector of the electroweak effective field theory we include all the operators upto mass dimension four in EWCL [2] which contribute to the two, three and four point functions. We confine to consider the set of operators consistent with discrete symmetries, P , T , and C. In a similar study, Bagger et. al.[3] considered operators contributing to two point functions only.In the framework of effective field theory, the dimensionless chiral coefficients of the EWCL, such as the precision parameter S and T , depend on the renormalization scale as) We evaluate β O by including all dimensionless chiral coefficients corresponding to O(p 4 ) operators in our renormalization group equation (RGE) analysis and take into account the bounds of the TGC from the LEP2 measurements as our input. We have extended our earlier study on computation of one loop RGE using background field technique for SU(2) case [4] to improve upon the existing RGE's [2, 5] for EWCL and are presented in reference [6]. Before presenting the β functions of two point chiral coefficients, we describe the experimental or theoretical bounds of all dimensionless chiral coefficients.Two point function chiral coefficients are extracted from data collected in Z factories. We perform the analysis with the three best measured quantities m W = 80.425 ± 0.038 GeV, sin 2 θ eff W = 0.23147 ± 0.00017 and the leptonic decay width of Z, Γ l = 83.984 ± 0.086 MeV for the S, T and U fitting. The other inputs used are 1/α em (m Z ) = 128.74, m Z = 91.18 GeV, and m t = 175 GeV.The central values with 1σ errors of the S, T, U parameters are found as S = (−0.06 ± 0.11) T = (−0.08 ± 0.14) U = (+0.17 ± 0.15)which roughly agrees with[3].The fit is based on one loop calculation and performed using the procedure in reference[7]. In order to make a correspondence with the definition of S − T in EWCL, we subtract the contribution of Higgs boson from the Standard Model (SM )at a reference value m ref H as given in[3]. The validity of the subtraction method is checked by observing the independence of Higgs mass in the fit.The relations among the S − T − U parameters with the chiral coefficients α 1 , α 0 , and α 8 of EWCL are found to be α 1 (µ) = − S(µ) 16π , α 0 (µ) = − α EM T (µ) 2 ,which are in complete agreement with those in [2] provided we take into account the sign difference for β parameter accounted for the calculation performed in Euclidean space[6]. We study the impact of LEP2 constraints on the dimensionless coefficients of the electroweak chiral Lagrangian on the precision observables using the improved renormalization group equations. We find that the current uncertainty in the triple and quartic gauge boson couplings can accommodate electroweak symmetry breaking models with S(Λ = 1 TeV) > 0. PACS numbers: 11.10.Gh, 11.10.Hi, 12.15.Ji,12.15.Lk There have been recent attempts to constrain the electroweak symmetry breaking (EWSB) models by analyzing the electroweak S, T , W and Y parameters [1]. However, these studies have not considered the uncertainty in the triple gauge couplings (TGC) from LEP2 and Tevatron. In this letter, we study implications of the TGC constraints from LEP2 and Tevatron on the parameter space of the nonlinearly realized electroweak chiral Lagrangian (EWCL) by taking into account the logarithmic scale dependence of the chiral coefficients. Restricting our study on the bosonic sector of the electroweak effective field theory we include all the operators upto mass dimension four in EWCL [2] which contribute to the two, three and four point functions. We confine to consider the set of operators consistent with discrete symmetries, P , T , and C. In a similar study, Bagger et. al. [3] considered operators contributing to two point functions only. In the framework of effective field theory, the dimensionless chiral coefficients of the EWCL, such as the precision parameter S and T , depend on the renormalization scale as O(m Z ) exp = O(Λ) New Phys. + β O ln Λ m Z .(1) We evaluate β O by including all dimensionless chiral coefficients corresponding to O(p 4 ) operators in our renormalization group equation (RGE) analysis and take into account the bounds of the TGC from the LEP2 measurements as our input. We have extended our earlier study on computation of one loop RGE using background field technique for SU(2) case [4] to improve upon the existing RGE's [2,5] for EWCL and are presented in reference [6]. Before presenting the β functions of two point chiral coefficients, we describe the experimental or theoretical bounds of all dimensionless chiral coefficients. Two point function chiral coefficients are extracted from data collected in Z factories. We perform the analysis with the three best measured quantities m W = 80.425 ± 0.038 GeV, sin 2 θ eff W = 0.23147 ± 0.00017 and the leptonic decay width of Z, Γ l = 83.984 which roughly agrees with [3]. The fit is based on one loop calculation and performed using the procedure in reference [7]. In order to make a correspondence with the definition of S − T in EWCL, we subtract the contribution of Higgs boson from the Standard Model (SM )at a reference value m ref H as given in [3]. The validity of the subtraction method is checked by observing the independence of Higgs mass in the fit. The relations among the S − T − U parameters with the chiral coefficients α 1 , α 0 , and α 8 of EWCL are found to be α 1 (µ) = − S(µ) 16π , α 0 (µ) = − α EM T (µ) 2 , α 8 (µ) = − U (µ) 16π .(3) which are in complete agreement with those in [2] provided we take into account the sign difference for β parameter accounted for the calculation performed in Euclidean space [6]. Since the S − T − U parameters are defined by Z pole data, Eq. (3) is to be read at µ = m Z , from which α 1 (m Z )-α 0 (m Z )-α 8 (m Z ) are determined. The three point chiral coefficients α 2 , α 3 and α 9 are extracted from the LEP2 W pair production measurements. These three chiral coefficients are related to the experimental observable δk γ , δk Z , δg 1 Z [8] as δk γ = −(α 1 + α 8 + α 2 + α 3 + α 9 )g 2 ,(4)δk Z = −(α 8 + α 3 + α 9 )g 2 + (α 1 + α 2 )g ′ 2 ,(5)δg 1 Z = −α 3 G 2 where G 2 = g 2 + g ′ 2 .(6) Due to the difference in the definition of the covariant differential operator, our triple chiral coefficients have extra signs compared with those in [2]. Current precision on TGC allows us to drop the negligible terms induced through the diagonalization and normalization between Z boson and photon. There are no experimental data relaxing the custodial symmetry except L3 collaboration [9] from where we take δk Z = −0.076 ± 0.064 as one of the inputs. Other inputs δk γ = −0.027 ± 0.045 and δg 1 Z = −0.016 ± 0.022 are taken from LEP Electroweak working group [10,11]. All these data are extracted from one-parameter TGC fits as the twoparameter fits on δg 1 Z and δk γ show larger errors while three parameter fits do not exist. We found TGC errors are quite large as reported in D0 collaboration [12] at Tevatron. Further the most stringent constraints data from LEP2 are preferably analyzed relaxing the custodial SU (2) gauge symmetry as it is natural in the framework of the EWCL to have a non-vanishing α 9 if the underlying dynamics break this symmetry explicitly [13]. Each of these data corresponds to a set of solution for α 2 (m Z ) , α 3 (m Z ), α 9 (m Z ) and are assumed to be extracted from independent measurements. Computing the anomalous TGC in EWCL from these data we get α 2 = (−0.09 ± 0.14) α 3 = (+0.03 ± 0.04) α 9 = (+0.12 ± 0.12) ρ co. =   1 0 1 −.7 −.3 1   . (7) Correlations among the experimental observables affects the ρ co. insignificantly, without changing their central values. We observe that α 3 (m Z ) is more tightly constrained than α 2 (m Z ) and α 9 (m Z ). Anomalous TGC are observed to be one order more constrained w.r.t. the tree level unitary bounds from f 1f2 → V 1 V 2 at Λ ≥ 1 (TeV) [14]. |δk γ | < 1.86 Λ 2 , |δk Z | < 0.85 Λ 2 , δg 1 Z < 0.87 Λ 2 . (8) The four point chiral coefficients or the quartic gauge couplings (QGC) have no experimental data and usually are assumed to be of order one. Partial wave unitary bounds of longitudinal vector boson scattering processes can be used to put bounds on the magnitude of those chiral coefficients. Absence of Higgs boson or other resonances below the UV cutoff Λ renders the form factor of these scattering amplitudes to be energy dependent. We use the following five conditions to constrain five chiral coefficients, α 4 , α 5 , α 6 , α 7 , and α 10 : |4α 4 + 2α 5 | < 3π v 4 Λ 4 , |3α 4 + 4α 5 | < 3π v 4 Λ 4 , |α 4 + α 6 + 3(α 5 + α 7 )| < 3π v 4 Λ 4 , |2(α 4 + α 6 ) + α 5 + α 7 | < 3π v 4 Λ 4 , |α 4 + α 5 + 2(α 6 + α 7 + α 10 )| < 6π 5 v 4 Λ 4 ,(9) where the bounds are obtained from W + L W + L → W + L W + L , W + L W − L → W + L W − L , W + L W − L → Z L Z L , W + L Z L → W + L Z L , and Z L Z L → Z L Z L , respec- tively. The α 1 and α 8 are small in magnitude and are dropped. The contributions of TGC and terms proportional to v 2 /Λ 2 are also dropped here, but are included in the numerical analysis. We have avoided a more strict procedure to derive unitary bounds as shown in [15]. Above is our current knowledge on those dimensionless chiral coefficients. Below we analyze how uncertainty in those dimensionless chiral coefficients can affect the value of S(Λ)-T (Λ)-U (Λ). In order to determine the values of S(Λ)-T (Λ)-U (Λ), we need the RGEs of α 1 -α 0 -α 8 , which are given as 8π 2 d α i /d t = β αi while the β α1,8,0 are β α1 = 1 12 + 4α 1 g 2 − α 8 g 2 + 5 2 α 2 g 2 − 5 6 α 3 g 2 + 1 2 α 9 g 2 (10) β α8 = − α 0 2 + α 1 g ′ 2 + 12α 8 g 2 + 5 6 α 2 g ′ 2 − 1 2 α 3 g ′ 2 + 17 6 α 9 g 2 (11) β α0 = − 3g ′ 2 8 + 9α 0 g 2 4 − 9α 0 g ′ 2 4 + α 1 3g 2 g ′ 2 4 − α 8 3 g 4 8 +α 2 3g 2 g ′ 2 2 − 3g ′ 4 4 + α 3 3g 2 g ′ 2 2 + α 9 − g 4 2 + 3g 2 g ′ 2 4 −α 4 15g 2 g ′ 2 4 + 15g ′ 4 8 − α 5 3g 2 g ′ 2 2 + 3g ′ 4 4 −α 6 3g 4 4 + 33G 4 8 − α 7 3g 4 + 3G 4 − α 10 9G 4 2 ,(12) where we observe that all TGC contributes to the β (1) In absence of TGC contribution ( red contours ), S(Λ) becomes more negative as Λ increases w.r.t. the reference LEP1 fit contour at Λ = m Z . This is in agreement with the observation of Ref. [3] and Ref. [16]. Inclusion of TGC contribution as obtained from LEP2 fit (Eq. 7), makes S(Λ) almost unchanged (the solid line). (2) We observe that when TGC with 1σ uncertainty at Λ = m Z are taken into account, S(1 TeV ) can vary between −0.3 ≤ S(1 TeV) ≤ 0.12 which is almost 3σ away from the prediction of S(1 TeV) without these uncertainties. Analysis with Tevatron data and LEP2 two dimensional TGC fit data would exceed this limit dramastically. (3) The TGC contributions can at most lower the value of T (1 TeV) by |∆T (1 TeV)| ≈ 0.1. Thus the contribution of TGC is not large enough to cancel the large leading contribution from 3g ′ 2 /8 in the β function of T parameter, which makes T (Λ) positive for high energy. Experimental data on the TGC allows the radiative mechanism to render large +ve T (Λ). To realize vanishing T (Λ) with QGC switched off would require T (m Z ) to be negative ≈ −0.4 or so, which is in confrontation to the global fit value given in Eq. (2). Whether is it possible to find a solution in the parameter space of the EWCL? To answer this question, it is worthwhile to understand the evolution of the beta functions of QGC affecting T (Λ) parameter. We observe that α 4 , α 5 terms come along with g ′ 2 , making them one order weaker w.r.t. those of α 6 , α 7 and α 10 . Assuming unitarity bounds on all anomalous QGC would be of the same order and α 10 to dominate among the total QGC contribution. We find that |α 10 | has to be ≥ 0.03 to switch the sign of T (1 TeV), which is contradictory to the unitary bound given in Eq. (9) at Λ = 1 TeV with v = 246 GeV. The reason for the subdominant behavior of QGC couplings with increasing energies can be explained from the Table 1. We realize that with the increasing Λ the TGC uncertainty δT TGC increases logarithmicly while the QGC uncertainty δT QGC decreases rapidly due to the power dependence in the unitary bounds given in Eq. (9). Consequently it is observed that δT QGC and δT TGC dominates the error of T (Λ) below and above Λ < 950 GeV, respectively. From Table. 1, we can conclude that in the constrained EWCL parameter space with 1σ error in TGC and with unitary bounds on QGC, it is unlikely to have a scenario with vanishing T (1 TeV) while keeping T (m Z ) = −0.08. It is worth mentioning that performing the analysis with twoparameter TGC fits δT TGC becomes larger while δT QGC changes insignificantly. However, there are possible ways to evade this situation: (1) Lowering the UV scale Λ down to 700 GeV or so, (2) Relaxing the error of T (1TeV) to 2σ or so, and (3) Generating a large enough positive T (Λ) from more fundamental dynamics, as proposed in most Technicolor models when matched with the effective theory. Λ T (Λ) ± We summarize our study and conclude that LEP2 data has constrained the anomalous TGC (three point chiral coefficients), but allows regions where the S(m Z parameter can be explained by the radiative corrections of the TGC accompanying with a positive S(Λ). This letter shows that the negative S(Λ) parameter problem can be related to the loosely constrained large anomalous TGC (α 2 and α 9 ). With the current experimental and theoretical knowledge, TGC's and QGC's uncertainty can undermine our prejudice for discarding or accepting a specific EWSB model. However the upcoming colliders, with higher sensitivity to the TGC, can reduce the parameter space and help to pinpoint the correct model of EWSB. We would like to thank Ulrich Parzefall for communication on the TGC measurements at LEP2, and Masaharu Tanabashi for stimulating discussion. SD thanks SERC, DST, India and the coreuniversity program of the JSPS for the partial financial support. Work of KH is supported in part by MET, Japan. QSY thanks the theory group of physics department, Tsinghua university for helpful discussion and thanks JSPS and NCTS (Hsinchu, Taiwan) for partial financial support. ± 0.086 MeV for the S, T and U fitting. The other inputs used are 1/α em (m Z ) = 128.74, m Z = 91.18 GeV, and m t = 175 GeV. The central values with 1σ errors of the S, T, U parameters are found as S = (−0.06 ± 0.11) T = (−0.08 ± 0.14) U = (+0.17 ± 0.15) α0,1,8 while QGC contributes only to the β α0 . This implies QGC do not contribute to the S parameter. The S(Λ), T (Λ), and U (Λ), are computed from the evolved α 1 (Λ)-α 0 (Λ)-α 8 (Λ) trough RGE. The S(Λ), T (Λ) and U (Λ) are the values of the parameters S, T , and U at the matching scale Λ, where the EWCL matches with fundamental theories, Technicolor models, extra dimension models, Higgsless models, etc.How does the uncertainty of TGC affect the value of S(Λ)? To answer this question we set all QGC to zero at Λ = m Z to the study the effect of TGC on S(Λ) − T (Λ) plane which is depicted inFig. 1. We highlight some features of this figure. FIG. 1 : 1S(Λ) − T (Λ) contours at Λ = mZ, 300 GeV, 1 TeV, and 3 TeV, respectively. TGC uncertainty is included in green contours while the red contours are without it. 1σ δTZ δT TGC δT QGC 0.3 TeV 0.25 ± 8.91 ±0.14 ±0.06 ±8.91 0.5 TeV 0.29 ± 1.16 ±0.14 ±0.08 ±1.15 1 TeV 0.40 ± 0.22 ±0.14 ±0.12 ±0.10 3 TeV 0.60 ± 0.25 ±0.14 ±0.17 ±0.04 TABLE I : IValues of T (Λ) and 1σ errors from δTZ, δT TGC and δT QGC . . R Barbieri, Nucl. Phys. B. 703127R. Barbieri et. al. , Nucl. Phys. B 703, 127 (2004). . A C Longhitano, Phys. Rev. D. 221166A. C. Longhitano, Phys. Rev. D 22, 1166 (1980); . T Appelquist, G H Wu, Nucl. Phys. B. 1883235Phys. Rev. DNucl. Phys. B 188, 118 (1981). T. Appelquist and G. H. Wu, Phys. Rev. D 48, 3235 (1993). . J A Bagger, Phys. Rev. Lett. 841385J. A. Bagger et. al. , Phys. Rev. Lett. 84, 1385 (2000). . Q S Yan, D S Du, Phys. Rev. D. 6985006Q. S. Yan and D. S. Du, Phys. Rev. D 69, 085006 (2004); . S Dutta, Nuc. Phys. B. 70475S. Dutta et. al, Nuc. Phys. B 704, 75 (2005). . M J Herrero, E. Ruiz Morales, Nucl. Phys. B. 418431M. J. Herrero and E. Ruiz Morales, Nucl. Phys. B 418, 431 (1994); . Nucl. Phys. B. 437319Nucl. Phys. B 437, 319 (1995). . S Dittmaier, C Grosse-Knetter, Nucl. Phys. B. 459497S. Dittmaier and C. Grosse-Knetter, Nucl. Phys. B 459, 497 (1996). . S Dutta, K Hagiwara, Q S Yan, K Yoshida, in prepS. Dutta, K. Hagiwara, Q.S. Yan, and K. Yoshida, in prep. . K Hagiwara, Annu. Rev. Nucl. Part. Sci. 48463K. Hagiwara, Annu. Rev. Nucl. Part. Sci. 48 463, (1998); . K Hagiwara, Z. Phys. C. 64352Erratum-ibid. CK. Hagiwara et. al. , Z. Phys. C 64, 559 (1994) [Erratum-ibid. C 68, 352 (1995). . K Hagiwara, Nucl. Phys. B. 282253K. Hagiwara et. al. , Nucl. Phys. B 282, 253 (1987). . P Achard, Phys. Lett. B. 586151P. Achard et al., Phys. Lett. B 586, 151 (2004). . A Heister, Eur. Phys. J. C. 21423A. Heister et al., Eur. Phys. J. C 21, 423 (2001); . G Abbiendi, Eur. Phys. J. C. 33463G. Abbiendi et al., Eur. Phys. J. C 33, 463 (2004); . S Schael, Phys. Lett. B. 6147S. Schael et al. Phys. Lett. B 614, 7 (2005). . V M Abazov, hep-ex/0504019V. M. Abazov et al., hep-ex/0504019. . P Sikivie, Nucl. Phys. 173189P. Sikivie et al., Nucl. Phys. B173, 189 (1980). . U Baur, D Zeppenfeld, Phys. Lett. B. 201383U. Baur and D. Zeppenfeld, Phys. Lett. B 201, 383 (1988). . G J Gounaris, Z. Phys. C. 66619G. J. Gounaris et. al., Z. Phys. C 66, 619 (1995); . Phys. Lett. B. 350212ibid, Phys. Lett. B 350, 212 (1995). . M E Peskin, J D Wells, Phys. Rev. D. 6493003M. E. Peskin and J. D. Wells, Phys. Rev. D 64, 093003 (2001).

In this note, we determine the representation content of the free, large N , SU (N ) Yang Mills theory on a sphere by decomposing its thermal partition function into characters of the irreducible representations of the conformal group SO(4,2). We also discuss the generalization of this procedure to finding the representation content of N = 4 Super Yang Mills.

10.1088/1126-6708/2006/01/160

hep-th/0501063

8b1ab10db37ea6ed8560adecad17474be1c290de

The Spectrum of Yang Mills on a Sphere arXiv:hep-th/0501063v2 6 Oct 2005 Alexander Barabanschikov Department of Physics NorthEastern University 02115BostonMAUSA Lars Grant Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Tata Institute of Fundamental Research 400005MumbaiIndia Lisa L Huang Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Suvrat Raju Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Tata Institute of Fundamental Research 400005MumbaiIndia The Spectrum of Yang Mills on a Sphere arXiv:hep-th/0501063v2 6 Oct 2005 In this note, we determine the representation content of the free, large N , SU (N ) Yang Mills theory on a sphere by decomposing its thermal partition function into characters of the irreducible representations of the conformal group SO(4,2). We also discuss the generalization of this procedure to finding the representation content of N = 4 Super Yang Mills. Introduction Almost thirty years ago t'Hooft, Polyakov, Migdal and Wilson suggested that large N Yang Mills theory could be recast as a string theory. Electric flux tubes of the confining gauge theory were expected to map to dual fundamental strings. This picture seemed crucially tied to confinement and strongly coupled gauge dynamics; in particular Yang Mills perturbation theory was not expected to capture the stringy behavior of Yang Mills theory. Some of these expectations were modified after one gauge-string duality was understood in detail via the celebrated AdS/CFT conjecture [1,2]. According to the Maldacena conjecture, maximally supersymmetric SU (N ) Yang Mills theory on an S 3 is dual to type IIB theory on AdS 5 × S 5 . N = 4 Yang Mills theory is a conformal (rather than confining) gauge theory, further it possesses a tunable coupling constant λ which may smoothly be taken to zero. Nonetheless, the AdS/CFT conjecture asserts that this gauge theory is dual to a string theory at all values of λ including λ = 0. It follows that stringy aspects of the N = 4 theory must be visible even at arbitrarily weak coupling. Evidence has been accumulating over recent years that this is indeed the case. First, the spectrum of weakly coupled N = 4 Yang Mills theory on an S 3 displays a Hagedorn like growth in its density of states [3,4] (a distinctly stringy feature). Second, perturbative contributions to the anomalous dimensions of certain long single trace operators are effectively encoded in a 1+1 dimensional field theory [5] . For recent developments in this story, please see [6] and references therein. With these lessons in mind, it is natural to wonder if the stringy spectrum of the simplest of all large N Yang Mills theories -pure SU (N ) Yang Mills -may also be computed (in some regime of parameters) in perturbation theory. Naively it seems that asymptotically free pure Yang Mills lacks a dimensionless parameter in which one might attempt a perturbative expansion. However Yang Mills theory on an S 3 of radius R does have a dimensionless coupling λ = ΛR, where Λ is the dynamically generated scale. When λ is taken to zero, the spectrum of pure Yang Mills may reliably be studied in perturbation theory, and turns out to display the same Hagedorn growth with energy as its supersymmetric counterpart [3,4]. It thus seems plausible that, like its supersymmetric counterpart, pure Yang Mills theory on an S 3 has a dual string description at all values of λ. In this note we will compute the spectrum of particles of the as yet unknown string dual to pure Yang Mills on a sphere at λ = 0. We do this by decomposing the partition function of our theory at λ = 0 (computed in [3,4] ) into a sum over characters of the conformal group (the conformal group is a good symmetry of pure Yang Mills theory precisely at λ = 0). According to the AdS/CFT dictionary, representations of the conformal group are in one to one correspondence with particles of the dual string theory; consequently our decomposition determines particle spectrum of interest. The plan of the rest of this note is as follows. In sections 1-4 we review the conformal algebra and calculate the characters of the representations. Since this involves some subtleties, in Section 5, we verify the calculation using the oscillator construction proposed in [7] . The integral we need to perform to obtain the character decomposition has poles. Section 6 discusses the appropriate treatment for these, while the actual calculation is performed in Section 7. We conclude with some comments on the supersymmetric case. In the Appendix we prove that the multiplicities obtained in Section 7 are integers. While we were in the process of writing this paper, we received [8] that has overlaps with Sections 3,4. Conformal Algebra Adding dilatations and special conformal transformations to a set of Lorentz generators in 4 dimensions gives the conformal algebra. [D, P µ ] = −iP µ , [D, K µ ] = iK µ , [K µ , P ν ] = 2i(η µν D + M µν ), [M µν , P ρ ] = i(η µρ P ν − η νρ P µ ), [M µν , K ρ ] = i(η µρ K ν − η νρ K µ ), [M µν , M ρσ ] = i (η µρ M νσ + η νσ M µρ − η µσ M νρ − η νσ M µρ ) . (2.1) We are interested in unitary representations of this algebra, where these generators are hermitian. However, it is convenient for the purposes of constructing the representations of this algebra, to choose a basis of generators which satisfies the euclidean conformal algebra [9] and in which the generators are no longer all hermitian. The generators (some D ′ , P ′µ , etc.) in this new basis will satisfy the same algebra as above with η µν → δ µν . The hermiticity properties of the generators in this basis are: M ′ † = M ′ , D ′ † = −D ′ , P ′ † = K ′ , K ′ † = P ′ . From now on, we will use this new set of generators and drop the primes for clarity. We can extract two sets of SU (2) generators from the Lorentz generators M µν . We define: J z 1 = 1/2(M 12 + M 03 ), J z 2 = 1/2(M 12 − M 03 ), J + 1 = 1/2(M 23 + M 01 + i(M 02 − M 13 )), J + 2 = 1/2(M 23 − M 01 − i(M 13 + M 02 )), J − 1 = J + † 1 , J − 2 = J + † 2 . (2.2) We will also choose to use an hermitian operator D ′′ = iD for convenience. We note that the set of generators M = {D ′′ , J z 1 , J + 1 , J − 1 , J z 2 , J + 2 , J − 2 } generate the maximal compact subgroup SO(2) × SO(4) ⊂ SO(4, 2) of the conformal group. We can divide the generators into three sets : G 0 = {D, J z 1 , J z 2 }, G + = {J + 1 , J + 2 , P µ }, G − = {K µ , J − 1 , J − 2 }. With this division, the Lie algebra above has the property that: [g 0 , g + ] = g ′ + , [g 0 , g ′ 0 ] = g ′′ 0 , [g + , g − ] = g 0 , [g 0 , g − ] = g ′ − . where anything with a subscript 0 belongs to linear combinations of operators in G 0 and similarly symbols with subscripts +(−) belong to linear combinations of operators in G + (G − ). These relations make it clear that G + and G − act like raising and lowering operators on the charges G 0 . The operators in G 0 commute and we will use these as Cartan generators for the algebra. It will be convenient to choose linear combinations of the operators in G + and G − that diagonalize G 0 . These combinations are: P w = P 1 + iP 2 , P w = P 1 − iP 2 , P z = P 3 + iP 4 , Pz = P 3 − iP 4 . (2.3) These generators all have well defined weights under the Cartan generators G 0 . D J 1 J 2 J + 1 0 1 0 J + 2 0 0 1 P w 1 1 2 1 2 Pw 1 − 1 2 − 1 2 P z 1 − 1 2 1 2 Pz 1 1 2 − 1 2 (2.4) Representations of the Conformal Group Any irreducible representation of the conformal group can be written as some direct sum of representations of the compact subgroup SO(4) × SO(2): R SO(4,2) = i R i comp . (3.1) We are ultimately interested in the occurrence of these representations in the partition function of the conformal Yang-Mills gauge theory quantized on S 3 × R; the hamiltonian of the theory is the dilatation operator D. The spectrum of this theory is bounded below and therefore we will be interested in representations of the conformal algebra where the values of the charge D are bounded below. Then there must be some term, R λ comp in the above sum that has the lowest dimension. This term has a highest weight state |λ > with weights λ = (d, j 1 , j 2 ). The K µ operators necessarily annihilate all the states in R λ comp because the K µ have negative weight under the operator D. If we consider the operation of the P µ on this set of states, we generate a whole representation of the conformal algebra with states: [λ] * = R λ SO(4,2) = n w nwn z nz P n w w P nw w P n z z P nz z × R λ comp . (3.2) We will denote this set of states by [d, j 1 , j 2 ] * . A careful analysis [10] shows that, barring the trivial case, this representation is unitary if one of the following conditions holds on the highest weight state |λ >: (i) d ≥ j 1 + j 2 + 2 j 1 = 0 j 2 = 0, (ii) d ≥ j 1 + j 2 + 1 j 1 j 2 = 0. (3.3) In the case where equality does not hold in these unitarity conditions, the representation is called long and all the states produced by the operation of the P µ are non-zero. If equality holds in one of the conditions, then the representation will be a truncated short representation in which some of the states listed in (3.2) are 0. A unitary representation is one where we can define a positive definite norm. To find the states that should be absent in a short representation, one can assume that the states in R λ comp are normalized in the standard way [9] . Calculating the norm of the states P µ |λ > will show that when equality holds in the unitarity conditions above, some of these states, say a µ P µ |λ > have norm 0. This should be interpreted as meaning that this state is 0 so that the operator a µ P µ annihilates |λ >. The descendants of a µ P µ |λ > then also do not occur in the representation. This last statement needs some care as we will see. We will list here the possible types of short representations: • In the generic short representation, j 1 = 0, j 2 = 0, d = j 1 + j 2 + 2 the states of norm 0 occur at level 1. The state |d + 1, j 1 − 1 2 , j 2 − 1 2 > is not found in the representation and its descendants are also not to be found in the representation. The set of all descendants of the state λ ′ = |d+1, j 1 − 1 2 , j 2 − 1 2 > is the same as [d+1, j 1 − 1 2 , j 2 − 1 2 ] * , so that we may write the generic short, irreducible representation as: [d, j 1 , j 2 ] = [d, j 1 , j 2 ] * − [d + 1, j 1 − 1 2 , j 2 − 1 2 ] * . (3.4) • In the case j 1 = j 2 = 0, d = 1, the state |3, 0, 0 > is not found. All its descendants are also absent, so we may write the irreducible representation as [1, 0, 0] = [1, 0, 0] * − [3, 0, 0] * . (3.5) • In the case j 1 > 0, j 2 = 0, the state |d + 1, j 1 − 1 2 , 1 2 > is absent. Note that the weights of this state satisfy the unitarity bound (i) in (3.3). When we delete the states [d + 1, j 1 − 1 2 , 1 2 ] * , we must delete it as a short representation, ie. we must not delete the states that do not occur in the short rep [d + 1, j 1 − 1 2 , 1 2 ]. We will do a calculation below, using an oscillator representation, showing that this is the correct way to remove the zero norm states in this case. We will have [d, j 1 , 0] = [d, j 1 , 0] * − [d + 1, j 1 − 1 2 , 1 2 ] * + [d + 2, j 1 − 1, 0] * . (3.6) Characters The characters for these representations are now easy to compute. First we compute a character of the set of states [d, j 1 , j 2 ] * . We will denote the character of this set of states by a bar on χ: χ [d,j 1 ,j 2 ] = T r [d,j 1 ,j 2 ] * exp[iDθ + iJ z 1 θ 1 + iJ z 2 θ 2 ] = n k ≥ 0 |m 1 | < j 1 |m 2 | < j 2 < adjoint|e iDθ+iJ z 1 θ 1 +iJ z 2 θ 2 P n 1 w P n 2 w P n 3 z P n 4 z |d, m 1 , m 2 > = χ SU(2) j 1 χ SU(2) j 2 e idθ 4 j=1 (1 − exp[i α j · θ]) (4.1) where θ = (θ, θ 1 , θ 2 ), and α j runs over the 4 generators P w , Pw, P z , Pz and refers to their weights taken from the table (2.4) , ie α 1 = (1, 1/2, 1/2). The characters of the possible representations are given by: (i) Long d > j 1 + j 2 + 2 χ [d,j 1 ,j 2 ] =χ [d,j 1 ,j 2 ] (ii) Short j 1 = j 2 = 0 d = 1 χ [1,0,0] =χ [1,0,0] −χ [3,0,0] (iii) Short j 1 > 0 j 2 = 0 d = j 1 + 1 χ [d,j 1 ,0] =χ [d,j 1 ,0] −χ [d+1,j 1 −1/2,1/2] +χ [d+2,j 1 −1,0] (iv) Short j 1 > 0 j 2 > 0 d = j 1 + j 2 + 2 χ [d,j 1 ,j 2 ] =χ [d,j 1 ,j 2 ] −χ [d+1,j 1 −1/2,j 2 −1/2] (4.2) We will note shortly that these characters are not orthogonal. Nevertheless, they can be used to decompose the spectrum of the Conformal Yang Mills theory we are interested in. Oscillator Construction Here we will discuss an oscillator construction [7] for the SO(4,2) algebra and use it to confirm the character of the short representations j 2 = 0 and d = j 1 + 1. The SO(4,2) algebra may be represented by 8 bosonic oscillators a I , b J , a I and b J (I, J = 1, 2) having the following commutation relations: [a I , a J ] = δ J I [b P , b Q ] = δ Q P . (5.1) The generators of the SO(4, 2) group are represented as: J i 1 = 1/2(σ i ) J I [a I a J − 1/2δ I J a K a K ], J i 2 = 1/2(σ i ) Q P [b P b Q − 1/2δ P Q b R b R ] D = 1/2(N a + N b + 2), P IJ = a I b J , K IJ = a I b J . (5.2) We note that a state constructed out of oscillators acting on a vacuum satisfying a I |0 >= b J |0 >= 0 has weights (1/2(N a + N b + 2), 1/2(n a 1 − n a 2 ), 1/2(n b 1 − n b 2 )) under D, J z 1 , J z 2 (n a 1 is the number of a 1 operators used to create the state and N a = n a 1 + n a 2 ). The unitarity constraints are built into this representation, so we may calculate with it without worrying about states that have norm zero. For example, we may compute the "blind" partition function of the short representation |λ >= (j 1 + 1, j 1 , 0). We first choose a state with the right weights to act as the primary: (a 2 ) 2j 1 |0 > . (5.3) Now we can easily generate from this state, a representation of the maximal compact subgroup SO(4) × SO (2): a I 1 a I 2 a I 3 . . . a I 2j 1 |0 > . (5.4) There are 2j 1 + 1 states here, all with dimension D = j 1 + 1 as we expect. Now we operate with all possible P µ : Z [j 1 +1,j 1 ,0] = n 1 ,n 2 ,m 1 ,m 2 I k < adjoint|x D a n 1 1 a n 2 2 b m 1 1 b m 2 2 a I 1 a I 2 a I 3 . . . a I 2j 1 |0 > = ∞ N=0 x N+j 1 N (N + 2j 1 ) = x j+1 (2j 1 + 1 − 4j 1 x + (2j 1 − 1)x 2 ) (1 − x) 4 . (5.5) This agrees with the result in (4.2) . In the second line, we have used the fact that n 1 + n 2 = m 1 + m 2 and that the number of as in (5.4) is 2j 1 . This calculation can easily be repeated with chemical potentials added for the angular momenta. Character Decomposition Character decomposition integrals are evaluated over the Haar measure of the group in question, in this case SO(4, 2). We can reduce these integrals to integrals over the maximal torus of the maximal compact subgroup SU (2) × SU (2) × SO(2) using the Weyl integration formula G f (g)dµ G = 1 |W | T f (t) α∈R (1 − exp(α(t)))dµ T . (6.1) where f (g) is a function satisfying f (hgh −1 ) = f (g) so that it only depends on the conjugacy class of g, and dµ G and dµ T are the Haar measures on the group G and the maximal torus T . α ∈ R means the product is over the roots of SO(4, 2). Each root corresponds to a generator in table (2.4) , for example the factor corresponding to K w is (1 − exp(−i(θ + θ 1 +θ 2 2 ))). The constant |W | is the order of the Weyl group in the compact case. In this non-compact case, it will diverge. We nevertheless obtain a useful orthogonality relation below where this constant is not relevant. An integral of characters over the group G becomes: G χ * [d,j 1 ,j 2 ] χ [d ′ ,j ′ 1 ,j ′ 2 ] dµ G = 1 |W | 2π 0 4π 0 4π 0 χ * [d,j 1 ,j 2 ] (θ, θ 1 , θ 2 )χ [d ′ ,j ′ 1 ,j ′ 2 ] (θ, θ 1 , θ 2 ) α∈R (1 − exp(i α · θ)) dθ 2π dθ 1 4π dθ 2 4π . (6.2) While the characters of the non-compact group SO(4, 2) are not orthogonal, the characters of the sets of states [d, j 1 , j 2 ] * can easily be shown to explicitly satisfy the following orthogonality relation: 1 4 χ * [d,j 1 ,j 2 ]χ [d ′ ,j ′ 1 ,j ′ 2 ] α∈R (1 − exp(i α · θ)))dµ T = δ d,d ′ δ j 1 ,j ′ 1 δ j 2 ,j ′ 2 . (6.3) This orthogonality is enough for us to decompose the partition function of YM into representations of the conformal group. In the case of non-compact groups, character decomposition integrals involve some dθ 2 4π ,(6.4) where α ∈ P means product only over the 4 roots corresponding to momentum generators P i as in (4.1) . It is clear that this integral has singularities along the contour of integration. To resolve this, we deform the contour inwards.This is equivalent to ignoring the contribution from the boundaries. To see why this is justified, expand 1 1 − xq i = x n q n i . (6.5) We have introduced new notation here. x = e iθ , q i = e i ±θ 1 ±θ 2 2 for i = 1, 2, 3, 4. Now recalling that x measures the scaling dimension or the energy, we see that that we should add a small imaginary part to θ which is equivalent to inserting an energy cutoff in the integral. With this pole prescription, the decomposition yields: χ [1,0,0] * [1,0,0] = ∞ d=2 χ [d, d−2 2 , d−2 2 ] . (6.6) These representations are generically short (barring [2, 0, 0]). We can count the operators in our theory manually to check this result. Using two scalar field representations, the primary operators in the tensor product at the first few levels are: φ 1 φ 2 [2, 0, 0], φ 2 ∂ µ φ 1 [3, 1 2 , 1 2 ], ∂ µ φ 1 ∂ ν φ 2 [4, 1, 1]. (6.7) which agrees with the decomposition. We will use this same pole prescription in performing the decomposition of the YM theory. The Integral The single trace partition function of Free Yang Mills on a sphere was calculated in [3,4]. The result was written as Z[θ, θ 1 , θ 2 ] = − φ(k) k ln(1 − z(kθ, kθ 1 , kθ 2 )), (7.1) where the 'single particle' partition function, z is given by: z = 1 + (x − x 3 ) i q i + x 4 − 1 i (1 − xq i ) . (7.2) We need to decompose this expression as a sum of characters of the conformal group. First, 1 − z = (1 − x 2 )(x 2 − ( q i )x + 1) i (1 − xq i ) . (7.3) So, the logarithm in (7.1) will separate the factors here into terms which we will integrate one at a time. Also, we have explicitlȳ χ d,j 1 ,j 2 = sin[(j 1 + 1/2)θ 1 ] sin[ θ 1 2 ] sin[(j 2 + 1/2)θ 2 ] sin[ θ 2 2 ] exp[idθ] i 1 (1 − xq i ) . (7.4) The measure of integration is: dM = 4 sin 2 [ θ 1 2 ] sin 2 [ θ 2 2 ] i (x − 1/q i )(x − q i ) d θ 2π d θ 1 4π d θ 2 4π . (7.5) Note that the θ 1 , θ 2 integrals go over 0, 4π. We will evaluate dM Z[θ, θ 1 , θ 2 ]χ * [d,j 1 ,j 2 ] . (7.6) Half of the measure cancels with the denominator of the character. The remaining part of the measure may be written as 4 sin θ 1 2 sin θ 2 2 i (x−q i ) = 4 sin θ 1 2 sin θ 2 2 {(x 4 +1)−4 cos θ 1 2 cos θ 2 2 (x 3 +x)+2x 2 (cos θ 1 +cos θ 2 +1)}. (7.7) We will do the integral over the 4 linear factors in the denominator of (7.3) first. The contribution from the partition function Z is − k,i φ(k) k log 1 1 − x k q k i = − k,i,n φ(k) x kn q kn i kn = − k,n φ(k) 4 cos knθ 1 2 cos knθ 2 2 x kn kn . (7.8) The integration over θ picks out coefficients of x d in the product of (7.8) and the measure (7.7) . The coefficient of x d in (7.8) is c(d) = − k|d φ(k) d 4 cos dθ 1 2 cos dθ 2 2 = −4 cos dθ 1 2 cos dθ 2 2 . (7.9) Hence, we need to deal with the integral c(d) + c(d − 4) − 4 cos θ 1 2 cos θ 2 2 (c(d − 1) + c(d − 3)) + 2(cos θ 1 + cos θ 2 + 1)c(d − 2) (cos j 1 θ 1 − cos(j 1 + 1)θ 1 )(cos j 2 θ 2 − cos(j 2 + 1)θ 2 ) d θ 1 4π d θ 2 4π . (7.10) With ∆ a b = δ a b + δ −a b the contribution from the factors in the denominator of (7.3) is given by I 1 [d, j 1 , j 2 ] = −∆ j 1 j 2 ∆ d 2j 1 + ∆ d−4 2j 1 + 2∆ d−2 2j 1 +∆ d−1±1 2j 1 ∆ d−1±1 2j 2 +∆ d−3±1 2j 1 ∆ d−3±1 2j 2 −∆ d−2±2 2j 1 ∆ d−2±2 2j 2 . (7.11) Next, we consider the (1 − x 2 ) factor in (7.3) . − log(1 − x 2k ) = x 2kn n . (7.12) This time, for the coefficient of x d , we have 2k|d 2φ(k) d = 1 d even 0 otherwise (7.13) Substituting this into the main integral, we find that for d > 4, we need to integrate 4 + 2(cos θ 1 + cos θ 2 + 1)dM d even, −4 cos θ 1 2 cos θ 2 2 dM d odd. (7.14) For d < 4 the expression above and below should be modified to drop terms that cannot contribute to the pole in x. Define I 0 [j 1 , j 2 ] = 4∆ 0 j 1 ∆ 0 j 2 + ∆ j 1 −1 0 ∆ j 2 0 + ∆ j 2 −1 0 ∆ j 1 0 . (7.15) With I = I 1 + I 0 , the contribution from the second term is I 2 [d, j 1 , j 2 ] = I[d, j 1 , j 2 ] − I[d, j 1 , j 2 + 1] − I[d, j 1 + 1, j 2 ] + I[d, j 1 + 1, j 2 + 1]. (7.16) Finally we consider the remaining quadratic term in (7.3) . We will call i q k i = 4 cos kθ 1 2 cos kθ 2 2 = α k , to save space. − log(1 − (α k x k − x 2k )) = n (α k x k − x 2k ) n n = p,q (−1) q α p k x (p+2q)k 1 p + q p + q q . (7.17) Again we will want to collect the coefficient of x d here. A term in the sum above contributes to this coefficient only if p + 2q = d/k. Also, this expression is summed over k against φ(k)/k. This means we need to consider the sum k|d d/k p=0 φ(k) k α p k (−1) ( d k −p)/2 2 d k + p d 2k + p/2 p . (7.18) We now look at a generic integral, integrating this term against cos Aθ 1 2 cos Bθ 2 2 . All terms occurring in the actual integral of the term in (7.17) may be reduced to this form. Use the identity 4 p cos kθ 1 2 p cos Aθ 1 2 cos kθ 2 2 p cos Bθ 2 2 = p 1 2 (p − A k ) p 1 2 (p − B k ) . (7.19) To shorten expressions, define p 2 = s, d 2k = x, A 2k = y, B 2k = z. This allows us to write the generic integral over the sum in (7.18)as S 1 [d, A, B] = k|(d,A,B) φ(k) x s=max(y,z) (−1) x−s s + x 2s 2s s − y 2s s − z 1 k(x + s) . (7.20) Now define S 2 [d, A, B] = S 1 [d, A, B] + S 1 [d − 4, A, B] − σ 1 ,σ 2 =±1/2 S 1 [d − 1, A + σ 1 , B + σ 2 ] + S 1 [d − 3, A + σ 1 , B + σ 2 ] + ρ 1 ,ρ 2 =±1 S 1 [d − 2, A + ρ 1 , B] + S 1 [d − 2, A, B + ρ 2 ] + 2S 1 [d − 2, A, B]. (7.21) We can now collect all the terms that appear in the integral over the quadratic term (7.17) I 3 [d, j 1 , j 2 ] = S 2 [d, j 1 , j 2 ] + S 2 [d, j 1 + 1, j 2 + 1] − S 2 [d, j 1 + 1, j 2 ] − S 2 [d, j 1 , j 2 + 1]. (7.22) Collecting the terms contributing, one finds the following enlightening result: dM Z[θ, θ 1 , θ 2 ]χ * [d,j 1 ,j 2 ] = I 2 [d, j 1 , j 2 ] + I 3 [d, j 1 , j 2 ]. (7.23) where I 2 is defined in (7.16) and I 3 in (7.22) . As we noted above, for d < 4 the expressions get modified. These sums are prohibitively difficult to evaluate by hand, but may be easily done with a computer. A Mathematica script which evaluates the sum above, with all special cases included, is available online at http://people.fas.harvard.edu/∼llhuang/conformalresult.nb. It is easy to list the operators in the theory at low scaling dimension: Z = − φ(k) k ln (1 + x)(x 2 − 4x + 1) (1 − x) 3 ≈ β d x d . (7.24) where β = 2 + √ 3 is the larger root of the quadratic term. This is the characteristic Hagedorn growth in the number of states. It is easy to verify, numerically, that (7.23)does reproduce the right growth in the density of states. As a final consistency check, it is necessary for (7.23)to sum to an integer for every value of d, j 1 , j 2 ; this is not at all apparent from the expression we have. Nevertheless, using some elementary number-theoretic results, we show, in the appendix, that our answer always sums to an integer. We list here the representation content of the theory up to dimension 9. N = 4 SYM In principle, it is not difficult to generalize the procedure above to the case of the N = 4 Yang Mills Theory. This theory has an exact superconformal symmetry. Representations of the Superconformal group are labeled by the highest weight under SO(4) × SO(2) and the R charges. These were originally classified in [11]. They were studied in [12], [13], [14]and are discussed in detail in [15]. It is easy to generalize the partition function by adding in chemical potentials for the R charges. This result can be read off from the appendix in [15]. Similarly, it is simple to generalize the result for the Haar measure and the characters [16]. Unfortunately, short representations in the superconformal case have a rather more intricate structure than in the conformal case and it is not always possible to write them as a difference of two long representations. This complication is not important in the spectrum of single trace operators in the N = 4 theory, because it is known that the only relevant short representations are the 1 2 , 1 2 BPS multiplets. The more serious complication is numerical. Performing the character decomposition involves finding the coefficient of a specified monomial in a given power series expansion. Since we have six chemical potentials in the supersymmetric case, the simple algorithms are O(d 6 ). Thus, the calculation quickly becomes unfeasible. In a set of papers [17,18,19,20,21], Bianchi et. al. conjectured that the spectrum of the free SYM theory may be obtained from the spectrum of type IIB theory on flat space through a specified algorithm. They verified their conjecture using a sieve procedure which allowed them to determine the spectrum up to scaling dimension 10. Further verification of this conjecture must await either a deeper understanding of their result or the development of more efficient numerical techniques. Conclusions Unitary representations of the Conformal Algebra must obey d ≥ j 1 + j 2 + 2, for j 1 j 2 = 0 and d ≥ j 1 + j 2 + 1 otherwise. Depending on whether either of these bounds is saturated, the characters of the conformal group fall into three classes. These are described in (4.2). The Free Yang Mills theory on a sphere has an exact conformal symmetry. Hence, its partition function may be written as a sum over the characters above. Formally, we have the result Z = N d,j 1 ,j 2 χ d,j 1 ,j 2 . (9.1) In this note, we performed this decomposition. We find that N d,j 1 ,j 2 is described by (7.23). Our formula demonstrates the correct asymptotics. Moreover, it is possible to prove that it always produces an integer. Appendix Consider (7.20), which has a sum over k and s. Schematically, the set of allowed k, s, x values is shown below. In the figure, for each value of k, s can range over the values demarcated by the horizontal line at the bottom and the outermost line. The critical point is to partition this large set of values correctly. We group the set of k, s values in subsets of the kind S x 0 ,s 0 = {s, k, x, y, z : x + s = p(x 0 + s 0 )} This foliation is indicated on the diagram. It is clear that in each partition, we have pk = g, where g is a constant. Second, we have gcd(s 0 + x 0 , x 0 − s 0 , s 0 − y 0 , s 0 − z 0 , s 0 ) = 1. Since (7.20) can be written as: α 1 g(x 0 + s 0 ) = α 2 2gs 0 = α 3 g(x 0 − s 0 ) = α 4 g(s 0 − y 0 ) ... where the α i are integral, it suffices to show that the sum is divisible by g to show that it is an integer. This leaves us to prove the following statement: For notational simplicity, we consider the simpler statement, k|g (−1) n 3 k n 1 k n 2 k φ(g/k) mod g = 0, (10.2) where n i are arbitrary integers with gcd n i = 1. Furthermore, we take g = p t , where p is prime. We take p = 2 so that (−1) n 3 k has the same sign. The generalization of this proof to generic g is straightforward. With t = 1, our sum is: n 1 n 2 (t − 1) + n 1 t n 2 t = 0( mod t). (10.3) where we have used n 1 t n 2 t = n 1 t n 2 t mod t. Assume the statement is true for t = n. For t = n + 1, our sum is: The first term is divisible by p n+1 by hypothesis. With n 3 = n 1 − n 2 , write the second term as: n 1 p n n 2 p n n 1 p n+1 (n 1 p n+1 − 1)...(n 1 p n + 1) − n 2 p n+1 ...(n 2 p n + 1)n 3 p n+1 ...(n 3 p n + 1) n 2 p n+1 ...n 2 p n n 3 p n+1 ..n 3 p n . We can cancel leading terms divisible by p n+1 in the numerator and denominator, but then we notice that subleading terms not divisible by p n+1 cancel in the numerator but not in the denominator. So the second term is also divisible by p n+1 . This proves our result. Fig. 1 : 1Grouping the Terms ) k(x 0 −s 0 ) (s 0 + x 0 00 subtleties. Written naively, these integrals have poles. To learn how to deal with thesepoles, consider the representation [1, 0, 0] × [1, 0, 0]. The decomposition of this tensor product by characters will involve integrals like (χ [1,0,0] ) 2χ * [d,j 1 ,j 2 ] dµ G = (cos j 1 θ 1 − cos(j 1 + 1)θ 1 )(cos j 2 θ 2 − cos(j 2 + 1)θ 2 ) exp[−idθ] exp[2iθ] α∈P (1 − e i α· θ ) dθ 2π dθ 1 4π AcknowledgementsThe authors would like to thank Shiraz Minwalla for suggesting this problem and providing guidance. We would like to thank Joseph Marsano, Andrew Neitzke and Xi Yin for several useful discussions. LLH would like to thank the Tata Institute of Fundamental Research, for its hospitality, while this work was being completed. The work of LLH was supported in part by an NSF Graduate Research Fellowship. This work was supported in part by DOE grant DE-FG03-91ER40654 and by the NSF career grant PHY-0239626. . J M Maldacena, arXiv:hep-th/9711200Adv. Theor. Math. Phys. 21113Int. J. Theor. Phys.J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200]. . O Aharony, S S Gubser, J M Maldacena, H Ooguri, Y Oz, arXiv:hep-th/9905111Phys. Rept. 323O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000) [arXiv:hep-th/9905111]. . O Aharony, J Marsano, S Minwalla, K Papadodimas, M Van Raamsdonk, arXiv:hep-th/0310285O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, arXiv:hep-th/0310285. The Hagedorn transition, deconfinement and N = 4 SYM theory. B Sundborg, arXiv:hep-th/9908001Nucl. Phys. B. 573349B. Sundborg, "The Hagedorn transition, deconfinement and N = 4 SYM theory," Nucl. Phys. B 573, 349 (2000) [arXiv:hep-th/9908001]. . D Berenstein, J M Maldacena, H Nastase, arXiv:hep-th/0202021JHEP. 020413D. Berenstein, J. M. Maldacena and H. Nastase, JHEP 0204, 013 (2002) [arXiv:hep- th/0202021]. . N Beisert, arXiv:hep-th/0407277Phys. Rept. 405N. Beisert, Phys. Rept. 405, 1 (2005) [arXiv:hep-th/0407277]. . I Bars, M Gunaydin, Commun. Math. Phys. 9131I. Bars and M. Gunaydin, Commun. Math. Phys. 91, 31 (1983). . N Beisert, G Ferretti, R Heise, K Zarembo, arXiv:hep-th/0412029N. Beisert, G. Ferretti, R. Heise and K. Zarembo, arXiv:hep-th/0412029. . S Minwalla, arXiv:hep-th/9712074Adv. Theor. Math. Phys. 2781S. Minwalla, Adv. Theor. Math. Phys. 2, 781 (1998) [arXiv:hep-th/9712074]. . G Mack, Commun. Math. Phys. 551G. Mack, Commun. Math. Phys. 55, 1 (1977). . V K Dobrev, V B Petkova, Phys. Lett. B. 162127V. K. Dobrev and V. B. Petkova, Phys. Lett. B 162, 127 (1985). . V K Dobrev, V B Petkova, Lett. Math. Phys. 9287V. K. Dobrev and V. B. Petkova, Lett. Math. Phys. 9, 287 (1985). . V K Dobrev, V B Petkova, Fortsch. Phys. 35537V. K. Dobrev and V. B. Petkova, Fortsch. Phys. 35, 537 (1987). V K Dobrev, V B Petkova, Proceedings. A.O. Barut and H.D. DoebnernullBerlinSpringer-Verlag261V. K. Dobrev and V. B. Petkova, Proceedings, eds. A.O. Barut and H.D. Doebner, Lecture Notes in Physics, Vol. 261 (Springer-Verlag, Berlin, 1986) pp. 300-308. . F A Dolan, H Osborn, arXiv:hep-th/0209056Annals Phys. 30741F. A. Dolan and H. Osborn, Annals Phys. 307, 41 (2003) [arXiv:hep-th/0209056]. . V K Dobrev, arXiv:hep-th/0406154V. K. Dobrev, arXiv:hep-th/0406154. . M Bianchi, J F Morales, H Samtleben, arXiv:hep-th/0305052JHEP. 030762M. Bianchi, J. F. Morales and H. Samtleben, JHEP 0307, 062 (2003) [arXiv:hep- th/0305052]. . N Beisert, M Bianchi, J F Morales, H Samtleben, arXiv:hep-th/0310292JHEP. 04021N. Beisert, M. Bianchi, J. F. Morales and H. Samtleben, JHEP 0402, 001 (2004) [arXiv:hep-th/0310292]. . N Beisert, M Bianchi, J F Morales, H Samtleben, arXiv:hep-th/0405057JHEP. 040758N. Beisert, M. Bianchi, J. F. Morales and H. Samtleben, JHEP 0407, 058 (2004) [arXiv:hep-th/0405057]. . M Bianchi, arXiv:hep-th/0409292M. Bianchi, arXiv:hep-th/0409292. . M Bianchi, arXiv:hep-th/0409304M. Bianchi, arXiv:hep-th/0409304.

We present a probabilistic model that uses both prosodic and lexical cues for the automatic segmentation of speech into topically coherent units. We propose two methods for combining lexical and prosodic information using hidden Markov models and decision trees. Lexical information is obtained from a speech recognizer, and prosodic features are extracted automatically from speech waveforms. We evaluate our approach on the Broadcast News corpus, using the DARPA-TDT evaluation metrics. Results show that the prosodic model alone is competitive with word-based segmentation methods. Furthermore, we achieve a significant reduction in error by combining the prosodic and word-based knowledge sources.

10.1162/089120101300346796

cs/0105037

5610e72291e24237514cd1f87d30826ada71a158

Integrating Prosodic and Lexical Cues for Automatic Topic Segmentation 31 May 2001 Gökhan Tür Bilkent University Bilkent University SRI International SRI International Dilek Hakkani-Tür Bilkent University Bilkent University SRI International SRI International Andreas Stolcke Bilkent University Bilkent University SRI International SRI International Elizabeth Shriberg Bilkent University Bilkent University SRI International SRI International Integrating Prosodic and Lexical Cues for Automatic Topic Segmentation 31 May 2001 We present a probabilistic model that uses both prosodic and lexical cues for the automatic segmentation of speech into topically coherent units. We propose two methods for combining lexical and prosodic information using hidden Markov models and decision trees. Lexical information is obtained from a speech recognizer, and prosodic features are extracted automatically from speech waveforms. We evaluate our approach on the Broadcast News corpus, using the DARPA-TDT evaluation metrics. Results show that the prosodic model alone is competitive with word-based segmentation methods. Furthermore, we achieve a significant reduction in error by combining the prosodic and word-based knowledge sources. Introduction Topic segmentation is the task of automatically dividing a stream of text or speech into topically homogeneous blocks. That is, given a sequence of (written or spoken) words, the aim of topic segmentation is to find the boundaries where topics change. Figure 1 gives an example of a topic change boundary from a broadcast news transcript. Topic segmentation is an important task for various language understanding applications, such as information extraction and retrieval, and text summarization. In this paper, we present our work on automatic detection of topic boundaries from speech input using both prosodic and lexical information. Other automatic topic segmentation systems have focused on written text and have depended largely on lexical information. This approach is problematic when segmenting speech. Firstly, relying on word identities can propagate automatic speech recognizer errors to the topic segmenter. Secondly, speech lacks typographic cues, as shown in Figure 1: there are no headers, paragraphs, sentence punctuation marks, or capitalized letters. Speech itself, on the other hand, provides an additional, nonlexical knowledge source through its durational, intonational, and energy characteristics, i.e., its prosody. Prosodic cues are known to be relevant to discourse structure in spontaneous speech (cf. Section 2.3) and can therefore be expected to play a role in indicating topic transitions. Furthermore, prosodic cues by their nature are relatively unaffected by word identity, and should therefore improve the robustness of lexical topic segmentation methods based on automatic speech recognition. . . . tens of thousands of people are homeless in northern china tonight after a powerful earthquake hit an earthquake registering six point two on the richter scale at least forty seven people are dead few pictures available from the region but we do know temperatures there will be very cold tonight minus seven degrees <TOPIC CHANGE> peace talks expected to resume on monday in belfast northern ireland former u. s. senator george mitchell is representing u. s. interests in the talks but it is another american center senator rather who was the focus of attention in northern ireland today here's a. b. c.'s richard gizbert the senator from america's best known irish catholic family is in northern ireland today to talk about peace and reconciliation a peace process does not mean asking unionists or nationalists to change or discard their identity or aspirations . . . Figure 1 An example of a topic boundary in a broadcast news word transcript. Topic segmentation research based on prosodic information has generally relied on hand-coded cues (with the notable exception of Hirschberg and Nakatani [1998]), or has not combined prosodic information with lexical cues (Litman and Passonneau [1995] is one example where lexical information was combined with hand-coded prosodic features for a related task). Therefore, the present work is the first that combines automatic extraction of both lexical and prosodic information for topic segmentation. The general framework for combining lexical and prosodic cues for tagging speech with various kinds of "hidden" structural information is a further development of our earlier work on sentence segmentation and disfluency detection for spontaneous speech (Shriberg, Bates, and Stolcke 1997;Stolcke and Shriberg 1996;), conversational dialog tagging , and information extraction from broadcast news (Hakkani-Tür et al. 1999). In the next section, we review previous work on topic segmentation. In Section 3, we describe our prosodic and language models as well as methods for combining them. Section 4 reports our experimental procedures and results. We close with some general discussion (Section 5) and conclusions (Section 6). Previous Work Work on topic segmentation is generally based on two broad classes of cues. On the one hand, one can exploit the fact that topics are correlated with topical content-word usage, and that global shifts in word usage are indicative of changes in topic. Quite independently, discourse cues, or linguistic devices such as discourse markers, cue phrases, syntactic constructions, and prosodic signals are employed by speakers (or writers) as generic indicators of endings or beginnings of topical segments. Interestingly, most previous work has explored either one or the other type of cue, but only rarely both. In automatic segmentation systems, word usage cues are often captured by statistical language modeling and information retrieval techniques. Discourse cues, on the other hand, are typically modeled with rule-based approaches or classifiers derived by machine-learning techniques (such as decision trees). Approaches Based on Word Usage Most automatic topic segmentation work based on text sources has explored topical word usage cues in one form or other. Kozima (1993) used mutual similarity of words in a sequence of text as an indicator of text structure. Reynar (1994) presented a method that finds topically similar regions in the text by graphically modeling the distribution of word repetitions. The method of Hearst (1994;1997) uses cosine similarity in a word vector space as an indicator of topic similarity. More recently, the U.S. Defense Advanced Research Projects Agency (DARPA) initiated the Topic Detection and Tracking (TDT) program to further the state of the art in finding and following new topics in a stream of broadcast news stories. One of the tasks in the TDT effort is segmenting a news stream into individual stories. Several of the participating systems rely essentially on word usage: model topics with unigram language models and their sequential structure with hidden Markov models (HMMs). Ponte and Croft (1997) extract related word sets for topic segments with the information retrieval technique of local context analysis, and then compare the expanded word sets. Approaches Based on Discourse and Combined Cues Previous work on both text and speech has found that cue phrases or discourse particles (items such as now or by the way), as well as other lexical cues, can provide valuable indicators of structural units in discourse (Grosz and Sidner 1986;Passonneau and Litman 1997, among others). In the TDT framework, the UMass "HMM" approach described in Allan et al. (1998) uses an HMM that models the initial, middle, and final sentences of a topic segment, capitalizing on discourse cue words that indicate beginnings and ends of segments. Aligning the HMM to the data amounts to segmenting it. Beeferman, Berger, and Lafferty (1999) combined a large set of automatically selected lexical discourse cues in a maximum-entropy model. They also incorporated topical word usage into the model by building two statistical language models: one static (topic independent) and one that adapts its word predictions based on past words. They showed that the log likelihood ratio of the two predictors behaves as an indicator of topic boundaries, and can thus be used as an additional feature in the exponential model classifier. Approaches Using Prosodic Cues Prosodic cues form a subset of discourse cues in speech, reflecting systematic duration, pitch, and energy patterns at topic changes and related locations of interest. A large literature in linguistics and related fields has shown that topic boundaries (as well as similar entities such as paragraph boundaries in read speech, or discourse-level boundaries in spontaneous speech) are indicated prosodically in a manner that is similar to sentence or utterance boundaries-only stronger. Major shifts in topic typically show longer pauses, an extra-high F0 onset or "reset", a higher maximum accent peak, greater range in F0 and intensity (Brown, Currie, and Kenworthy 1980;Grosz and Hirschberg 1992;Nakajima and Allen 1993;Geluykens and Swerts 1993;Ayers 1994;Hirschberg and Nakatani 1996;Nakajima and Tsukada 1997;Swerts 1997) and shifts in speaking rate (Brubaker 1972;Koopmans-van Beinum and van Donzel 1996;Hirschberg and Nakatani 1996). Such cues are known to be salient for human listeners; in fact, subjects can perceive major discourse boundaries even if the speech itself is made unintelligible via spectral filtering (Swerts, Geluykens, and Terken 1992). Work in automatic extraction and computational modeling of these characteristics has been more limited, with most of the work in computational prosody modeling dealing with boundaries at the sentence level or below. However, there have been some studies of discourse-level boundaries in a computational framework. They differ in various ways, such as type of data (monolog or dialog, human-human or human-computer), type of features (prosodic and lexical versus prosodic only), which features are considered available (e.g., utterance boundaries or no boundaries), to what extent features are automatically extractable and normalizable, and the machine learning approach. Be-cause of these vast difference the overall results cannot be compared directly to each other or to our work, but we describe three of the approaches briefly here. An early study by Litman and Passonneau (1995) used hand-labeled prosodic boundaries and lexical information, but applied machine learning to a training corpus and tested on unseen data. The researchers combined pause, duration, and hand-coded intonational boundary information with lexical information from cue phrases (such as and and so). Additional knowledge sources included complex relations, such as coreference of noun phrases. Work by Swerts and Ostendorf (1997) used prosodic features that in principle could be extracted automatically, such as pitch range, to classify utterances from human-computer task-oriented dialog into two categories: initial or noninitial in the discourse segment. The approach used CART-style decision trees to model the prosodic features, as well as various lexical features that could also in principle be estimated automatically. In this case, utterances were presegmented, so the task was to classify segments rather than find boundaries in continuous speech; the features also included some (such as type of boundary tone) that may not be easy to extract robustly across speaking styles. Finally, Hirschberg and Nakatani (1998) proposed a prosodyonly front end for tasks such as audio browsing and playback, which could segment continuous audio input into meaningful information units. They used automatically extracted pitch, energy, and "other" features (such as the cross-correlation value used by the pitch tracker in determining the estimate of F0) as inputs to CART-style trees, and aimed to predict major discourse-level boundaries. They found various effects of frame window length and speakers, but concluded overall that prosodic cues could be useful for audio browsing applications. The Approach Topic segmentation in the paradigm used in this study and others (Allan et al. 1998) proceeds in two phases. In the first phase, the input is divided into contiguous strings of words assumed to belong to the same topic. We refer to this step as chopping. For example, in textual input, the natural units for chopping are sentences (as can be inferred from punctuation and capitalization), since we can assume that topics do not change in mid-sentence. 1 For continuous speech input, the choice of chopping criteria is less obvious; we compare several possibilities in our experimental evaluation. Here, for simplicity, we will use sentence to refer to units of chopping, regardless of the criterion used. In the second phase, the sentences are further grouped into contiguous stretches belonging to one topic, i.e., the sentence boundaries are classified into topic boundaries and nontopic boundaries. 2 Topic segmentation is thus reduced to a boundary classification problem. We will use B to denote the string of binary boundary classifications. Furthermore, our two knowledge sources are the (chopped) word sequence W and the stream of prosodic features F . Our approach aims to find the segmentation B with highest probability given the information in W and F argmax B P (B|W, F )(1) using statistical modeling techniques. 1 Similarly, it is sometimes assumed for topic-segmentation purposes that topics change only at paragraph boundaries (Hearst 1997). 2 We do not consider the problem of detecting recurring, discontinuous instances of the same topic, a task known as topic tracking in the TDT paradigm (Doddington 1998). In the following subsections, we first describe the prosodic model of the dependency between prosody F and topic segmentation B; then, the language model relating words W and B; and finally, two approaches for combining the models. Prosodic Modeling The job of the prosodic model is to estimate the posterior probability (or, alternatively, likelihood) of a topic change at a given word boundary, based on prosodic features extracted from the data. For the prosodic model to be effective, one must devise suitable, automatically extractable features. Feature values extracted from a corpus can then be used in training probability estimators and to select a parsimonious subset of features for modeling purposes. We discuss each of these steps in turn in the following sections. Features. We started with a large collection of features capturing two major aspects of speech prosody, similar to our previous work (Shriberg, Bates, and Stolcke 1997): • Duration features: duration of pauses, duration of final vowels and final rhymes, 3 and versions of these features normalized both for phone durations and speaker statistics. • Pitch features: fundamental frequency (F0) patterns preceding and following the boundary, F0 patterns across the boundary, and pitch range relative to the speaker's baseline. We processed the raw F0 estimates (obtained with ESPS signal processing software from Entropic Research Laboratory [1993]), with robustness-enhancing techniques developed by Sönmez et al. (1998). We did not use amplitude-or energy-based features since exploratory work showed these to be much less reliable than duration and pitch and largely redundant given the above features. One reason for omitting energy features is that, unlike duration and pitch, energy-related measurements vary with channel characteristics. Since channel properties vary widely in broadcast news, features based on energy measures can correlate with shows, speakers, and so forth, rather than with the structural locations in which we were interested. We included features that, based on the descriptive literature, should reflect breaks in the temporal and intonational contour. We developed versions of such features that could be defined at each interword boundary, and that could be extracted by completely automatic means (no human labeling). Furthermore, the features were designed to be as independent of word identities as possible, for robustness to imperfect recognizer output. A brief characterization of the informative features for the segmentation task is given with our results in Section 4.6. Since the focus here is on computational modeling we refer the reader to a companion paper for a detailed description of the acoustic processing and prosodic feature extraction. Decision trees. Any of a number of probabilistic classifiers (such as neural networks, exponential models, or naïve Bayes networks) could be used as posterior probability estimators. As in past prosodic modeling work (Shriberg, Bates, and Stolcke 1997), we chose CART-style decision trees (Breiman et al. 1984), as implemented by the IND package (Buntine and Caruana 1992), because of their ability to model feature interactions, to deal with missing features, and to handle large amounts of training data. The foremost reason for our preference for decision trees, however, is that the learned models can be inspected and diagnosed by human investigators. This ability is crucial for understanding what and how features are used, and for debugging the feature extraction process itself. 4 Let F i be the features extracted from a window around the ith potential topic boundary (chopping boundary), and let B i be the boundary type (boundary/no-boundary) at that position. We trained decision trees to predict the ith boundary type, i.e., to estimate P (B i |F i , W ). The decision is only weakly conditioned on the word sequence W , insofar as some of the prosodic features depend on the phonetic alignment of the word models (which we will denote with W t ). We can thus expect the prosodic model estimates to be robust to recognition errors. The decision tree paradigm also allows us to add, and automatically select, other (nonprosodic) features that might be relevant to the task. Feature selection. The greedy nature of the decision tree learning algorithm implies that larger initial feature sets can give worse results than smaller subsets. Furthermore, it is desirable to remove redundant features for computational efficiency and to simplify the interpretation of results. For this purpose we developed an iterative feature selection "wrapper" algorithm (John, Kohavi, and Pfleger 1994) that finds useful, task-specific feature subsets. The algorithm combines elements of a brute-force search with previously determined heuristics about good groupings of features. The algorithm proceeds in two phases: In the first phase, the number of features is reduced by leaving out one feature at a time during tree construction. A feature whose removal increases performance is marked as to be avoided. The second phase then starts with the reduced feature set and performs a beam search over all possible subsets to maximize tree performance. We used entropy reduction in the overall tree (after cross-validation pruning) as a metric for comparing alternative feature subsets. Entropy reduction is the difference in entropy between the prior class distribution and the posterior distribution estimated by the tree, as measured on a held-out set; it is a more fine-grained metric than classification accuracy, and is also more relevant to the model combination approach described later. Training data. To train the prosodic model, we automatically aligned and extracted features from 70 hours (about 700,000 words) of the Linguistic Data Consortium (LDC) 1997 Broadcast News (BN) corpus. Topic boundary information determined by human labelers was extracted from the SGML markup that accompanies the word transcripts of this corpus. The word transcripts were aligned automatically with the acoustic waveforms to obtain pause and duration information, using the SRI Broadcast News recognizer (Sankar et al. 1998). Lexical Modeling Lexical information in our topic segmenter is captured by statistical language models (LMs) embedded in an HMM. The approach is an extension of the topic segmenter developed by Dragon Systems for the TDT2 effort , which was based purely on topical word distributions. We extend it to also capture lexical and (as described in Section 3.3) prosodic discourse cues. Model structure. The overall structure of the model is that of an HMM (Rabiner and Juang 1986) in which the states correspond to topic clusters T j , and the observations are sentences (or chopped units) W 1 , . . . , W N . The resulting HMM, depicted in Figure 2, forms a complete graph, allowing for transitions between any two topic clusters. Note that it is not necessary that the topic clusters correspond exactly to the actual topics to be located; for segmentation purposes it is sufficient that two adjacent actual topics are unlikely to be mapped to the same induced cluster. The observation likelihoods for the HMM states, P (W i |T j ), represent the probability of generating a given sentence W i in a particular topic cluster T j . We automatically constructed 100 topic cluster LMs, using the multipass k-means algorithm described in . Since the HMM emissions are meant to model the topical usage of words, but not topic-specific syntactic structures, the LMs consist of unigram distributions that exclude stop words (high-frequency function and closed-class words). To account for unobserved words we interpolate the topic clusterspecific LMs with the global unigram LM obtained from the entire training data. The observation likelihoods of the HMM states are then computed from these smoothed unigram LMs. All HMM transitions within the same topic cluster are given probability one, whereas all transitions between topics are set to a global topic switch penalty (TSP) that is optimized on held-out training data. The TSP parameter allows trading off between false alarms and misses. Once the HMM is trained, we use the Viterbi algorithm (Viterbi 1967;Rabiner and Juang 1986) to search for the best state sequence and corresponding segmentation. Note that the transition probabilities in the model are not normalized to sum to one; this is convenient and permissible since the output of the Viterbi algorithm depends only on the relative weight of the transition weights. We augmented the Dragon segmenter with additional states and transitions to also capture lexical discourse cues. In particular, we wanted to model the initial and final sentences in each topic segment, as these often contain formulaic phrases and keywords used by broadcast speakers (From Washington, this is . . . , And now . . . ). We added two additional states BEGIN and END to the HMM ( Figure 3) to model these sentences. Likelihoods for the BEGIN and END states are obtained as the unigram language model probabilities of the initial and final sentences, respectively, of the topic segments in the training data. Note that a single BEGIN and END state are shared for all topics. Best results were obtained by making traversal of these states optional in the HMM topology, presumably because some initial and final sentences are better modeled by the topicspecific LMs. The resulting model thus effectively combines the Dragon and UMass HMM topic segmentation approaches described in Allan et al. (1998). In preliminary experiments, we observed a 5% relative reduction in segmentation error with initial and final states over the baseline HMM topology of Figure 2. Therefore, all results reported later use an HMM topology with initial and final states. Note that, since the topic-initial and topicfinal states are optional, our training of the model is suboptimal. Instead of labeling all topic-initial and topic-final training sentences as data for the corresponding state, we would expect further improvements by training the HMM in unsupervised fashion using the Baum-Welch algorithm (Baum et al. 1970;Rabiner and Juang 1986). Training data. Topic unigram language models were trained from the pooled TDT Pilot and TDT2 training data (Cieri et al. 1999), covering transcriptions of broadcast news from January 1992 through June 1994 and from January 1998 through February 1998, respectively. These corpora are similar in style, but do not overlap with the 1997 LDC BN corpus from which we selected our prosodic training data and the evaluaton . . test set. For training the language models we removed stories with fewer than 300 and more than 3000 words, leaving 19,916 stories with an average length of 538 words (including stop words). Model Combination We are now in a position to describe how lexical and prosodic information can be combined for topic segmentation. As discussed before, the LMs in the HMM capture topical word usage as well as lexical discourse cues at topic transitions, whereas a decision tree models prosodic discourse cues. We expect that these knowledge sources are largely independent, so their combination should yield significantly improved performance. Below we present two approaches for building a combined statistical model that performs topic segmentation using all available knowledge sources. For both approaches it is convenient to associate a "boundary" pseudotoken with each potential topic boundary (i.e., with each sentence boundary). Correspondingly, we introduce, into the HMM, new states that emit these boundary tokens. No other states emit boundary tokens; Structure of an HMM with topic BEGIN and END states. TSP represents the topic switch penalty. therefore each sentence boundary must align with one of the boundary states in the HMM. As shown in Figure 4, there are two boundary states for each topic cluster, one representing a topic transition and the other representing a topic-internal transition between sentences. Unless otherwise noted, the observation likelihoods for the boundary states are set to unity. The addition of boundary states allows us to compute the model's prediction of topic changes as follows: Let B 1 , . . . , B C denote the topic boundary states and, similarly, let N 1 , . . . , N C denote the nontopic boundary states, where C is the number of topic clusters. Using the forward-backward algorithm for HMMs (Rabiner and Juang 1986), we can compute P (q i = B j |W ) and P (q i = N j |W ), the posterior probabilities that one of these states is occupied at boundary i. The model's prediction of a topic boundary is simply the sum over the corresponding state posteriors: P HMM (B i = yes|W ) = C j=1 P (q i = B j |W )(2) BEGIN Figure 4 Structure of the final HMM with fictitious boundary states used for combining language and prosodic models. In the figure, states B1, B2, . . . , B100 represent the presence of a topic boundary, whereas states N1, N2, . . . , N100 represent topic-internal sentence boundaries. TSP is the topic switch penalty. P HMM (B i = no|W ) = C j=1 P (q i = N j |W ) (3) = 1 − P HMM (B i = yes|W ) Model combination in the decision tree. Decision trees allow the training of a single classifier that takes both lexical and prosodic features as input, provided we can compactly encode the lexical information for the decision tree. We compute the posterior probability P HMM (B i = yes|W ) as shown above, to summarize the HMM's belief in a topic boundary based on all available lexical information W . The posterior value is then used as an additional input feature to the prosodic decision tree, which is trained in the usual manner. During testing, we declare a topic boundary whenever the tree's overall posterior estimate P DT (B i |F i , W ) exceeds some threshold. The threshold may be varied to trade off false alarms for miss errors, or to optimize an overall cost function. Using HMM posteriors as decision tree features is similar in spirit to the knowledge source combination approaches used by Beeferman, Berger, and Lafferty (1999) and Reynar (1999), who also used the output of a topical word usage model as input to an overall classifier. In previous work we used the present approach as one of the knowledge source combination strategies for sentence and disfluency detection in spontaneous speech. Model combination in the HMM. An alternative approach to knowledge source combination uses the HMM as the top-level model. In this approach, the prosodic decision tree is used to estimate likelihoods for the boundary states of the HMM, thus integrating the prosodic evidence into the HMM's segmentation decisions. More formally, let Q = (r 1 , q 1 , . . . , r i , q i , . . . , r N , q N ) be a state sequence through the HMM. The model is constructed such that the states r i representing topic (or BE-GIN/END) clusters alternate with the states q i representing boundary decisions. As in the baseline model, the likelihoods of the topic cluster states T j account for the lexical observations: P (W i |r i = T j ) = P (W i |T j )(4) as estimated by the unigram LMs. Now, in addition, we let the likelihood of the boundary state at position i reflect the prosodic observation F i . Recall that, like W i , F i refers to complete sentence units; specifically, F i denotes the prosodic features of the ith boundary between such units. P (F i |q i = B j , W ) = P (F i |B i = yes, W ) P (F i |q i = N j , W ) = P (F i |B i = no, W ) for all j = 1, . . . , C(5) Using this construction, the product of all state likelihoods will give the overall likelihood, accounting for both lexical and prosodic observations: N i=1 P (W i |r i ) N i=1 P (F i |q i , W ) = P (W, F |Q)(6) Applying the Viterbi algorithm to the HMM will thus return the most likely segmentation conditioned on both words and prosody, which is our goal. Although decomposing the likelihoods as shown allows prosodic observations to be conditioned on the words W , we use only the phonetic alignment information W t from the word sequence W in our prosodic models, ignoring the word identities, so as to make them more robust to recognition errors. The likelihoods P (F i |B i , W t ) for the boundary states can now be obtained from the prosodic decision tree. Note that the decision tree estimates posteriors P DT (B i |F i , W t ). These can be converted to likelihoods using Bayes rule as in P (F i |B i , W t ) = P (F i |W t )P DT (B i |F i , W t ) P (B i |W t ) .(7) The term P (F i |W t ) is a constant for all decisions B i and can thus be ignored when applying the Viterbi algorithm. Next, we approximate P (B i |W t ) ≈ P (B i ), justified by the fact that the W t contains only information about start and end times of phones and words, but not directly about word identities. Instead of explicitly dividing the posteriors we prefer to downsample the training set to make P (B i = yes) = P (B i = no) = 1 2 . A beneficial side effect of this approach is that the decision tree models the lower-frequency events (topic boundaries) in greater detail than if presented with the raw, highly skewed class distribution. As is often the case when combining probabilistic models of different types, it is advantageous to weight the contributions of the language models and the prosodic trees relative to each other. We do so by introducing a tunable model combination weight (MCW), and by using P DT (F i |B i , W t ) MCW as the effective prosodic likelihoods. The value of MCW is optimized on held-out data. Experiments and Results To evaluate our topic segmentation models we carried out experiments in the TDT paradigm. We first describe our test data and the evaluation metrics used to compare model performance. We then give results obtained with individual knowledge sources, followed by results using the combined models. Test Data We evaluated our system on three hours (6 shows, about 53,000 words) of the 1997 LDC BN corpus. The threshold for the model combination in the decision tree and the topic switch penalty were optimized on the larger development training set of 104 shows, which includes the prosodic model training data. The MCW for the model combination in the HMM was optimized using a smaller held-out set of 10 shows of about 85,000 words total size, separate from the prosodic model training data. We used two test conditions: forced alignments using the true words, and recognized words as obtained by a simplified version of the SRI Broadcast News recognizer (Sankar et al. 1998), with a word error rate of 30.5%. Our aim in these experiments was to use fully automatic recognition and processing wherever possible. For practical reasons, we departed from this strategy in two areas. First, for word recognition, we used the acoustic waveform segmentations provided with the corpus (which also included the location of nonnews material, such as commercials and music). Since current BN recognition systems perform this segmentation automatically with very good accuracy and with only a few percentage points penalty in word error rate (Sankar et al. 1998), we felt the added complication in experimental setup and evaluation was not justified. Second, for prosodic modeling, we used information from the corpus markup concerning speaker changes and the identity of frequent speakers (e.g., news anchors). Automatic speaker segmentation and labeling is possible, though with errors (Przybocki and Martin 1999). Nevertheless, our use of speaker labels was motivated by the fact that meaningful prosodic features may require careful normalization by speaker, and unreliable speaker information would have made the analysis of prosodic feature usage much less meaningful. Evaluation Metrics We have adopted the evaluation paradigm used by the TDT2-Topic Detection and Tracking Phase 2 (Doddington 1998) program, allowing fair comparisons of various approaches both within this study and with respect to other recent work. Segmentation accuracy was measured using TDT evaluation software from NIST, which implements a variant of an evaluation metric suggested by Beeferman, Berger, and Lafferty (1999). The TDT segmentation metric is different from those used in most previous topic segmentation work, and therefore merits some discussion. It is designed to work on data streams without any potential topic boundaries, such as paragraph or sentence boundaries, being given a priori. It also gives proper partial credit to segmentation decisions that are close to actual boundaries; for example, placing a boundary one word from an actual boundary is considered a lesser error than if the hypothesized boundary is off by, say, 100 words. The evaluation metric reflects the probability that two positions in the corpus probed at random and separated by a distance of k words are correctly classified as belonging to the same story or not. If the two words belong to the same topic segment, but are erroneously claimed to be in different topic segments by the segmenter, then this will increase the system's false alarm probability. Conversely, if the two words are in different topic segments, but are erroneously marked to be in the same segment, this will contribute to the miss probability. The false alarm and miss rates are defined as averages over all possible probe positions with distance k. Formally, miss and false alarm rates are computed as 5 Here sys can be ref to denote the reference (correct) segmentation, or hyp to denote the segmenter's decision. P Miss = s Ns−k i=1 d s hyp (i, i + k) × (1 − d s ref (i, i + k)) s Ns−k i=1 (1 − d s ref (i, i + k)) (8) P FalseAlarm = s Ns−k i=1 (1 − d s hyp (i, i + k)) × d s ref (i, i + k) s Ns−k i=1 d s ref (i, i + k)(9) For audio sources an analogous metric is defined where segmentation decisions (same or different topic) are probed at a time-based distance ∆: P Miss = s Ts−∆ t=0 d s hyp (t, t + ∆) × (1 − d s ref (t, t + ∆))dt s Ts−∆ t=0 (1 − d s ref (t, t + ∆))dt (10) P FalseAlarm = s Ts−∆ t=0 (1 − d s hyp (t, t + ∆)) × d s ref (t, t + ∆)dt s Ts−∆ t=0 d s ref (t, t + ∆)dt(11) where the integration is over the entire duration of all stories of the shows in the test corpus, and where d s sys (t 1 , t 2 ) =    1 if times t 1 and t 2 in show s are deemed by sys to be within the same story 0 otherwise We used the same parameters as used in the official TDT2 evaluation: k = 50 and ∆ = 15 seconds. Furthermore, again following NIST's evaluation procedure, we combine miss and false alarm rates into a single segmentation cost metric where the C Miss = 1 is the cost of a miss, C FalseAlarm = 1 is the cost of a false alarm, and P Seg = 0.3 is the a priori probability of a segment being within an interval of k words or ∆ seconds on the TDT2 training corpus. 6 C Seg = C Miss × P Miss × P seg + C FalseAlarm × P FalseAlarm × (1 − P seg )(12) Chopping Unlike written text, the output of the automatic speech recognizer contains no sentence boundaries. Therefore, chopping text into (pseudo)sentences is a nontrivial problem when processing speech. Some presegmentation into roughly sentence-length units is necessary since otherwise the observations associated with HMM states would comprise too few words to give robust likelihoods of topic choice, causing poor performance. We investigated chopping criteria based on a fixed number of words (FIXED), at speaker changes (TURN), at pauses (PAUSE), and, for reference, at actual sentence boundaries (SENTENCE) obtained from the transcripts. Table 1 gives the error rates for the four conditions, using the true word transcripts of the larger development data set. For the PAUSE condition, we empirically determined an optimal minimum pause duration threshold to use. Specifically, we considered pauses exceeding 0.575 of a second as potential topic boundaries in this (and all later) experiments. For the FIXED condition, a block length of 10 words was found to work best. We conclude that a simple prosodic feature, pause duration, is an excellent criterion for the chopping step, giving comparable or better performance than standard sentence boundaries. Therefore, we used pause duration as the chopping criterion in all further experiments. Source-specific Model Tuning As mentioned earlier, the segmentation models contain global parameters (the topic transition penalty of the HMM and the posterior threshold for the combined decision tree) to trade false alarms for miss errors. Optimal settings for these parameters depend on characteristics of the source, in particular on the relative frequency of topic changes. Since broadcast news programs come from identified sources it is useful and legitimate to optimize these parameters for each show type. 7 We therefore optimized the global parameter for each model to minimize the segmentation cost on the training corpus (after training all other model parameters in a source-independent fashion). Compared to a baseline using source-independent global TSP and threshold, the 6 Another parameter in the NIST evaluation is the deferral period, i.e., the amount of look-ahead before a segmentation decision is made. In all our experiments we allowed unlimited deferral, effectively until the end of the news show being processed. 7 Shows in the 1997 BN corpus come from eight sources: ABC World News Tonight, CNN Headline News, CNN Early Prime, PRI The World, CNN Prime News, CNN The World Today, C-SPAN Public Policy, and C-SPAN Washington Journal. Six of these occurred in the test set. Table 2 Summary of error rates with the language model only (LM), the prosody model only (PM), the combined decision tree (CM-DT), and the combined HMM (CM-HMM). (a) shows word-based error metrics, (b) shows time-based error metrics. In both cases a "chance" classifier that labels all potential boundaries as nontopic would achieve 0.3 weighted segmentation cost. source-dependent models showed between 5 and 10% relative error reduction. All results reported below use the source-dependent approach. Table 2 shows the results for both individual knowledge sources (words and prosody), as well as for the combined models (decision tree and HMM). It is worth noting that the prosody-only results were obtained by running the combined HMM without language model likelihoods; this approach gave better performance than using the prosodic decision trees directly as classifiers. Segmentation Results Both word-and time-based metrics are given; they exhibit generally very similar results. Another dimension of the evaluation is the use of correct word transcripts (forced alignments) versus automatically recognized words. Again, results along this dimension are very similar, with some exceptions noted below. Comparing the individual knowledge sources, we observe that prosody alone does somewhat better than the word-based HMM alone. The types of errors made differ consistently: the prosodic model has a higher false alarm rate, while the word-LMs have more miss errors. The prosodic model shows more false alarms because many regular sentence boundaries often show characteristics similar to those of topic boundaries. It also suggests that both models could be combined by letting the prosodic model selects candidate topic boundaries that are then filtered using lexical information. The combined models generally improve on the individual knowledge sources. 8 In the word-based evaluation, the combined decision tree (DT) reduced overall segmentation cost by 19% over the language model on true words (17% on recognized words). The combined HMM gave even better results: 27% and 24% improvement in the error rate over the language model for the true and recognized words, respectively. Looking again at the breakdown of errors, we can see that the two model combination approaches work quite differently: the combined DT has about the same miss rate as the LM, but lower false alarms. The combined HMM, by contrast, combines a miss rate as low as (or lower than) that of the prosodic model with the lower false alarm rate of the LM, suggesting that the functions of the two knowledge sources are complementary, as discussed above. Furthermore, the different error patterns of the two combination approaches suggest that further error reductions could be achieved by combining the two hybrid models. 9 The trade-off between false alarms and miss probabilities is shown in more detail in Figure 5, which plots the two error metrics against each other. Note that the false alarm rate does not reach 1 because the segmenter is constrained by the chopping algorithm: the pause criterion prevents the segmenter from hypothesizing topic boundaries everywhere. Figure 6. We can identify four different kinds of features used in the tree, listed below. For each feature type, we give the feature names found in the tree and the relative feature usage, an approximate measure of feature importance (Shriberg, Bates, and Stolcke 1997). Relative feature usage is computed as the relative frequency with which features of a given type are queried in the tree, over a held-out test set. Decision Tree for the Prosody-only Model 1.Pause duration (PAU DUR, 42.7% usage). This feature is the duration of the nonspeech interval occurring at the boundary. The importance of pause duration is underestimated here because, as explained earlier, pause durations are already used during the chopping process, so that the decision tree is applied only to boundaries exceeding a certain duration. Separate experiments using boundaries below our chopping threshold show that the tree also distinguishes shorter pause durations for segmentation decisions. 2.F0 differences across the boundary (F0K LR MEAN KBASELN and F0K WRD DIFF MNMN NG, 35.9% usage). These features compare the mean F0 of the word preceding the boundary (measured from voiced regions within that word) to either the speaker's estimated baseline F0 (F0K LR MEAN KBASELN) or to the mean F0 of the word following the boundary (F0K WRD DIFF MNMN N). Both features were computed based on a log-normal scaling of F0. Other measures (such as minimum or maximum F0 in the word or preceding window) as well as other normalizations (based on F0 toplines, or non-log-based scalings) were included in the initial feature set, but were not selected in the best-performing tree. The baseline feature captures a pitch range effect, and is useful at boundaries where the speaker changes (since range here is compared only within-speaker). The second feature captures the relative size of the pitch change at the boundary, but of course is not meaningful at speaker boundaries. 3.Turn features (TURN F and TURN TIME, 14.6% usage). These features reflect the change of speakers. TURN F indicates whether a speaker change occurred at the boundary, while TURN TIME measures the time passed since the start of the current turn. 4.Gender (GEN, 6.8% usage). This feature indicates the speaker gender right before a potential boundary. Inspection of the tree reveals that the purely prosodic features (pause duration and F0 differences) are used as the prosody literature suggests. The longer the observed pause, the more likely a boundary corresponds to a topic change. Also, the closer a speaker comes to his or her F0 baseline, or the larger the difference to the F0 following a boundary, the more likely a topic change occurs. These features thus correspond to the well-known phenomena of boundary tones and pitch reset that are generally associated with sentence boundaries (Vaissière 1983). We found these indicators of sentences boundaries to be particularly pronounced at topic boundaries. While turn and gender features are not prosodic features per se, they do interact closely with them since prosodic measurements must be informed by and carefully normalized for speaker identity and gender, 10 and it is therefore natural to include them in Figure 7 The decision tree of the combination model. a prosodic classifier. Not surprisingly, we find that turn boundaries are positively correlated with topic boundaries, and that topic changes become more likely the longer a turn has been going on. Interestingly, speaker gender is used by the decision tree for several reasons. One reason is stylistic differences between males and females in the use of F0 at topic boundaries. This is true even after proper normalization, e.g., equating the gender-specific nontopic boundary distributions. In addition, we found that nontopic pauses (i.e., chopping boundaries) are more likely to occur in male speech. It could be that male speakers in BN are assigned longer topic segments on average, or that male speakers are more prone to pausing in general, or that male speakers dominate the spontaneous speech portions, where pausing is naturally more frequent. The details of this gender effect await further study. Figure 7 depicts the decision tree that combines the HMM language model topic decisions with prosodic features (see Section 3.3.1). Again, we list the features used with their relative feature usages. 1.Language model posterior (POST TOPIC,49.3% usage). This is the posterior probability P (B i = yes|W ) computed from the HMM. Decision Tree for the Combined Model 2.Pause duration (PAU DUR,49.3% usage). This feature is the same as described for the prosody-only model. 3.F0 differences across the boundary (F0K WRD DIFF HILO N and F0K LR MEAN KBASELN, 1.4% usage). These features are similar to those found for the prosody-only tree. The only difference is that for the first feature, the comparison of F0 values across the boundary is done by taking the maximum F0 of the previous word and the minimum F0 of the following word-rather than the mean for both cases. The decision tree found for the combined task is smaller and uses fewer features than the one trained with prosodic features only, for two reasons. First, the LM posterior feature is found to be highly informative, superseding the selection of many of the lowfrequency features previously found. Furthermore, as explained in Section 3.3.2, the prosody-only tree was trained on a downsampled dataset that equalizes the priors for topic and nontopic boundaries, as required for integration into the HMM. A welcome side effect of this procedure is that it forces the tree to model the less frequent class (topic boundaries) in much greater detail than if the tree were trained on the raw class distribution, as is the case here. Because of its small size, the tree in Figure 7 is particularly easy to interpret. The toplevel split is based on the LM posterior. The right branch handles cases where words are highly indicative of a topic boundary. However, for short pauses the tree queries further prosodic features to prevent false alarms. Specifically, short pauses must be accompanied both by an F0 close to the speaker's baseline and by a large F0 reset to be deemed topic boundaries. Conversely, if the LM posteriors are low (left top-level branch), but the pause is very long, the tree still outputs a topic boundary. Comparison of Model Combination Approaches Results indicate that the model combination approach using an HMM as the top-level model works better than the combined decision tree. While this result deserves more investigation we can offer some preliminary insights. We found it difficult to set the posterior probability thresholds for the combined decision tree in a robust way. As shown by the "CM-DT" curve in Figure 5, there is a large jump in the false alarm/miss trade-off for the combined tree, in contrast to the combined HMM approach, which controls the trade-off by a changing topic switch penalty. This occurs because posterior probabilities from the decision tree do not vary smoothly; rather, they vary in steps corresponding to the leaves of the tree. The discontinuous character of the thresholded variable makes it hard to estimate a threshold on the training data that performs robustly on the test data. This could account for the poor result on the time-based metrics for the combined tree (where the threshold optimized on the training data was far from optimal on the test set; see footnote 8). The same phenomenon is reflected in the fact that the prosody-only tree gave better results when embedded in an HMM without LM likelihoods than when used by itself with a posterior threshold. Contributions of Different Feature Types We saw in Section 4.6 that pause duration is by far the single most important feature in the prosodic decision tree. Furthermore, speaker changes are queried almost as often as the F0-related features. Pause durations can be obtained using standard speech recognizers, and are in fact used by many current TDT systems (see Section 4.10). Speaker changes are not prosodic features per se, and would be detected independently from the prosodic features proper. To determine if prosodic measurements beyond pause and speaker information improve topic segmentation accuracy, we tested systems that consisted of the HMM with the usual topic LMs, plus a decision tree that had access only to various subsets of pause-and speaker-related features, without using any of the F0based features. Decision tree and HMM were combined as described in Section 3.3.2. Table 3 shows the results of the system using only topic language models (LM) as well as combined systems using all prosodic features (CM-HMM-all), only pause duration and turn features (CM-HMM-pause-turn), and using only pause duration, turn, and gender features (CM-HMM-pause-turn-gender). These results show that by using only pause duration, turn, and gender features, it is indeed possible to obtain better results (20% reduced segmentation cost) than with the lexical model alone, with gen- Table 3 Segmentation error rates with the language model only (LM), the combined HMM using all prosodic features (CM-HMM-all), the combined HMM using only pause duration and turn features (CM-HMM-pause-turn), and using only pause-duration, turn, and gender features (CM-HMM-pause-turn-gender). Model Cseg LM 0.1895 CM-HMM-pause-turn 0.1519 CM-HMM-pause-turn-gender 0.1511 CM-HMM-all 0.1377 der making only a minor contribution. However, we also see that a substantial further improvement (9% relative) is obtained by adding F0 features into the prosodic model. Results Compared to Other Approaches Because our work focused on the use of prosodic information and required detailed linguistic annotations (such as sentence punctuation, turn boundaries, and speaker labels), we used data from the LDC 1997 BN corpus to form the training set for the prosodic models and the (separate) test set used for evaluation. This choice was crucial for the research, but unfortunately complicates a quantitative comparison of our results to other TDT segmentation systems. The recent TDT2 evaluation used a different set of broadcast news data that postdated the material used by us, and was generated by a different speech recognizer (although with a similar word error rate) (Cieri et al. 1999). Nevertheless we have attempted to calibrate our results with respect to these TDT2 results. 11 We have not tried to compare our results to research outside the TDT evaluation framework. In fact, other evaluation methodologies differ too much to allow meaningful quantitative comparisons across publications. We wanted to ensure that the TDT2 evaluation test set was comparable in segmentation difficulty to our test set drawn from the 1997 BN corpus, and that the TDT2 metrics behaved similarly on both sets. To this end, we ran an early version of our words-only segmenter on both test sets. As shown in Table 4, not only are the results on recognized words quite close, but the optimal false alarm/miss trade-off is similar as well, indicating that the two corpora have roughly similar topic granularities. While the full prosodic component of our topic segmenter was not applied to the TDT2 test corpus, we can compare the performance of a simplified version of SRI's segmenter to other evaluation systems (Fiscus et al. 1999). The two best-performing systems in the evaluation were those of CMU (Beeferman, Berger, and Lafferty 1999) with C Seg = 0.1463, and Dragon van Mulbregt et al. 1999) with C Seg = 0.1579. The SRI system achieved C Seg = 0.1895. All systems in the evaluation, including ours, used only information from words and pause durations determined by a speech recognizer. A good reference to calibrate our performance is the Dragon system, from which we borrowed the lexical HMM segmentation framework. Dragon made adjustments in its lexical modeling that account for the improvements relative to the basic HMM structure on which our system is based. As described by van Mulbregt et al. (1999), a significant segmentation error reduction was obtained from optimizing the number of topic clusters (kept fixed at 100 in our system). Second, Dragon introduced more supervision into the model training by building separate LMs for segments that had been hand-labeled as not related to news (such as sports and commercials) in the TDT2 training corpus, which also resulted in substantial improvements. Finally, Dragon used some of the TDT2 training data for tuning the model to the specifics of the TDT2 corpus. In summary, the performance of our combined lexical-prosodic system with C Seg = 0.1438 is competitive with the best word-based systems reported to date. More importantly, since we found the prosodic and lexical knowledge sources to complement each other, and since Dragon's improvements for TDT2 were confined to a better modeling of the lexical information, we would expect that adding these improvements to our combined segmenter would lead to a significant improvement in the state of the art. Discussion Results so far indicate that prosodic information provides an excellent source of information for automatic topic segmentation, both by itself and in conjunction with lexical information. Pause duration, a simple prosodic feature that is readily available as a byproduct of speech recognition, proved highly effective in the initial chopping phase, as well as being the most important feature used by prosodic decision trees. Additional, pitch-based prosodic features are also effective as features in the decision tree. The results obtained with recognized words (at 30% word error rate) did not differ greatly from those obtained with correct word transcripts. No significant degradation was found with the words-only segmentation model, while the best combined model exhibited about a 5% error increase with recognized words. The lack of degradation on the words-only model may be partly due to the fact that the recognizer generally outputs fewer words than contained in the correct transcripts, biasing the segmenter toward a lower false alarm rate. Still, part of the appeal of prosodic segmentation is that it is inherently robust to recognition errors. This characteristic makes it even more attractive for use in domains with higher error rates due to poor acoustic conditions or more conversational speaking styles. It is especially encouraging that the prosody-only segmenter achieved competitive performance. It was fairly straightforward to modify the original Dragon HMM segmenter , which is based purely on topical word usage, to incorporate discourse cues, both lexical and prosodic. The addition of these discourse cues proved highly effective, especially in the case of prosody. The alternative knowledge source combination approach, using HMM posterior probabilities as decision tree inputs, was also effective, although less so than the HMM-based approach. Note that the HMM-based integration, as implemented here, makes more stringent assumptions about the independence of lexical and prosodic cues. The combined decision tree, on the other hand, has some ability to model dependencies between lexical and prosodic cues. The fact that the HMM-based combination approach gave the best results is thus indirect evidence that lexical and prosodic knowledge sources are indeed largely independent. (a). . . we have a severe thunderstorm watch two severe thunderstorm watches and a tornado watch in effect the tornado watch in effect back here in eastern colorado the two severe thunderstorm watches here indiana over into ohio those obviously associated with this line which is already been producing some hail i'll be back in a moment we'll take a look at our forecast weather map see if we can cool it off in the east will be very cold tonight minus seven degrees <TOPIC CHANGE> LM probability: 0.018713 PM probability: 0.937276 karen just walked in was in the computer and found out for me that national airport in washington d. c. did hit one hundred degrees today it's a record high for them it's going to be uh hot again tomorrow but it will begin to cool off the que question is what time of day is this cold front going to move by your house if you want to know how warm it's going to be tomorrow comes through early in the day won't be that hot at all midday it'll still be into the nineties but not as hot as it was today comes through late in the day you'll still be in the upper nineties but some relief is on the way . . . (b). . . you know the if if the president has been unfaithful to his wife and at this point you know i simply don't know any of the facts other than the bits and pieces that we hear and they're simply allegations at this point but being unfaithful to your wife isn't necessarily a crime lying in an affidavit is a crime inducing someone to lie in an affidavit is a crime but that occurred after this apparent taping so i'll tell you there are going to be extremely thorny legal issues that will have to be sorted out white house spokesman mike mccurry says the administration will cooperate in starr's investigation <TOPIC CHANGE> LM probability: 1.000000 PM probability: 0.134409 cubans have been waiting for this day for a long time after months of planning and preparation pope john paul the second will make his first visit to the island nation this afternoon it is the first pilgrimage ever by a pope to cuba judy fortin joins us now from havana with more . . . Figure 8 Examples of true topic boundaries where lexical and prosodic models make opposite decisions. (a) The prosodic model correctly predicts a topic change, the LM does not. (b) The LM predicts a topic change, the prosodic model does not. Apart from the question of probabilistic independence, it seems that lexical and prosodic models are also complementary in the errors they make. This is manifested in the different distributions of miss and false alarm errors discussed in Section 4.5. It is also easy to find examples where the two models make complementary errors. Figure 8 shows two topic boundaries that are missed by one model but not the other. Several aspects of our model are preliminary or suboptimal in nature and can be improved. Even when testing on recognized words, we used parameters optimized on forced alignments. This is suboptimal but convenient, since it avoids the need to run word recognition on the relatively large training set. Since results on recognized words are very similar to those on true words we can conclude that not much was lost with this expedient. Also, we have not yet optimized the chopping stage relative to the combined model (only relative to the words-only segmenter). The use of prosodic features other than pause duration for chopping should further improve the overall performance. The improvement obtained with source-dependent topic switch penalties and posterior thresholds suggests that more comprehensive source-dependent modeling would be beneficial. In particular, both prosodic and lexical discourse cues are likely to be somewhat source specific (e.g., because of different show formats and different speakers). Given enough training data, it is straightforward to train source-dependent models. Conclusion We have presented a probabilistic approach for topic segmentation of speech, combining both lexical and prosodic cues. Topical word usage and lexical discourse cues are represented by language models embedded in an HMM. Prosodic discourse cues, such as pause durations and pitch resets, are modeled by a decision tree based on automatically extracted acoustic features and alignments. Lexical and prosodic features can be combined either in the HMM or in the decision tree framework. Our topic segmentation model was evaluated on broadcast news speech, and found to give competitive performance (around 14% error according to the weighted TDT2 segmentation cost metric). Notably, the segmentation accuracy of the prosodic model alone is competitive with a word-based segmenter, and a combined prosodic/lexical HMM achieves a substantial error reduction over the individual knowledge sources. Figure 2 2Structure of the basic HMM developed by Dragon for the TDT Pilot Project. The labels on the arrows indicate the transition probabilities. TSP represents the topic switch penalty. Figure 3 Structure of an HMM with topic BEGIN and END states. TSP represents the topic switch penalty. where the summation is over all broadcast shows s and word positions i in the test corpus and where d s sys (i, words i and j in show s are deemed by sys to be within the same story 0 otherwise Figure 5 5False alarm versus miss probabilities (word-based metrics) for automatic topic segmentation from known words (forced alignments). The segmenters used were a words-only HMM (LM), a prosody-only HMM (PM), a combined decision tree (CM-DT), and a combined HMM (CM-HMM). Feature subset selection was run with an initial set of 73 potential features, which the algorithm reduced to a set of 7 nonredundant features helpful for the topic segmentation Table 1 1Segmentation error rates for various chopping criteria, using true words of the larger development data set.Chopping Criterion PMiss P FalseAlarm CSeg FIXED 0.5688 0.0639 0.2153 TURN 0.6737 0.0436 0.2326 SENTENCE 0.5469 0.0557 0.2030 PAUSE 0.5111 0.0688 0.2002 Table 4 4Word-based segmentation error rates for different corpora. Note that a hand-transcribed (forced alignment) version of the TDT2 test set was not available.Error Rates on Forced Alignments Error Rates on Forced Alignments Test Set PMiss P FalseAlarm CSeg PMiss P FalseAlarm CSeg TDT2 NA NA NA 0.5509 0.0694 0.2139 BN'97 0.4685 0.0817 0.1978 0.5128 0.0683 0.2017 The rhyme is the part of a syllable that comprises the nuclear phone (typically a vowel) and any following phones. This is the part of the syllable most typically affected by lengthening. Interpreting large trees can be a daunting task. However, the decision questions near the tree root are usually interpretable, or, when nonsensical, usually indicate problems with the data. Furthermore, as explained in Section 4.6, we have developed simple statistics that give an overview of feature usage throughout the tree. The definitions are those fromDoddington (1998), but have been simplified and edited for clarity. The exception is the time-based evaluation of the combined decision tree. We found that the posterior probability threshold optimized on the training set works poorly on the test set for this model architecture and the time-based evaluation. The threshold that is optimal on the test set achieves Cseg = 0.1651. Section 4.7 gives a possible explanation for this result. Such a combination of combined models was suggested by one of the reviewers; we hope to pursue it in future research. For example, the features that measure F0 differences across boundaries do not make sense if the speaker changes at the boundary. Accordingly, we made such features undefined for the decision tree at turn boundaries. Since our study was conducted, a third round of TDT benchmarks (TDT3) has taken place (NIST 1999). However, for TDT3 the topic segmentation evaluation metric was modified and the most recent results are thus not directly comparable with those from TDT2 or the present study. AcknowledgmentsWe thank Becky Bates, Madelaine Plauché, Ze'ev Rivlin, Ananth Sankar, and Kemal Sönmez for invaluable assistance in preparing the data for this study. The paper was greatly improved as a result of comments by Andy Kehler, Madelaine Plauché, and the anonymous reviewers. This research was supported by DARPA and NSF under NSF grant IRI-9619921 and DARPA contract no. N66001-97-C-8544. 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Aims. We study the spatial clustering of 632 (1130) XMM-COSMOS Active Galactic Nuclei (AGNs) with known spectroscopic (spectroscopic or photometric) redshifts in the range z = [0.1 − 2.5] in order to measure the AGN bias and estimate the typical mass of the hosting dark matter (DM) halo as a function of AGN host galaxy properties. We create AGN subsamples in terms of stellar mass M * and specific black hole accretion rate L X /M * , to probe how AGN environment depends on these quantities. Further, we derive the M * − M halo relation for our sample of XMM-COSMOS AGNs and compare it to results in literature for normal non-active galaxies. Methods. We measure the projected two-point correlation function w p (r p ) using both the classic and the generalized clustering estimator based on photometric redshifts as probability distribution functions in addition to any available spectroscopic redshifts. We measure the large-scale (r p 1 h −1 Mpc) linear bias b by comparing the clustering signal to that expected of the underlying DM distribution. The bias is then related to the typical mass of the hosting halo M halo of our AGN subsamples. Since M * and L X /M * are correlated, we match the distribution in terms of one quantity, while split the distribution in the other. Results. For the full spectroscopic AGN sample, we measure a typical DM halo mass of log(M halo /h −1 M ⊙ ) = 12.79 +0.26 −0.43 , similar to galaxy group environments and in line with previous studies for moderate-luminosity X-ray selected AGN. We find no significant dependence on specific accretion rate L X /M * , with log(M halo /h −1 M ⊙ ) = 13.06 +0.23 −0.38 and log(M halo /h −1 M ⊙ ) = 12.97 +0.39 −1.26 for low and high L X /M * subsamples, respectively. We also find no difference in the hosting halos in terms of M * with log(M halo /h −1 M ⊙ ) = 12.93 +0.31 −0.62 (low) and log(M halo /h −1 M ⊙ ) = 12.90 +0.30 −0.62 (high). By comparing the M * − M halo relation derived for XMM-COSMOS AGN subsamples with what is expected for normal non-active galaxies by abundance matching and clustering results, we find that the typical DM halo mass of our high M * AGN subsample is similar to that of non-active galaxies. However, AGNs in our low M * subsample are found in more massive halos than non-active galaxies. By excluding AGNs in galaxy groups from the clustering analysis, we find evidence that the result for low M * may be due a larger fraction of AGNs as satellites in massive halos.

10.1051/0004-6361/201935186

1906.07911

28c3c4847ec1e3782ad8a00a9f606bb3e8ff129d

The XMM-NEWTON WIDE FIELD SURVEY IN THE COSMOS FIELD: CLUSTERING DEPENDENCE OF X-RAY SELECTED AGN ON HOST GALAXY PROPERTIES 26 Jun 2019 June 27, 2019 A Viitanen [email protected] Department of Physics University of Helsinki Gustaf Hällströmin katu 2aFI-00014HelsinkiFinland V Allevato Department of Physics University of Helsinki Gustaf Hällströmin katu 2aFI-00014HelsinkiFinland Scuola Normale Superiore Piazza dei Cavalieri 7, I56126PisaItaly A Finoguenov Department of Physics University of Helsinki Gustaf Hällströmin katu 2aFI-00014HelsinkiFinland A Bongiorno INAF -Osservatorio Astronomico di Roma Monte Porzio Catone (Roma) Via Frascati 3300078Italy N Cappelluti Physics Department University of Miami Knight Physics Building, Coral Gables33124FLUSA R Gilli INAF -Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, via Gobetti 93/3 40129BolognaItaly T Miyaji Instituto de Astronomía Universidad Nacional Autónoma de México 22860EnsenadaMexico M Salvato Max-Planck-Institut für extraterrestrische Physik Giessenbachstrasse 185748GarchingGermany The XMM-NEWTON WIDE FIELD SURVEY IN THE COSMOS FIELD: CLUSTERING DEPENDENCE OF X-RAY SELECTED AGN ON HOST GALAXY PROPERTIES 26 Jun 2019 June 27, 2019Received Nameofmonth dd, yyyy; accepted Nameofmonth dd, yyyyAstronomy & Astrophysics manuscript no. ms c ESO 2019dark matter -galaxies: active -galaxies: evolution -large-scale structure of Universe -quasars: general -surveys Aims. We study the spatial clustering of 632 (1130) XMM-COSMOS Active Galactic Nuclei (AGNs) with known spectroscopic (spectroscopic or photometric) redshifts in the range z = [0.1 − 2.5] in order to measure the AGN bias and estimate the typical mass of the hosting dark matter (DM) halo as a function of AGN host galaxy properties. We create AGN subsamples in terms of stellar mass M * and specific black hole accretion rate L X /M * , to probe how AGN environment depends on these quantities. Further, we derive the M * − M halo relation for our sample of XMM-COSMOS AGNs and compare it to results in literature for normal non-active galaxies. Methods. We measure the projected two-point correlation function w p (r p ) using both the classic and the generalized clustering estimator based on photometric redshifts as probability distribution functions in addition to any available spectroscopic redshifts. We measure the large-scale (r p 1 h −1 Mpc) linear bias b by comparing the clustering signal to that expected of the underlying DM distribution. The bias is then related to the typical mass of the hosting halo M halo of our AGN subsamples. Since M * and L X /M * are correlated, we match the distribution in terms of one quantity, while split the distribution in the other. Results. For the full spectroscopic AGN sample, we measure a typical DM halo mass of log(M halo /h −1 M ⊙ ) = 12.79 +0.26 −0.43 , similar to galaxy group environments and in line with previous studies for moderate-luminosity X-ray selected AGN. We find no significant dependence on specific accretion rate L X /M * , with log(M halo /h −1 M ⊙ ) = 13.06 +0.23 −0.38 and log(M halo /h −1 M ⊙ ) = 12.97 +0.39 −1.26 for low and high L X /M * subsamples, respectively. We also find no difference in the hosting halos in terms of M * with log(M halo /h −1 M ⊙ ) = 12.93 +0.31 −0.62 (low) and log(M halo /h −1 M ⊙ ) = 12.90 +0.30 −0.62 (high). By comparing the M * − M halo relation derived for XMM-COSMOS AGN subsamples with what is expected for normal non-active galaxies by abundance matching and clustering results, we find that the typical DM halo mass of our high M * AGN subsample is similar to that of non-active galaxies. However, AGNs in our low M * subsample are found in more massive halos than non-active galaxies. By excluding AGNs in galaxy groups from the clustering analysis, we find evidence that the result for low M * may be due a larger fraction of AGNs as satellites in massive halos. Introduction Supermassive black holes (SMBH) with M ∼ 10 6−9 M ⊙ reside at the centers of virtually every massive galaxy. SMBHs reach these masses by growing via matter accretion and simultaneously shine luminously as an active galactic nucleus (AGN). Interestingly, BHs and their host galaxies seem to coevolve, as suggested by the correlation between the SMBH and the host galaxy properties (velocity dispersion, luminosity, stellar mass). However, the co-evolution scenario, AGN feedback and accretion mechanisms are still poorly known (e.g. Alexander & Hickox 2012). AGNs and their host galaxies reside in collapsed dark matter (DM) structures i.e. halos. In the concordance ΛCDM cosmology these halos form hierarchially 'bottom up' from the smallest structures (density fluctuations in the CMB) that grow via gravitational instability to the largest (galaxy groups and clus-ters). AGNs and DM halos they reside in are both biased tracers of the underlying DM distribution. By measuring the clustering of AGN, and comparing that to the underlying DM distribution, the AGNs may be linked to their hosting DM halos (e.g. Cappelluti et al. 2012;Krumpe et al. 2014). Recent AGN clustering measurements have not been able to paint a coherent picture of the complex interplay of AGN and their environment. It seems that optically selected luminous quasars prefer to live in halos few × 10 12 h −1 M ⊙ over a wide range in redshift (Croom et al. 2005;da Ângela et al. 2008;Ross et al. 2009) while moderate luminosity X-ray selected AGN prefer larger halos 10 12.5−13 h −1 M ⊙ at similar redshifts (Coil et al. 2009;Allevato et al. 2011;Koutoulidis et al. 2013). Mendez et al. (2016) suggest that the clustering of AGN could be understood as the clustering of galaxies with matched properties in terms of stellar mass and star-formation rate and redshift, and AGN selection effects. This would indicate that in-A&A proofs: manuscript no. ms stead of the properties of the AGN itself, the properties of the host galaxy, such as, stellar mass M * or specific black hole accretion rate L X /M * have a more significant role in driving the clustering of AGN. Many authors have investigated the relation between the stellar mass and the DM halo mass, the so-called M * − M halo relation, for normal non-active galaxies via abundance matching (Moster et al. 2013;Behroozi et al. 2013), clustering measurements and HOD modeling (Zheng et al. 2007;Wake et al. 2011) or weak lensing (Coupon et al. 2015). For X-ray selected AGNs, the M * − M halo relation has only recently been studied observationally. Georgakakis et al. (2014) argue that AGN environment is closely related to M * . However, they do not measure M * directly, but use the rest frame absolute magnitude in the J band as a proxy for M * . Very recently, Mountrichas et al. (2019) measured the AGN clustering dependence directly in terms of M * and found that the environments of X-ray AGN at z = 0.6 − 1.4 are similar to normal galaxies with matched SFR and redshift. In this study, we wish to build upon the previous Xray selected AGN clustering measurements in XMM-COSMOS (Miyaji et al. 2007;Gilli et al. 2009;Allevato et al. 2011), to investigate the clustering dependence on host galaxy properties (M * , L X /M * ). We compare this to the M * − M halo relation for normal non-active galaxies. In our clustering measurements, we also investigate the new generalized estimator which has been introduced (Georgakakis et al. 2014;Allevato et al. 2016), where photometric redshifts are included in the clustering analysis as probability density functions. Clustering measurements using photometric redshifts will be important in future X-ray AGN surveys, where spectroscopic redshifts are not available either due to AGN being optically faint, or because no extensive spectroscopic follow-up campaigns are available. In eROSITA, for example, spectroscopic redshifts will be available only for a certain portion of the sky, and only at later stages of the survey (Merloni et al. 2019). We adopt a flat ΛCDM cosmology with Ω m = 0.3, Ω Λ = 0.7, σ 8 = 0.8 and h = 0.7. Distances reported are comoving distances and the dependence in h is shown explicitly. The symbol 'log' signifies base 10 logarithm. DM halo masses are defined as the enclosed mass within the Virial radius, within which the mean density is 200 times more than the background density. DM halo masses scale as h −1 , while M * scales as h −2 . XMM-COSMOS Multiwavelength Data Set To study the dependence of AGN clustering in terms of host galaxy properties, we use the Cosmic Evolution Survey (COS-MOS, Scoville et al. 2007). COSMOS is a multiwavelength survey over 1.4 × 1.4 deg 2 field designed to study the evolution of galaxies and AGNs up to redshift z ∼ 6. To date the field has been covered by a wide variety of instruments from radio to Xray bands. XMM-Newton surveyed 2.13 deg 2 of the sky in the COSMOS field in the 0.5 − 10 keV band for a total of 1.55 Ms Cappelluti et al. 2007Cappelluti et al. , 2009, providing an unprecedented large sample of point-like X-ray sources (1822). Brusa et al. (2010) carried out the optical identification and presented the multiwavelength properties (24µm to UV) of ∼1800 sources with a spectroscopic completeness of ∼50% (e.g. Hasinger et al. 2018). Salvato et al. (2009Salvato et al. ( , 2011 derived accurate photometric redshifts with σ ∆z/(1+z spec ) ∼ 0.015. Bongiorno et al. (2012) to z 3. The quantity L X /M * corresponds to the rate of accretion onto the central SMBH scaled relative to the stellar mass of the host galaxy. Assuming a M * − M BH relation and a constant bolometric correction to convert from L X to L bol , then Eddington ratio (λ Edd ≡ L bol /L Edd ) can be expressed as: λ Edd = A × k bol 1.3 × 10 38 × L X M * .(1) With A = 500 and k bol = 25, L X /M * = 10 34 erg s −1 M −1 ⊙ corresponds to accretion at Eddington luminosity i.e. λ Edd = 1 (Bongiorno et al. 2012). In this paper we use the catalog presented in Bongiorno et al. (2012), and we focus on 1130 AGN in the redshift range 0.1 < z < 2.5, with mean z ∼ 1.2. The redshifts are either spectroscopic (632) or high quality photometric (498) ones. The 2-10 keV luminosity L X spans log(L X / erg s −1 ) = 42.3 − 45.5 with a mean log(L X / erg s −1 ) = 43.7. The typical host galaxy of our AGN is a red and massive galaxy with mean log (M * / M ⊙ ) = 10.7. However, the host galaxies also span a wide range of stellar masses with log (M * / M ⊙ ) = 7.6 − 12.3. The L X and M * distributions for our sample of 1130 XMM-COSMOS AGN are shown in Figure 2. It would be of interest to also study the clustering as a function of host galaxy SFR or specific SFR (SFR/M * ) as recently done by Mountrichas et al. (2019). However, Bongiorno et al. (2012) conclude for XMM-COSMOS that while stellar masses from SED fitting are relatively robust for both type 1 and type 2 AGNs, SFRs are more sensitive to AGN contamination from type 1 AGN and are unreliable. Thus in order to increase statistics in our clustering analysis, we will not consider the host galaxy SFR, available only for type 2 AGN in XMM-COSMOS. The recent Chandra COSMOS Legacy Survey (CCLS; Civano et al. 2016;Marchesi et al. 2016) contains the largest sample of X-ray selected AGNs to date. However, for CCLS AGN, host galaxy properties have only been estimated for type 2 AGNs, while Bongiorno et al. (2012) provide the estimates for both type 1 and 2 AGNs. Further, the clustering of XMM-COSMOS AGNs is well studied (Miyaji et al. 2007;Gilli et al. 2009;Allevato et al. 2011Allevato et al. , 2012Allevato et al. , 2014, but not in terms of host galaxy properties as in this work. For CCLS AGN, Allevato et al. (2016) measured the clustering at 2.9 ≤ z ≤ 5.5, and Koutoulidis et al. (2018) used multiple fields including COSMOS to measure the clustering. Thus, there are no clustering measurements for CCLS AGN at the redshift of interest (z < 2.5). AGN Subsamples The full AGN sample with known spectroscopic redshifts consists of N = 632 AGNs with mean z = 1.19. For AGNs with only known photometric redshifts, we take into account the full probability distribution function Pdf(z). In this picture, the total weight of an AGN is the integral over z. We limit ourselves to z < 2.5 and the combined weighted number of AGNs with photometric redshifts is N = 488.64 with weighted mean z = 1.44 To study the dependence on host galaxy properties, we divide our AGN sample effectively in two bins of M * and L X /M * which we refer to as the low and high subsamples. In detail, first we bin the distribution of host galaxy stellar mass log M * of the sample with binsize 0.1 dex. Then, each bin is split individually exactly in half based on the logarithm of the specific BH accretion rate log L X /M * to create the low and high L X /M * subsamples. The low and high L X /M * subsamples consist of 309 objects each. We find the average values for the low (high) L X /M * subsamples to be mean log L X /M * = 32.53 (33.49), while the difference in mean log M * is 0.01. We then repeat this process by binning the log L X /M * and splitting in terms of log M * . The number of objects in the low and high M * subsamples is 309. The average values for the low (high) M * subsamples are mean log M * = 10.39 (11.05) and the difference in mean log(L X /M * ) is 0.01. COSMOS is known to be affected by cosmic variance that influences the clustering measurements (e.g. Gilli et al. 2009;Mendez et al. 2016). This means that it is also important to take into account how our low and high M * AGN subsamples relate to the large structures in the field. To this end, as an additional test, we associate the AGN sample with known spectroscopic redshifts with the co-added COSMOS galaxy group catalog (see Finoguenov et al. 2007;Leauthaud et al. 2010;George et al. 2011). An AGN is taken to belong to a galaxy group if the AGNgroup angular separation on the sky is < R 200,deg (radius of the group in degrees enclosing 200 times the critical density), and the radial comoving distance separation is < π max (see Section 3). We find 22 (17) AGNs in our low (high) M * AGN subsamples with spectroscopic redshifts in galaxy groups with a total number of 39 AGNs. We summarize the properties of the different AGN subsamples in Table 1, and the L X and M * distributions are shown in Figures 2 and 3. Methods Two-Point Statistics In clustering studies, a widely used measure to quantify clustering is the two-point correlation function ξ(r) which is defined as the excess probability above random of finding a pair of AGNs in a volume element dV at physical separation r, so that dP = n 1 + ξ(r) dV,(2) where n is the mean number density of AGNs. To estimate ξ(r), we use the Landy & Szalay (1993) estimator ξ(r) = DD ′ − 2DR ′ + RR ′ RR ′ ,(3) where DD ′ = DD N d (N d − 1)/2 (4) DR ′ = DR N d N r (5) RR ′ = RR N r (N r − 1)/2 ,(6) and DD, DR and RR are the number of data-data, data-random and random-random pairs with physical separation r, respectively. N d and N r are the total number of sources in the data and random catalogs. This estimator requires the creation of a random catalog to act as an unclustered distribution of AGNs with the same selection effects in terms of RA, Dec, and redshift, as present in the data catalog (see Section 3.4). As the distances between AGN are inferred from their redshifts, the estimates are affected by distortions due to peculiar motions of AGNs. To avoid this effect, we express pair separations in terms of distance parallel (π) and perpendicular (r p ) to the line-of-sight of the observer, defined with respect to the mean distance to the pair. Then, the projected 2PCF, which is insensitive to redshift space distortions, is defined as (Davis & Peebles 1983) w p (r p ) = 2 ∞ 0 ξ(r p , π)dπ.(7) In practice, the integration is not carried out to infinity, but to finite value π max . The estimation of the π max is a balance between including all of the correlated pairs and not including noise to the signal by uncorrelated pairs. For the estimation of the 2PCFs, we use CosmoBolognaLib 1 (Marulli et al. 2016), which is a free (as in freedom) software library for numerical cosmological calculations. We note that another common way to measure the clustering is to use the cross-correlation function where positions of both an AGN sample and a complete galaxy sample are used to decrease statistical uncertainties (e.g. Coil et al. 2009;Krumpe et al. 2015;Powell et al. 2018;Mountrichas et al. 2019). At our redshift of interest in COSMOS, especially at 1 z 2.5, it is difficult to build a complete galaxy sample with known spectroscopic redshifts (see Sec. 3.2 for discussion on the effect of photometric redshift in clustering measurements) to measure the clustering with, and thus we are limited to the AGN auto-correlation function. Generalized Estimator Motivated by recent progress in utilizing photometric redshifts in AGN clustering studies (Georgakakis et al. 2014;Allevato et al. 2016), we use the full probability distribution function Pdf(z) for AGNs with no known spectroscopic redshifts. In this approach, the classic Landy & Szalay (1993) estimator is replaced by a generalized one, where pairs are weighted based on Pdf(z) of the two objects. For the details, we refer the reader to Georgakakis et al. (2014, Section 3). For the 498 AGNs with photometric redshifts, we discretize the Pdf(z) by integrating the Pdfs in terms of z with an accuracy of δz = 0.01, and normalize the Pdfs to unity. Further, we only consider the part of the Pdf with Pdf(z) > 10 −5 . Using our redshift limit, we only use the part of the Pdfs with z < 2.5. This means that the AGNs with Pdfs that span over this redshift limit A&A proofs: manuscript no. ms are cut, and for these AGNs, the Pdf does not necessarily sum to unity i.e. i Pdf(z i ) ≤ 1. i pdf i z log (M * /M ⊙ ) log L X /M * erg s −1 M ⊙ type1/type2 χ 2 min dof b log M halo h −1 M ⊙ Large uncertainties in photometric redshifts may lead to loss of not only accuracy, but also not being able to recover the full clustering signal. This is highlighted by the use of large values of π max 200 h −1 Mpc (Georgakakis et al. 2014;Allevato et al. 2016) versus studies with only spectroscopic redshifts with π max 100 h −1 Mpc (e.g. Coil et al. 2009;Allevato et al. 2011;Mountrichas et al. 2016). Therefore, we select only Pdfs based on the following quality criteria: the comoving distance separation between the z min and z max may not exceed a critical value of ∆d = 100 h −1 Mpc. We define z min and z max separately for each AGN so that Pdf(z) < 10 −5 for z < z min and z > z max . In detail, from the total of 498 AGN with photometric redshifts, 32 AGN pass the quality criterion and are included in the subsample including spectroscopic and photometric redshifts. In terms of our L X /M * (M * ) AGN subsamples, a total of 32 (28) AGN with photometric redshifts are kept and divided equally between the low and high subsamples in both cases. The number of AGN in each of our subsamples including photometric redshifts are shown in Table 1. This quality cut is suggested by the fact that including all phot-z Pdfs will lead to large uncertainties in the measured clustering signal for all the AGN subsamples. The investigation of quality criteria for studies including phot-z Pdfs is beyond the scope of this work. However, given the importance of photometric redshifts in future large surveys such as eROSITA, we will explore clustering photz Pdfs in a future study (Viitanen et al., in prep.). Halo model In the halo model (e.g. Cooray & Sheth 2002), the AGN clustering signal is the sum of the 1-halo and 2-halo terms, which arise from the clustering of AGN that occupy the same halo, and two distinct halos, respectively. On large scales (r p 1 h −1 Mpc), the 2-halo term is the dominant term, and the AGN projected 2PCF may be related to the underlying DM projected 2PCF w 2−halo Fig. 3. Distribution in terms of M * , L X /M * , and redshift for XMM-COSMOS AGN with known spec-z (left panels) and spec-z + phot-z Pdfs (right panels). The low and high M * subsamples are created so that they have exactly the same specific BH accretion rate distribution (upper panels). A similar approach is used in terms of specific BH accretion rate (lower panels). For clarity, when the histograms match exactly, we have slightly offset the bins visually for the high subsample. the linear bias b w 2−halo p (r p ) = b 2 w 2−halo DM (r p ),(8) where w 2−halo DM is estimated at the mean redshift of the corresponding AGN subsample and integrated to the same value of π max . The DM projected 2PCF is related to the DM one-dimensional 2PCF ξ 2−halo DM w 2−halo DM (r p ) = 2 ∞ r p ξ 2−halo DM (r)rdr r 2 − r 2 p ,(9) where ξ 2−halo DM (r) is in turn estimated using the linear power spectrum P 2−halo (k): ξ 2−halo DM (r) = 1 2π 2 P 2−halo (k)k 2 sin kr kr dk. We base our estimation of the linear power spectrum on Eisenstein & Hu (1999), which is also implemented in CosmoBolognaLib. The 1-halo term (r p 1 h −1 Mpc) also contains important information on the AGN halo occupation and could be contributing towards the clustering signal up to scales r p ∼ 3 h −1 Mpc. However, due to low number counts of pairs especially at small scales r p 3 h −1 Mpc in our XMM-COSMOS subsamples (see Fig. 5), we are not able to constrain the AGN 1-halo term and excluding the 1-halo term from the modeling does not affect our results significantly at large scales. Random catalog and error estimation The random catalog consists of an unclustered set of AGNs with the same selection effects and observational biases. To this end, we follow Miyaji et al. (2007). In detail, for each random object, we draw right ascension and declination at random in the COS-MOS field. In detail, right ascension is drawn uniformly, while for declination we draw sin(Dec) uniformly. Then, we draw a 0.5 − 2 keV flux from the data catalog, and if the drawn flux is above the limit given by the sensitivity map (Cappelluti et al. 2009, see also Figure 1), we keep the object. Otherwise we discard it. Each kept random object is given a redshift drawn from the smoothed redshift distribution of the data catalog with gaussian smoothing using σ z = 0.3. For each of the data catalogs, we create a random catalog with N r = 100N d . We show the red-A&A proofs: manuscript no. ms shift distribution of the data and random catalogs for our AGN subsamples in Figure 4. Poissonian errors are readily assigned to the projected 2PCF, but are known to underestimate the errors. For this reason we adopt a Bootstrap resampling technique by dividing the XMM-COSMOS survey into N region = 18 subregions (3 × 3 × 2 for RA, Dec, and comoving distance, respectively) of roughly equal comoving volumes. We resample the regions N rs = 100 times. In each of the resamples, the regions are assigned different weights based on the number of times they are selected (Norberg et al. 2009). The elements of the covariance matrix C are then defined as C i j = 1 N rs N rs k=1 w p,k (r p,i ) − w p (r p,i ) w p,k (r p, j ) − w p (r p, j ) ,(11) where i and j refer to the ith and jth r p bins and the bar denotes the mean over N region resamples. The 1σ error for w p (r p,i ) is the square root of the corresponding diagonal element i.e. √ C ii . Results For each of the AGN subsamples, we estimate the projected 2PCF w p (r p ) with r p = 1.0 − 100 h −1 Mpc using 12 logarithmic bins. We use one bin in the π direction, where the upper limit of this bin is dictated by π max . In order to set π max , we try out all the values in the range π max = 20 − 75 h −1 Mpc with an accuracy of ∆π max = 5 h −1 Mpc. For the full spectroscopic AGN sample, we found that the signal converges at π max = 40 h −1 Mpc, which is adopted for all the subsamples. This value is similar to previous clustering studies involving XMM-COSMOS AGNs (Gilli et al. 2009;Allevato et al. 2011). The AGN projected 2PCF w p (r p ) is then estimated using Eq. 7 and the 1σ bootstrap errors are estimated using Eq. 11. We show the estimated projected 2PCF for our subsamples in Figure 5. Comparison between the spectroscopic subsamples and the specz+photz subsamples are shown in Figures 5 (full) and 6 (M * and L X /M * subsamples). We derived the best-fit large-scale bias (Eq. 8) using χ 2 minimization for r p = 1−30 h −1 Mpc. In detail, we utilize the inverse of the full covariance matrix C −1 and minimize χ 2 = ∆ T C −1 ∆, where ∆ is a with the same number of elements as the number of r p bins used in the fit. ∆ is defined explicitly as ∆ = w 2−halo p,AGN − b 2 w 2−halo DM . With one free parameter, we estimate the 1σ errors on the best-fit bias, given by the lower and upper bounds of the region (χ 2 − χ 2 min )/ν ≤ 1.0, where ν = N − 1 is the number of degrees of freedom. To exclude noisy bins in the fit, we require that the number of pairs in each bin is > 16. The largescale bias derived for all the XMM-COSMOS AGN subsamples are summarized in Table 1 and shown in Figure 7. For the full spectroscopic AGN subsample ( Figure 5), we find a best-fit bias of b = 2.20 +0.37 −0.45 . Following the biasmass relation described in van den Bosch (2002) and Sheth et al. (2001), this corresponds to a typical mass of the hosting halo of log(M halo /h −1 M ⊙ ) = 12.79 +0.26 −0.43 . Notice that in this work we define the typical mass explicitly as the DM halo mass which satisfies b = b(M halo ) (e.g. Hickox et al. 2009;Allevato et al. 2016;Mountrichas et al. 2019). Albeit with large uncertainties, we find a small 1σ difference in the biases of the spectroscopic AGN subsamples split in terms of stellar mass ( Figure 5). The biases are b = 2.11 +0.45 −0.58 for the low stellar mass and b = 2.69 +0.61 −0.79 for the high stellar mass. However, it is worth noting that the two subsamples peak at different redshifts (z ∼ 1.0 versus z ∼ 1.4). In terms of the typical masses of the hosting halos, we find no difference. For the M * subsamples, we find that excluding AGNs that are associated with groups has a greater effect on the measured best-fit bias of the low M * subsample. We measure b = 1.69 +0.49 −0.72 (b = 2.48 +0.55 −0.71 ) for the low (high) M * AGN subsample. This lower value for the bias could be an indication that AGNs in galaxies with lower stellar mass are more preferably satellites in their DM halos. Moreover, we derive an AGN bias b = 2.14 +0.35 −0.41 (at z ∼ 0.9) and b = 2.95 +0.93 −1.42 (z ∼ 1.5) for the low and high L X /M * subsamples, respectively ( Figure 5). No significant difference is observed in the typical masses of the hosting halos. Similar results in terms of bias dependence on M * and L X /M * are found when using phot-z Pdfs in addition to any available spectroscopic redshifts. In particular, in our full AGN subsamples, an increase of ∼5% in the weighted number of AGNs introduces no systematic error in the estimation of the bias, but decreases the 1σ error of the bias by (δb 1 − δb 2 )/δb 1 ∼ 10%, where δb i is the average error derived from the lower and upper limits of the bias (see Table 1). However, since including photometric redshifts do not change the conclusions drawn from our clustering measurements, in the following sections we focus on the results from the AGN subsamples with known spectroscopic redshifts. Discussion We have performed clustering measurements of 1130 X-ray selected AGN in XMM-COSMOS at 0.1 < z < 2.5 (mean z ∼ 1.2) in order to study AGN clustering dependence on host galaxy stellar mass and specific BH accretion rate L X /M * . For our AGN subsamples we find a typical DM halo mass ∼ 10 13 h −1 M ⊙ that roughly correspond to group-sized environments. This is in agreement with similar studies using X-ray selected AGNs at similar redshifts (Coil et al. 2009;Allevato et al. 2011;Fanidakis et al. 2013;Koutoulidis et al. 2013), as well as at lower redshifts z < 0.1 (e.g. Krumpe et al. 2018;Powell et al. 2018). We have also investigated including photometric redshifts as Pdfs in the analysis in addition to any available spectroscopic redshifts. In COSMOS, Leauthaud et al. (2015) use weak lensing measurements on X-ray COSMOS AGN at z < 1 with log L X /erg s −1 = [41.5 − 43.5] and log M * /M ⊙ = [10.5 − 12]. They infer that 50 per cent of AGN reside in halos with log M halo /M ⊙ < 12.5 in tension with the claim that X-ray AGN inhabit group-sized environments with masses ∼ 10 13 M ⊙ . However, they also underline that due to the skewed tail in the halo mass distribution, the effective/typical halo mass derived from clustering measurements may be markedly different from the median of the distribution. In fact, they found an effective mass of M eff ∼ 10 12.7 M ⊙ , which is close to the typical halo masses derived in this work. It is worth noticing that they derived the effective halo mass from modelling the AGN halo occupation (see Eq. (4) in Leauthaud et al. 2015), which may differ from the typical halo mass inferred from the 2-halo term as in this work. Also, they found that the effective DM halo mass of their AGN sample lies between the median and the mean values of the DM halo mass distribution, which are lower and higher than the effective DM halo mass, respectively. Given the statistics in our XMM-COSMOS AGN sample, we are not able to constrain the median or the mean of the DM halo mass distribution. In the future this could be done through HOD modelling, provided the 1-halo term is constrained. Moreover, different cuts in luminosity and host galaxy mass may reflect in different hosting DM halo mass distributions. For instance, our sample of XMM-COSMOS AGN spans a range of host galaxy stellar masses log M * /M ⊙ = [8 − 12], including also low-mass systems with masses < 10 10.5 M ⊙ (that are likely satellite galaxies in galaxy groups), and probes higher redshifts up to z = 2.5. Clustering in terms of specific BH accretion rate We divided the full sample in low and high specific BH accretion rate subsamples with the same M * distributions and find no significant clustering dependence on L X /M * , and thus Eddington ratio. Krumpe et al. (2015) also found no dependence on λ Edd for their sample of local (0.16 < z < 0.36) X-ray and optically selected AGN in the Rosat All-Sky Survey. They concluded that high accretion rates in AGN are not necessarily linked to high density environments where galaxy interactions would be frequent. Our result provides further evidence that this is also true for non-local AGN at intermediate redshifts z ∼ 1. Mendez et al. (2016) studied the clustering of AGN in the PRIMUS and DEEP2 surveys (including the COSMOS field) at z ∼ 0.7 based on multiple selection criteria. In their X-ray selected AGN sample, they did not find a significant dependence on clustering in terms of specific BH accretion rate, in line with our results. Clustering in terms of host galaxy stellar mass We also studied the AGN clustering dependence on host galaxy stellar mass, probing the M * − M halo relation for active galaxies. In Figure 8, we compare our results for XMM-COSMOS AGN with recent studies in literature using normal (non-active) galaxies. For our comparison, we convert the results to our adopted h = 0.7 cosmology. DM halo masses defined with respect to 200 times ρ crit have been re-defined to be with respect to mean density of the background. The blue curve shows the Moster et al. (2013) The solid lines show the squared best-fit bias times the projected DM correlation function estimated at the mean redshift of the particular sample. The grey datapoints are not used in the fit due to low number of pairs. The excess correlation at r p ∼ 15 h −1 Mpc is likely driven by large structure in the COSMOS field. timated using a multi-epoch abundance matching method which we have calculated at the mean redshift z ∼ 1.2 of our AGN sample. The orange curve shows the galaxy M * − M halo relation of Behroozi et al. (2013) at z ∼ 1.2. Coupon et al. (2015) estimated the M * − M halo relation in the CFHTLenS/VIPERS field at z ∼ 0.8 using constraints from several different meth-ods including galaxy clustering. Compared to our AGN sample, their sample has a similar range in stellar mass and a slightly lower redshift. Results from HOD modeling of galaxy clustering in DEEP2 (Zheng et al. 2007) ied a sample of moderate-luminosity AGN in COSMOS at a lower redshift z ∼ 0.66 than our sample. At M * > 10 10.5 M ⊙ , they suggest that AGN populate similar DM halos as normal galaxies. Similarly, we found that high M * ( 10 10.5 M ⊙ ) XMM-COSMOS AGN follow the same M * − M halo relation as normal non-active galaxies. On the contrary, we estimated that low M * ( 10 10.5 M ⊙ ) AGN are more clustered than normal galaxies. Mountrichas et al. (2019) measured clustering of AGN from the XMM-XXL survey in terms of host galaxy properties (M * , SFR, sSFR) at z ∼ 0.8 and find a positive dependence on the environment with respect to M * . Within errors, our results at slightly higher redshift are in agreement with their measurements (see Figure 8). The M * − M halo relation obtained from our clustering analysis of XMM-COSMOS AGN is not consistent with results inferred for normal galaxies at similar redshifts, at least for the low M * bin. In fact, we found that AGN host galaxies with low M * reside in slightly more massive halos than normal galaxies of similar stellar mass. On the other hand, at high M * , our results are in good agreement with the M * − M halo relation of normal galaxies. Following Figure 8, we do not expect the observed discrepancy at low M * to be due to the different mean redshift of the two subsamples (z ∼ 1 and z ∼ 1.4). If we exclude AGN that are associated with galaxy groups from our M * subsamples, we see that this affects our low M * bin more, while leaving the high M * bin relatively unchanged. This could indicate that XMM-COSMOS AGN with higher M * are more preferably found in central galaxies of their respective halos. For lower M * , the fraction of AGNs as satellites would be higher. Nevertheless, excluding the galaxy groups from the analysis brings our result for the low M * closer to the M * − M halo of normal non-active galaxies. It is important to note that our results for the M * subsamples include both type1 and type2 AGNs i.e. AGNs regardless of obscuration are considered in the same subsample. With the limited sample size of XMM-COSMOS, we are not able to further divide the subsamples and examine the M * − M halo relation for type1 and type2 AGNs separately, to see whether there are any differences between these to populations. However, this issue can be revisited with Chandra COSMOS Legacy Survey AGNs. Conclusions We have measured the clustering of XMM-COSMOS AGN in terms of host galaxy stellar mass M * and specific BH accretion rate L X /M * . Using these two quantities, we created AGN subsamples by splitting the full sample in terms of one quantity, while matching the distribution in the other. In addition, we investigated including AGNs with photometric redshifts as Pdfs in addition to AGNs with known spectroscopic redshifts. From our analysis, we make the following conclusions: 1. XMM-COSMOS AGNs are highly biased with a typical DM halo mass of M halo ∼ 10 13 h −1 M ⊙ , characteristic to group-sized environments and in broad agreement with previous results for moderate-luminosity X-ray selected AGN. accretion rates in AGNs do not necessarily correspond to more dense environments. 3. Also we find no significant clustering dependence in terms of host galaxy stellar mass. By comparing our results with various M * − M halo relations found for normal non-active galaxies, we find that our low M * AGN subsample is more clustered than what is expected of normal galaxies at similar M * . We investigate this further by excluding AGNs that are associated with galaxy groups. We find that excluding objects in galaxy groups results in a lower AGN bias for the low M * AGN subsamples, but does not affect high M * . This could be due to a higher fraction of satellites for the lower stellar mass systems. 4. Our selected quality criterion for including additional photometric redshifts as Pdfs decreases the errors on the measured best-fit bias and does not introduce a bias to the clustering signal. Optimal quality cuts for including photometric redshifts will be studied in a future work. Fig. 1 . 1XMM-COSMOS sensitivity map in the soft band 0.5 − 2.0 keV (Cappelluti et al. 2009). Orange points mark the positions of 1130 AGN with z = [0.1 − 2.5] used in this study. Fig. 2 . 2Distribution of 2-10 keV luminosity (left) and host galaxy stellar mass (right) as a function of redshift for our sample of 1130 AGNs. Blue (orange) points show 632 (498) AGN with known spectroscopic (photometric) redshifts. Fig. 4 . 4Redshift distributions of the data and random catalogs for our AGN subsamples. The random redshifts are drawn from the smoothed redshift distribution of the data catalog using a gaussian smoothing technique with σ z = 0.3. Fig. 5 . 5M * − M halo relation for central galaxies es-The measured projected 2PCF for the full sample and AGN subsamples. The errorbars correspond to 1σ estimated via the bootstrap method. Fig. 6 . 6and the NMBS(Wake et al. 2011) at comparable redshifts (z ∼ 1.0 − 1.1) are shown as well. Using weak lensing methods,Leauthaud et al. (2015) stud-Effect of including photometric redshifts as Pdfs in the estimation of the projected 2PCF (crosses). Different symbols have the same meaning as inFigure 5. The bins have been slightly offset in the r p direction for clarity. Fig. 7 .Fig. 8 . 78Left: Redshift evolution of the bias for the different XMM-COSMOS AGN subsamples. The grey dashed lines correspond to constant halo mass bias evolution b(z, M halo = const) for log M halo = 11.5, 12.0, 12.5, 13.0, 13.5, where M halo is given in units of h −1 M ⊙ . Right: Corresponding typical AGN hosting halo mass evolution with redshift. For visual guidance, the dashed lines show the estimated mass of the halo for the full spectroscopic AGN sample. The M * − M halo relationship for our spectroscopic redshift AGN sample (stars) compared to previous studies in literature according to the legend. For each of the M * subsamples, the horizontal errorbars represent one standard deviation of log M * of the sample. used a Spectral Energy Distribution (SED) fitting technique based on AGN+Galaxy template SEDs and estimated the host galaxy properties, i.e. stellar mass M * and star-formation rate (SFR) of ∼1700 AGN in COSMOS up150°40' 20' 00' 149°40' 20' 3°00' 2°40' 20' 00' 1°40' RA (J2000) DEC (J2000) 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 Limiting flux (0.5-2.0 keV) [erg s −1 cm −2 ] 1e−15 Table 1 . 1XMM-COSMOS AGN subsamples. https://github.com/federicomarulli/CosmoBolognaLib Article number, page 3 of 11 . We find no significant clustering dependence in terms of specific BH accretion rate, consistent with a picture that higherArticle number, page 10 of 11 A. Viitanen et al.: XMM-COSMOS AGN Clustering and Host Galaxy Properties Acknowledgements. We thank the referee for helpful comments that have improved this paper. 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Many Embedded Systems are indeed Software Based Control Systems (SBCSs), that is control systems whose controller consists of control software running on a microcontroller device. This motivates investigation on Formal Model Based Design approaches for automatic synthesis of SBCS control software. In previous works we presented an algorithm, along with a tool QKS implementing it, that from a formal model (as a Discrete Time Linear Hybrid System, DTLHS) of the controlled system (plant), implementation specifications (that is, number of bits in the Analog-to-Digital, AD, conversion) and System Level Formal Specifications (that is, safety and liveness requirements for the closed loop system) returns correct-by-construction control software that has a Worst Case Execution Time (WCET) linear in the number of AD bits and meets the given specifications. In this technical report we present full experimental results on using it to synthesize control software for two versions of buck DC-DC converters (single-input and multi-input), a widely used mixed-mode analog circuit.

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Quantized Feedback Control Software Synthesis from System Level Formal Specifications for Buck DC/DC Converters January 26, 2013 20 Jun 2012 Federico Mari [email protected] Department of Computer Science Sapienza University of Rome via Salaria 11300198Rome Igor Melatti [email protected] Department of Computer Science Sapienza University of Rome via Salaria 11300198Rome Ivano Salvo [email protected] Department of Computer Science Sapienza University of Rome via Salaria 11300198Rome Enrico Tronci [email protected] Department of Computer Science Sapienza University of Rome via Salaria 11300198Rome Quantized Feedback Control Software Synthesis from System Level Formal Specifications for Buck DC/DC Converters January 26, 2013 20 Jun 20121 Many Embedded Systems are indeed Software Based Control Systems (SBCSs), that is control systems whose controller consists of control software running on a microcontroller device. This motivates investigation on Formal Model Based Design approaches for automatic synthesis of SBCS control software. In previous works we presented an algorithm, along with a tool QKS implementing it, that from a formal model (as a Discrete Time Linear Hybrid System, DTLHS) of the controlled system (plant), implementation specifications (that is, number of bits in the Analog-to-Digital, AD, conversion) and System Level Formal Specifications (that is, safety and liveness requirements for the closed loop system) returns correct-by-construction control software that has a Worst Case Execution Time (WCET) linear in the number of AD bits and meets the given specifications. In this technical report we present full experimental results on using it to synthesize control software for two versions of buck DC-DC converters (single-input and multi-input), a widely used mixed-mode analog circuit. Read AD conversionx of plant sensor outputs x 3. If (x is not in the Controllable_Region) 4. Then // Exception (Fault Detected): 5. Start Fault Isolation and Recovery (FDIR) 6. Else // Nominal case: 7. Compute (Control_Law) commandû fromx 8. Send DA conversion u ofû to plant actuators Introduction Many Embedded Systems are indeed Software Based Control Systems (SBCSs). An SBCS consists of two main subsystems: the controller and the plant. Typically, the plant is a physical system consisting, for example, of mechanical or electrical devices whereas the controller consists of control software running on a microcontroller. In an endless loop, the controller reads sensor outputs from the plant and sends commands to plant actuators in order to guarantee that the closed loop system (that is, the system consisting of both plant and controller) meets given safety and liveness specifications (System Level Formal Specifications). Software generation from models and formal specifications forms the core of Model Based Design of embedded software [2]. This approach is particularly interesting for SBCSs since in such a case system level (formal) specifications are much easier to define than the control software behavior itself. Fig. 1 shows the typical control loop skeleton for an SBCS. Measures from plant sensors go through an AD (analog-to-digital ) conversion (quantization) before being processed (line 2) and commands from the control software go through a DA (digital-to-analog) conversion before being sent to plant actuators (line 8). Basically, the control software design problem for SBCSs consists in designing software implementing functions Control_Law and Controllable_Region computing, respectively, the command to be sent to the plant (line 7) and the set of states on which the Control_Law function works correctly (Fault Detection in line 3). In [5] we presented an algorithm and a tool QKS that from the plant model (as a hybrid system), from formal specifications for the closed loop system behaviour (System Level Formal Specifications) and from implementation specifications (that is, number of bits used in the quantization process) can generate correct-by-construction control software satisfying the given specifications. In this technical report we present full experimental results on using it to synthesize control software for two versions of buck DC-DC converters (single-input and multi-input), a widely used mixed-mode analog circuit. Background We denote with [n] an initial segment {1, . . . , n} of the natural numbers. We denote with X = [x 1 , . . . , x n ] a finite sequence (list) of variables. By abuse of language we may regard sequences as sets and we use ∪ to denote list concatenation. Each variable x ranges on a known (bounded or unbounded) interval D x either of the reals or of the integers (discrete variables). We denote with D X the set x∈X D x . To clarify that a variable x is continuous (i.e. real valued) we may write x r . Similarly, to clarify that a variable x is discrete (i.e. integer valued) we may write x d . Boolean variables are discrete variables ranging on the set B = {0, 1}. We may write x b to denote a boolean variable. Analogously X r (X d , X b ) denotes the sequence of real (integer, boolean) variables in X. Finally, if x is a boolean variable we writē x for (1 − x). Predicates A linear expression over a list of variables X is a linear combination of variables in X with real coefficients. A linear constraint over X (or simply a constraint) is an expression of the form L(X) ≤ b, where L(X) is a linear expression over X and b is a real constant. Predicates are inductively defined as follows. A constraint C(X) over a list of variables X is a predicate over X. If A(X) and B(X) are predicates over X, then (A(X) ∧ B(X)) and (A(X) ∨ B(X)) are predicates over X. Parentheses may be omitted, assuming usual associativity and precedence rules of logical operators. A conjunctive predicate is a conjunction of constraints. For linear constraints we write: L(X) ≥ b for −L(X) ≤ −b, L(X) = b for ((L(X) ≤ b) ∧ (−L(X) ≤ −b)) and a ≤ x ≤ b for x ≥ a ∧ x ≤ b, being x ∈ X. A valuation over a list of variables X is a function v that maps each variable x ∈ X to a value v(x) in D x . We denote with X * ∈ D X the sequence of values [v(x 1 ), . . . , v(x n )]. By abuse of language, we call valuation also the sequence of values X * . A satisfying assignment to a predicate P over X is a valuation X * such that P (X * ) holds. Abusing notation, we may denote with P the set of satisfying assignments to the predicate P (X). Two predicates P and Q over X are equivalent, notation P ≡ Q, if they have the same set of satisfying assignments. A variable x ∈ X is said to be bounded in P if there exist a, b ∈ D x such that P (X) implies a ≤ x ≤ b. A predicate P is bounded if all its variables are bounded. Given a constraint C(X) and a fresh boolean variable (guard) y ∈ X, the guarded constraint y → C(X) (if y then C(X)) denotes the predicate ((y = 0) ∨ C(X)). Similarly, we useȳ → C(X) (if not y then C(X)) to denote the predicate ((y = 1) ∨ C(X)). A guarded predicate is a conjunction of either constraints or guarded constraints. When a guarded predicate is bounded, it can be easily transformed into a conjunctive predicate, as stated by the following proposition. Discrete Time Linear Hybrid Systems In this section we introduce our class of Discrete Time Linear Hybrid Systems (DTLHS for short). • X = X r ∪ X d ∪ X b is a finite sequence of real (X r ), discrete (X d ) and boolean (X b ) present state variables. We denote with X the sequence of next state variables obtained by decorating with all variables in X. • U = U r ∪ U d ∪ U b is a finite sequence of input variables. • Y = Y r ∪ Y d ∪ Y b is a finite sequence of auxiliary variables. Auxiliary variables are typically used to model modes (e.g., from switching elements such as diodes) or uncontrollable inputs (e.g., disturbances). V C i +v v r R v u + D L + O C +v C i L r C i D D i u i u L Figure 2: Single-input buck DC-DC converter • N (X, U, Y, X ) is a conjunctive predicate over X ∪ U ∪ Y ∪ X defining the transition relation ( next state) of the system. A DTLHS is bounded if predicate N is bounded. By Prop. 1, any bounded guarded predicate can be transformed into a conjunctive predicate. For the sake of readability, we will use bounded guarded predicates to describe the transition relation of bounded DTLHSs. Note that DTLHSs can effectively model linear algebraic constraints involving both continuous as well as discrete variables. Therefore many embedded control systems may be modeled as DTLHSs. Single-input Buck DC-DC Converter The buck DC-DC converter (Fig. 2) is a mixed-mode analog circuit converting the DC input voltage (V in in Fig. 2) to a desired DC output voltage (v O in Fig. 2). As an example, buck DC-DC converters are used off-chip to scale down the typical laptop battery voltage (12-24) to the just few volts needed by the laptop processor (e.g. [8]) as well as on-chip to support Dynamic Voltage and Frequency Scaling (DVFS) in multicore processors (e.g. [3,7]). Because of its widespread use, control schemas for buck DC-DC converters have been widely studied (e.g. see [3,7,8,9]). The typical software based approach (e.g. see [8]) is to control the switch u in Fig. 2 (typically implemented with a MOSFET) with a microcontroller. Designing the software to run on the microcontroller to properly actuate the switch is the control software design problem for the buck DC-DC converter in our context. The circuit in Fig. 2 can be modeled as a DTLHS H = (X, U , Y , N ). The circuit state variables are i L and v C . However we can also use the pair i L , v O as state variables in H model since there is a linear relationship between i L , v C and v O , namely: v O = r C R r C +R i L + R r C +R v C . Such considerations lead to use the following sets of variables to model H: X = X r = [i L , v O ], U = U b = [u], Y = Y r ∪ Y b with Y r = [i u , v u , i D , v D ] and Y b = [q] . Note how H auxiliary variables Y stem from the constitutive equations of the switching elements (i.e. the switch u and the diode D in Fig. 2). From a simple circuit analysis (e.g. see [4]) we have the following equations: i L = a 1,1 i L + a 1,2 v O + a 1,3 v D (1) v O = a 2,1 i L + a 2,2 v O + a 2,3 v D(2) where the coefficients a i,j depend on the circuit parameters R, r L , r C , L and C in the following way: a 1,1 = − r L L , a 1,2 = − 1 L , a 1,3 = − 1 L , a 2,1 = R rc+R [− rcr L L + 1 C ], a 2,2 = −1 rc+R [ rcR L + 1 C ], a 2,3 = − 1 L rcR rc+R . Using a discrete time model with sampling time T (writing x for x(t + 1)) we have: i L = (1 + T a 1,1 )i L + T a 1,2 v O + T a 1,3 v D (3) v O = T a 2,1 i L + (1 + T a 2,2 )v O + T a 2,3 v D .(4) The algebraic constraints stemming from the constitutive equations of the switching elements are the following: q → v D = R on i D (5) q → i D ≥ 0 (6) u → v u = R on i u (7) v D = v u − V in (8)q → v D = R off i D (9) q → v D ≤ 0 (10) u → v u = R off i u (11) i D = i L − i u (12) The transition relation N of H is given by the conjunction of the constraints in Eqs. (3)-(12) and the following explicit (safety) bounds: −4 ≤ i L ≤ 4 ∧ −1 ≤ v O ≤ 7 ∧ −10 3 ≤ i D ≤ 10 3 ∧ −10 3 ≤ i u ≤ 10 3 ∧ −10 7 ≤ v u ≤ 10 7 ∧ −10 7 ≤ v D ≤ 10 7 . Modelling Robustness on Input V in and Load R In this section we address the problem of refining the model given in Sect. 4 so as to require a controller for our single-input buck to be robust to foreseen variations in the load R and in the power supply V in . That is, given tolerances ρ R and ρ V in , we want the controller output by QKS for our single-input buck to work for any R ∈ [max{0, R(1 − ρ R )}, R(1 + ρ R )] and any V in ∈ [max{0, V in (1 − ρ V in )}, V in (1 + ρ V in )]. Variations in the power supply are modeled by replacing Eq. (8) in Sect. 4 with the following: v D ≤ v u − V in (1 − ρ V in ) (13) v D ≥ v u − V in (1 + ρ V in )(14) Along the same lines, we may model also variations in the load R. However, since N dynamics is not linear in R, much more work is needed (along the lines of [1]). To this aim, we proceed as follows. The only equation depending on R is Eq. (4) of Sect. 4. Consider constants a 2, 1 (R) = R rc+R [− rcr L L + 1 C ], a 2,2 (R) = −1 rc+R [ rcR L + 1 C ],v O ≥ T a 2,1 (R − i L )i L +(1+T a 2,2 (R − v O ))v O + T a 2,3 (R − v D )v D and v O ≤ T a 2,1 (R + i L )i L + (1 + T a 2,2 (R + v O ))v O + T a 2,3 (R + v D )v D , being • R − w = if w ≥ 0 then R(1 − ρ R ) else R(1 + ρ R ) and R + w = if w ≥ 0 then R(1 + ρ R ) else R(1 − ρ R ) for w ∈ {i L , v O }; • R − v D = if v D ≥ 0 then R(1 + ρ R ) else R(1 − ρ R ) and R + v D = if v D ≥ 0 then R(1 − ρ R ) else R(1 + ρ R ). This leads us to replace Eq. (4) of Sect. 4 with the equations in Fig. 3. Note that, w.r.t. the model in Sect. 4, in Fig. 3 we add to Y b 11 auxiliary boolean variables z i L , z v O , z v D , z ppp , z ppp , z ppn , z ppn , z pnp , z pnp , z pnn , z pnn , z npp , z npp , z npn , z npn , z nnp , z nnp , z nnn , z nnn with the following meaning. The boolean variable z i L [z v O , z v D ] is true iff i L [v O , v D ]∈ {p, n}, is true iff (if a = p then i L ≥ 0 else i L ≤ 0) ∧ (if z i L → i L ≥ 0 (15) z v O → v O ≥ 0 (16) z v D → v D ≥ 0 (17) z i L → i L ≤ 0 (18) z v O → v O ≤ 0 (19) z v D → v D ≤ 0 (20) z ppp → 1 − z i L + 1 − z v O + 1 − z v D ≥ 1 (21) z pnp → 1 − z i L + z v O + 1 − z v D ≥ 1 (22) z ppn → 1 − z i L + 1 − z v O + z v D ≥ 1 (23) z pnn → 1 − z i L + z v O + z v D ≥ 1 (24) z npp → z i L + 1 − z v O + 1 − z v D ≥ 1 (25) z nnp → z i L + z v O + 1 − z v D ≥ 1 (26) z npn → z i L + 1 − z v O + z v D ≥ 1 (27) z nnn → z i L + z v O + z v D ≥ 1 (28) z ppp → v O ≤ T a (M ) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (m) 2,3 v D (29) z ppp → v O ≥ T a (m) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (M ) 2,3 v D (30) z ppn → v O ≤ T a (M ) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (M ) 2,3 v D (31) z ppn → v O ≥ T a (m) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (m) 2,3 v D (32) z pnp → v O ≤ T a (M ) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (m) 2,3 v D (33) z pnp → v O ≥ T a (m) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (M ) 2,3 v D (34) z pnn → v O ≤ T a (M ) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (M ) 2,3 v D (35) z pnn → v O ≥ T a (m) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (m) 2,3 v D (36) z npp → v O ≤ T a (m) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (m) 2,3 v D (37) z npp → v O ≥ T a (M ) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (M ) 2,3 v D (38) z npn → v O ≤ T a (m) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (M ) 2,3 v D (39) z npn → v O ≥ T a (M ) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (m) 2,3 v D (40) z nnp → v O ≤ T a (m) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (m) 2,3 v D (41) z nnp → v O ≥ T a (M ) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (M ) 2,3 v D (42) z nnn → v O ≤ T a (m) 2,1 i L + (T a (m) 2,2 + 1)v O + T a (M ) 2,3 v D (43) z nnn → v O ≥ T a (M ) 2,1 i L + (T a (M ) 2,2 + 1)v O + T a (m) 2,3 v D(44)b = p then v O ≥ 0 else v O ≤ 0) ∧ (if c = p then v D ≥ 0 else v D ≤ 0) . This is stated by Eqs. (21)-(28). Finally, we use boolean variables z abc as guards for the inequalities replacing Eq. (4) as stated before. This is done in Eqs. (29)-(44). Multi-input Buck DC-DC Converter A multi-input buck DC-DC converter [6] (Fig. 4) . . , u n , thus a control software for the n-input buck dc-dc converter has to properly actuate the switches u 1 , . . . , u n . We model our n-input buck DC-DC converter with DTLHS H = (X, U, Y, N ), where X = X r = [i L , v O ], U = U b = [u 1 , . . . , u n ], and Y = Y r ∪ Y b with Y r = [v D , v D 1 , . . . , v D n−1 , i D , I u 1 , . . . , I u n , v u 1 , . . . , v u n ] and Y b = [q 0 , . . . , q n−1 ]. As for the predicate N , from a simple circuit analysis (e.g. see [4]) we have that state variables constraints are the same as Eqs. (3) and (4) q 0 → v D = R on i D (45) q 0 → i D ≥ 0 (46) q i → v D i = R on I u i (47) q i → I u i ≥ 0 (48) u j → v u j = R on I u j (49) i L = i D + n i=1 I u i (50)q 0 → v D = R off i D (51) q 0 → v D ≤ 0 (52) q i → v D i = R off I u i (53) q i → v D i ≤ 0 (54) u j → v u j = R off I u j (55) v D = v u i + v D i − V i (56) v D = v u n − V n (57) Finally, N is given by the conjunction of Eqs. (3) and (4) of Sect. 4, Eqs. (45)-(57) and the following explicit (safety) bounds: −4 ≤ i L ≤ 4∧−1 ≤ v O ≤ 7 ∧ −10 3 ≤ i D ≤ 10 3 ∧ n i=1 −10 3 ≤ I u i ≤ 10 3 ∧ n i=1 −10 7 ≤ v u i ≤ 10 7 ∧ n−1 i=0 −10 7 ≤ v D i ≤ 10 7 . Modelling Robustness on Inputs V i and Load R In this section we address the problem of refining the model given in Sect. 5 so as to require a controller for our multi-input buck to be robust to foreseen variations in the load R and in the power supplies V i (for i ∈ [n]). As it is explained in Sect. 4.1, given tolerances ρ R and ρ V i (for i ∈ [n]), we want the controller output by QKS for our multi-input buck to work for any R ∈ [max{0, R(1 − ρ R )}, R(1 + ρ R )] and any V i ∈ [max{0, V i (1 − ρ V i )}, V i (1 + ρ V i )] (for i ∈ [n]). Variations in the power supplies are modeled by replacing Eqs. (56) and (57) in Sect. 5 with the following (where i ranges in [n − 1]): v D ≤ v u i + v D i − V i (1 − ρ V i ) (58) v D ≥ v u i + v D i − V i (1 + ρ V i ) (59) v D ≤ v u n − V n (1 − ρ Vn ) (60) v D ≥ v u n − V n (1 + ρ Vn )(61) As for the robustness w.r.t. the load R, since the only equation depending on R is Eq. (4) of Sect. 4, which also holds for the multi-input buck, the same reasoning of Sect. 4.1 may be applied. Thus, we have to replace Eq. (4) of Sect. 4 with the equations in Fig. 3. Experimental Results In this section we present our experimental results about running QKS [5] on the buck models described in Sects. 4 and 5. Namely, we will present experimental results on the robust model for the single-input buck described in Sect. 4.1 (Sect. 6.1) and on the (non-robust) model for the multi-buck described in Sect. 5 (Sect. 6.2). All experiments run on an Intel 3.0 GHz hyperthreaded Quad Core Linux PC with 8 GB of RAM. Single-input Buck We run QKS on the single-input buck model taking into account foreseen variations in the load R and in the power supply V in (see Sect. 4.1). Since QKS also require as input the number of AD bits b (see [5] for details), we run multiple times QKS for different values of b, each time obtaining a controller K b . All other constants introduced in Sect. 4 are fixed as follows: T = 10 −6 secs, L = 2 · 10 −4 H, r L = 0.1 Ω, r C = 0.1 Ω, R = 5 Ω, C = 5 · 10 −5 F, V i = 15 V, ρ R = ρ V in = 25%, R on = 0 Ω, R off = 10 4 Ω. Tabs. 1, 2 and 3 show our experimental results. Columns in Tab. 1 have the following meaning. Column b shows the number of AD bits (see [5] for details). Columns labeled Control Abstraction show performance for control abstraction computation (see [5] for details) and they show running time (column CPU, in secs), memory usage (MEM, in bytes), the number of transitions in the generated control abstraction (Arcs), the number of self-loops in the maximum control abstraction (MaxLoops), and the fraction of loops that are kept in the minimum control abstraction w.r.t. the number of loops in the maximum control abstraction (LoopFrac). Columns labeled Controller Synthesis show the computation time (column CPU, in secs) for the generation of K b , and the size of its OBDD representation (OBDD, number of nodes). The latter is also the size (number of lines) of K b C code synthesized implementation. Finally, columns labeled Total show the total computation time (column CPU, in secs) and the memory (MEM, in bytes) for the whole process (i.e., control abstraction plus controller source code generation), as well as the final outcome µ ∈ {Sol, NoSol, Unk} of QKS (see [5] for details). For each MILP problem solved in QKS (see [5] for details), Tabs. 2 and 3 show (as a function of b) the total and the average CPU time (in seconds) spent solving MILP problem instances, together with the number of MILP Num is the number of times that the MILP problem of the given type is called, Time is the total CPU time (in secs) needed to solve all the Num instances of the MILP problem of the given type, and Avg is the average CPU time (in secs), i.e. the ratio between columns Time and Num. Each row in Tabs. 2 and 3 refer to a type of MILP problem solved, see [5] for details. Finally, in Figs. 5-8 we show the guaranteed operational range (controlled regions, see [5] for details) of the controllers generated for the single-input buck by QKS. Multi-input Buck We run QKS on the multi-input buck model described in Sect. 5. Differently from Sect. 6.1, we fix the number of AD bits b for QKS, namely b = 10. On the other hand, we run multiple times QKS by varying the number n Finally, in Figs. 9-12 we show the guaranteed operational range (controlled regions, see [5] for details) of the controllers generated for the multiinput buck by QKS. Conclusions We presented experimental results on using the QKS tool [5], to support a Formal Model Based Design approach to control software. Our experiments have been carried out on two versions of the buck DC-DC converter, namely the single-input and the multi-input versions. We also showed how robust controllers may be generated for such bucks, namely by taking into account also foreseen variations on some important buck parameters such as load and input power supplies. Figure 11: Multi-input buck: controlled region with n = 3 inputs Figure 1 : 1A typical control loop skeleton Proposition 1 . 1For each bounded guarded predicate P (X), there exists an equivalent bounded conjunctive predicate Q(X). Definition 1 . 1A Discrete Time Linear Hybrid System is a tuple H = (X, U, Y, N ) where: as (nonlinear) functions of R. It is easy to see that a 2,1 (R), a 2,2 (R) are monotonically increasing functions for R ∈ R + , while a 2,3 (R) is monotonically decreasing for R ∈ R + . Thus, if signs of i L , v O , v D are known, it is possible to replace Eq. (4) with two inequalities is positive (see Eqs. (15) and (18) [Eqs. (16) and (19), Eqs. (17) and (20)]). The boolean variable z abc , with a, b, c Figure 3 : 3DTLHS Figure 4 : 4Multi-input Buck DC-DC converter , consists of n power supplies with voltage values V 1 < . . . < V n , n switches with voltage values v u 1 , . . . , v u n and current values I u 1 , . . . , I u n , and n input diodes D 0 , . . . , D n−1 with voltage values v D 0 , . . . , v D n−1 and current values i D 0 , . . . , i D n−1 (in the following, we will also write v D for v D 0 and i D for i D 0 ). As for the converter in Sect. 4, the state variables are i L and v O . Differently from the converter in Sect. 4, the action variables are u 1 , . of the converter in Sect. 4. The algebraic constraints stemming from the constitutive equations of the switching elements are the following (where i and j range in [n − 1] and [n] respectively): Figure 5 : 5Single-input robust buck: controlled region with b = 8 bits instances solved. Columns in Tabs. 2 and 3 have the following meaning: Figure 8 : 8Single-input robust buck: controlled region with b = 11 bits of inputs for the multi-input buck. As for input voltages, we have V i = 10i V for all i ∈ [n]. All other constants introduced in Sect. 5 are fixed as in Sect. 6.1.Tabs. 4, 5 and 6 show our experimental results. Columns in Tab. 4 have the following meaning. Column n shows the number of inputs of the multiinput buck (see Sect. 5 for details). All other columns of Tab. 4, as well as of Tabs. 5 and 6 have the same meaning of the same columns of Tabs. 1, 2 and 3. 1 . 1Every T seconds (sampling time) do2. Table 1 : 1Single-input buck DC-DC converter: control abstraction and controller synthesis results.Control Abstraction Controller Synthesis Total b CPU MEM Arcs MaxLoops LoopFrac CPU |K| CPU MEM µ 8 1.95e+03 4.41e+07 6.87e+05 2.55e+04 0.00333 2.10e-01 1.39e+02 1.96e+03 4.46e+07 Unk 9 9.55e+03 5.67e+07 3.91e+06 1.87e+04 0.00440 2.64e+01 3.24e+03 9.58e+03 7.19e+07 Sol 10 1.42e+05 8.47e+07 2.61e+07 2.09e+04 0.00781 7.36e+01 1.05e+04 1.42e+05 1.06e+08 Sol 11 8.76e+05 1.11e+08 2.15e+08 2.26e+04 0.01435 2.94e+02 2.88e+04 8.76e+05 2.47e+08 Sol Table 2 : 2Single-input buck DC-DC converter: number of MILPs and time to solve themb = 8 b = 9 MILP Num Avg Time Num Avg Time 1 6.6e+04 7.0e-05 4.6e+00 2.6e+05 7.0e-05 1.8e+01 2 4.0e+05 1.5e-03 3.3e+02 1.6e+06 1.4e-03 1.1e+03 3 2.3e+05 9.1e-04 2.1e+02 9.2e+05 9.2e-04 8.4e+02 4 7.8e+05 9.9e-04 7.7e+02 4.4e+06 1.0e-03 4.5e+03 5 4.3e+05 2.8e-04 1.2e+02 1.7e+06 2.8e-04 4.9e+02 Table 3 : 3Single-input buck DC-DC converter: number of MILPs and time to solve them (continuation of Tab. 2)b = 10 b = 11 MILP Num Avg Time Num Avg Time 1 1.0e+06 2.7e-04 2.8e+02 4.2e+06 2.3e-04 9.7e+02 2 6.4e+06 3.8e-03 1.3e+04 2.5e+07 3.3e-03 4.6e+04 3 3.7e+06 3.0e-03 1.1e+04 1.5e+07 2.6e-03 3.8e+04 4 3.0e+07 2.6e-03 7.8e+04 2.6e+08 2.2e-03 5.7e+05 5 6.8e+06 1.8e-03 1.3e+04 2.7e+07 1.6e-03 4.2e+04 Table 4 : 4Multi-input buck DC-DC converter: control abstraction and controller synthesis results .88e+04 6.41e+07 7.38e+06 1.91e+04 0.00377 1.97e+01 1.21e+04 2.88e+04 8.35e+07 Sol 2 8.94e+04 7.63e+07 1.47e+07 1.91e+04 0.00743 2.66e+01 2.52e+04 8.94e+04 8.25e+07 Sol 3 2.46e+05 9.47e+07 2.93e+07 1.90e+04 0.01162 3.66e+01 3.47e+04 2.46e+05 1.05e+08 Sol 4 6.43e+05 9.51e+07 5.84e+07 1.88e+04 0.00330 5.32e+01 4.31e+04 6.43e+05 0.00e+00Control Abstraction Controller Synthesis Total n CPU MEM Arcs MaxLoops NoLoopsPerc CPU |K| CPU MEM µ 1 2Sol Table 5 : 5Multi-input buck DC-DC converter: number of MILPs and time to solve themn = 1 n = 2 MILP Num Avg Time Num Avg Time 1 1.0e+06 2.0e-04 2.1e+02 1.0e+06 2.1e-04 2.2e+02 2 6.4e+06 1.4e-03 5.1e+03 1.3e+07 1.9e-03 1.6e+04 3 3.7e+06 8.8e-04 3.2e+03 7.4e+06 1.6e-03 1.1e+04 4 8.7e+06 1.0e-03 8.9e+03 1.7e+07 1.7e-03 2.8e+04 5 6.9e+06 6.8e-04 4.6e+03 1.4e+07 1.1e-03 1.5e+04 Table 6 : 6Multi-input buck DC-DC converter: number of MILPs and time to solve them (continuation of Tab. 5)n = 3 n = 4 MILP Num Avg Time Num Avg Time 1 1.0e+06 2.1e-04 2.2e+02 1.0e+06 2.2e-04 2.3e+02 2 2.5e+07 3.0e-03 4.6e+04 5.1e+07 4.5e-03 1.2e+05 3 1.5e+07 2.2e-03 3.2e+04 2.9e+07 2.9e-03 8.6e+04 4 3.2e+07 2.4e-03 7.9e+04 6.3e+07 3.2e-03 2.0e+05 5 2.7e+07 1.6e-03 4.3e+04 5.5e+07 2.1e-03 1.1e+05 Figure 6: Single-input robust buck: controlled region with b = 9 bits Figure 7: Single-input robust buck: controlled region with b = 10 bits Figure 9: Multi-input buck: controlled region with n = 1 inputs Figure 10: Multi-input buck: controlled region with n = 2 inputs Figure 12: Multi-input buck: controlled region with n = 4 inputs Beyond hytech: Hybrid systems analysis using interval numerical methods. Thomas A Henzinger, Benjamin Horowitz, Rupak Majumdar, Howard Wong-Toi, HSCC. 1790Thomas A. Henzinger, Benjamin Horowitz, Rupak Majumdar, and Howard Wong-Toi. Beyond hytech: Hybrid systems analysis using in- terval numerical methods. In HSCC, LNCS 1790, pages 130-144, 2000. The embedded systems design challenge. A Thomas, Joseph Henzinger, Sifakis, FM. 4085Thomas A. Henzinger and Joseph Sifakis. The embedded systems design challenge. In FM, LNCS 4085, pages 1-15, 2006. Enabling on-chip switching regulators for multi-core processors using current staggering. W Kim, M S Gupta, G.-Y. Wei, D M Brooks, ASGI. W. Kim, M. S. Gupta, G.-Y. Wei, and D. M. Brooks. Enabling on-chip switching regulators for multi-core processors using current staggering. In ASGI, 2007. Type-2 fuzzy logic controller design for buck dc-dc converters. Ping-Zong Lin, Chun-Fei Hsu, Tsu-Tian Lee, FUZZ. Ping-Zong Lin, Chun-Fei Hsu, and Tsu-Tian Lee. Type-2 fuzzy logic controller design for buck dc-dc converters. In FUZZ, pages 365-370, 2005. Synthesis of quantized feedback control software for discrete time linear hybrid systems. Federico Mari, Igor Melatti, Ivano Salvo, Enrico Tronci, CAV. 6174Federico Mari, Igor Melatti, Ivano Salvo, and Enrico Tronci. Synthesis of quantized feedback control software for discrete time linear hybrid systems. In CAV, LNCS 6174, pages 180-195, 2010. A multiple-input digitally controlled buck converter for envelope tracking applications in radiofrequency power amplifiers. M Rodriguez, P Fernandez-Miaja, A Rodriguez, J Sebastian, IEEE Trans on Pow El. 252M. Rodriguez, P. Fernandez-Miaja, A. Rodriguez, and J. Sebastian. A multiple-input digitally controlled buck converter for envelope tracking applications in radiofrequency power amplifiers. IEEE Trans on Pow El, 25(2):369-381, 2010. A 480-mhz, multi-phase interleaved buck dc-dc converter with hysteretic control. G Schrom, P Hazucha, J Hahn, D S Gardner, B A Bloechel, G Dermer, S G Narendra, T Karnik, V De, PESC. IEEE6G. Schrom, P. Hazucha, J. Hahn, D.S. Gardner, B.A. Bloechel, G. Der- mer, S.G. Narendra, T. Karnik, and V. De. A 480-mhz, multi-phase interleaved buck dc-dc converter with hysteretic control. In PESC, pages 4702-4707 vol. 6. IEEE, 2004. Development of a fuzzy logic controller for dc/dc converters: design, computer simulation, and experimental evaluation. Wing-Chi, C K So, Yim-Shu Tse, Lee, IEEE Trans. on Power Electronics. 111Wing-Chi So, C.K. Tse, and Yim-Shu Lee. Development of a fuzzy logic controller for dc/dc converters: design, computer simulation, and experi- mental evaluation. IEEE Trans. on Power Electronics, 11(1):24-32, 1996. Proximate time-optimal digital control for synchronous buck dc-dc converters. V Yousefzadeh, A Babazadeh, B Ramachandran, E Alarcon, L Pao, D Maksimovic, IEEE Trans. on Power Electronics. 234V. Yousefzadeh, A. Babazadeh, B. Ramachandran, E. Alarcon, L. Pao, and D. Maksimovic. Proximate time-optimal digital control for syn- chronous buck dc-dc converters. IEEE Trans. on Power Electronics, 23(4):2018-2026, 2008.

A new class of enhanced non-perturbative corrections to the inclusiveB → Xsγ decay rate is identified, which contribute first at order Λ/m b in the heavy-quark expansion and cannot be described using a local operator product expansion. Instead, these effects are described in terms of hadronic matrix elements of non-local operators with component fields separated by light-like distances. They contribute to the high-energy part of the photon-energy spectrum but do not reduce to local operators when an integral over energy is taken to obtain the total inclusive decay rate. The dominant corrections depend on the flavor of the B-meson spectator quark and are described by tri-local fourquark operators. Their contribution is estimated using the vacuum insertion approximation. The corresponding uncertainty in the total decay rate is found to be at the few percent level. This new effect accounts for the leading contribution to the rate difference between B − andB 0 mesons.

10.1103/physrevd.75.114005

hep-ph/0609224

35669cd91bd26bb0f192765fa7e33e17884c55cb

Enhanced Non-local Power Corrections to theB → X s γ Decay Rate Jun 2007 Seung J Lee Institute for High-Energy Phenomenology Laboratory for Elementary-Particle Physics Cornell University 14853IthacaNYU.S.A Matthias Neubert Institute for High-Energy Phenomenology Laboratory for Elementary-Particle Physics Cornell University 14853IthacaNYU.S.A Institut für Physik (ThEP) Johannes Gutenberg-Universität D-55099MainzGermany Gil Paz School of Natural Sciences Institute for Advanced Study 08540PrincetonNJU.S.A Enhanced Non-local Power Corrections to theB → X s γ Decay Rate Jun 2007arXiv:hep-ph/0609224v2 15 A new class of enhanced non-perturbative corrections to the inclusiveB → Xsγ decay rate is identified, which contribute first at order Λ/m b in the heavy-quark expansion and cannot be described using a local operator product expansion. Instead, these effects are described in terms of hadronic matrix elements of non-local operators with component fields separated by light-like distances. They contribute to the high-energy part of the photon-energy spectrum but do not reduce to local operators when an integral over energy is taken to obtain the total inclusive decay rate. The dominant corrections depend on the flavor of the B-meson spectator quark and are described by tri-local fourquark operators. Their contribution is estimated using the vacuum insertion approximation. The corresponding uncertainty in the total decay rate is found to be at the few percent level. This new effect accounts for the leading contribution to the rate difference between B − andB 0 mesons. I. INTRODUCTION Precision studies of inclusive B-meson decays are a cornerstone of quark flavor physics. Detailed measurements of various kinematical distributions in the semileptonic decaysB → X lν, when combined with elaborate theoretical calculations, provide the currently most precise measurements of the elements |V cb | and |V ub | of the quark mixing matrix (see [1] for a comprehensive recent analysis of charmless inclusive decays). Studies of the rare decaysB → X s γ andB → X s l + l − allow sensitive tests of the flavor sector and provide constraints on extensions of the Standard Model. The theoretical description of inclusive B-meson decay rates is based on the operator product expansion (OPE) [2,3], by which total decay rates can be expressed in terms of forward B-meson matrix elements of local operators. Only two non-trivial matrix elements appear up to order (Λ/m b ) 2 in the expansion, one of which can be extracted from spectroscopy. The OPE breaks down when one tries to calculate differential inclusive decay distributions near phase-space boundaries. A twist expansion involving forward matrix elements of non-local light-cone operators (so-called shape functions) is then required to properly account for non-perturbative effects [4,5]. Recently, these non-local structures have been analyzed systematically beyond the leading order in Λ/m b [6][7][8]. It is generally believed that the non-local operators reduce to local ones when the differential decay distributions are integrated over all of phase-space. Here we show that this is not always the case. A precise control of hadronic power corrections is particularly important in the case of the inclusive radiative decayB → X s γ, which is the prototype of all flavorchanging neutral current processes. A significant effort is currently underway to complete the calculation of the leading-power (in Λ/m b ) contribution to the decay rate at next-to-next-to-leading order in renormalization-group improved perturbation theory. This leaves nonperturbative power corrections as the potentially largest source of theoretical uncertainty. It is well-known that inB → X s γ decay the OPE faces some limitations, which result from the fact that the photon has a partonic substructure. For instance, there exists a contribution to the total decay rate involving the interference of the b → sγ transition amplitude mediated by the electro-magnetic dipole operator Q 7γ with the charm-penguin amplitude (b → ccs followed by cc → γg) mediated by the current-current operator Q 1 (see [9] for the definition of the operators in the effective weak Hamiltonian). When the charm-quark is treated as a heavy quark (m c ∼ m b ), this contribution can be expanded in local operators [10][11][12][13], and it is believed to be a good approximation to keep only the first term in this expansion. Its contribution to the totalB → X s γ decay rate can be written as ∆Γ Γ 77 = − C 1 C 7γ λ 2 9m 2 c ≈ 0.03 , where λ 2 = 1 4 (m 2 B * − m 2 B ), and Γ 77 = G 2 F α 32π 4 |V tb V * ts | 2 m 5 b |C 7γ | 2 is the leading-order contribution to the decay rate from the electro-magnetic dipole operator. The ratio ∆Γ/Γ 77 therefore provides an estimate of the relative magnitude of the non-perturbative effect. We stress that when the scaling m 2 c ∼ Λm b is adopted instead of m c ∼ m b , then the charm-loop contribution must be described by the matrix element of a non-local operator [11][12][13][14]. It has been noted in [11] that the OPE forB → X s γ decay breaks down when one includes diagrams from operators other than Q 7γ , in which the photon couples to light quarks. An example of such a contribution, resulting from the decay b → sg mediated by the chromo-magnetic dipole operator Q 8g followed by photon emission from the light partons, was studied in [15]. It was argued that the corresponding effect can be estimated in terms of the parton fragmentation functions of a quark or gluon into a photon. Since this contribution is numerically very small, it has not received much further attention in the literature. A more careful analysis reveals that the correct interpretation of this effect is in terms of a subleading shape function [14]. In this Letter, we identify and analyze a novel class of non-local power corrections to theB → X s γ decay rate, which were not considered before. We argue that they can affect the total decay rate at the few percent level, and that they give the dominant flavor-specific contribution to the rate difference between charged and neutral B mesons. The presence of this effect leads to a dominant and irreducible source of theoretical uncertainty in the prediction for the totalB → X s γ branching fraction. II. NON-LOCAL POWER CORRECTIONS Power corrections to the high-energy part of theB → X s γ photon spectrum can be systematically parameterized in terms of subleading shape functions defined in terms of forward B-meson matrix elements of non-local light-cone string operators [6][7][8]. Some of these operators -the ones considered so far in the literature -reduce to local operators when one considers the total decay rate (i.e., the integral over the photon spectrum); however, a detailed analysis shows that several of them do not [14]. Representative diagrams giving rise to such operators are depicted in Figure 1. The graphs show different contributions to the discontinuity of the hadronic tensor W µν , which determines theB → X s γ photon-energy spectrum via the optical theorem. The total decay rate is obtained by an integration over the photon energy. In this Letter we focus on the two graphs shown in the first row (and two mirror graphs, in which the order of the weak vertices is interchanged). They describe the interference of the b → sγ transition amplitude mediated by the electromagnetic dipole operator Q 7γ with the b → sg amplitude mediated by the chromo-magnetic dipole operator Q 8g followed by the fragmentation of the gluon into an energetic photon and a soft quark-antiquark pair. While other diagrams, such as the first graph in the second row in the figure, give rise to four-quark operators containing strange quarks, the graphs in the first row produce all light-quark flavors. We expect that the resulting fourquark operators will have a larger overlap with the Bmeson states and thus give rise to the dominant power corrections. For simplicity, we also do not consider loopsuppressed effects such as the second graph in the second row of the figure. This diagram would match onto a nonlocal operator containing a soft gluon field, which mixes with the operators we consider under renormalization. The top two diagrams in Figure 1 affect theB → X s γ photon spectrum in the high-energy region, E γ ≈ m b /2, where it is most accessible to experiment. This is enforced by the fact that the amplitude mediated by the insertion of Q 8g interferes with the two-body decay amplitude mediated by the insertion of Q 7γ . The effect can therefore not be eliminated using kinematical cuts. We find that the corresponding contribution to the total decay rate can be parameterized in terms of forward Bmeson matrix elements of tri-local light-cone operators (with C F = 4/3 for N c = 3 colors): ∆Γ = −Γ 77 C 8g C 7γ 4πα s N c m b 0 −∞ ds 0 −∞ dt × B | C F (O 1 + O 2 ) − (T 1 + T 2 ) |B ,(1) where we have used that the Wilson coefficients C i are real in the Standard Model. The relevant factorization scale to use in this result is of order µ 2 ∼ m b Λ. We use a mass-independent normalization of meson states, such that B |h v h v |B = 2. The four-quark operators are defined as O 1 = q e qhv (0) Γ R q(tn)q(sn) Γ R h v (0) , O 2 = q e q 2h v (0) Γ R γ ⊥α q(tn)q(sn) γ α ⊥ Γ R h v (0) , T 1 = q e qhv (0) Γ R t a q(tn)q(sn) Γ R t a h v (0) , T 2 = q e q 2h v (0) Γ R γ ⊥α t a q(tn)q(sn) γ α ⊥ Γ R t a h v (0) , where e q is the electric charge of the soft light quark in units of e, Γ R = / n (1 + γ 5 )/2 is a right-handed Dirac structure, and t a are the generators of color SU(3). Here h v are the two-component heavy-quark fields defined in heavy-quark effective theory, while q are soft light-quark fields (q = u, d, s). The light-quark fields are located on the light-cone defined by the direction of the emitted photon with momentum q µ = E γn µ (withn 2 = 0). The non-local operators are made gauge invariant by the insertion of soft Wilson lines Sn in then-direction. The Wilson lines are absent in light-cone gaugen · A = 0, which we adopt implicitly to simplify the notation. Evaluating the hadronic matrix elements in (1) using a systematic non-perturbative approach is a very challenging task. In particular, lattice QCD is unable to handle operators with component fields separated by light-like distances. Naive dimensional analysis suggests that ∆Γ/Γ 77 ∼ (C 8g /C 7γ ) πα s (Λ/m b ), which could easily amount to a 5% correction to the decay rate. In more traditional applications of the OPE to inclusive B-meson decays, four-quark operators contribute at order (Λ/m b ) 3 in the heavy-quark expansion. The non-local operators in (1) lead to enhanced power corrections of order Λ/m b , because the two "vertical" propagators in Figure 1 have virtualities of order m b Λ and so introduce two powers of soft scales in the denominator. This mechanism was first studied in [16]. The existence of such enhanced power corrections in the total decay rate may seem puzzling at first sight. Consider the diagram in Figure 2, which represents the contribution to the total rate derived from the first graph in Figure 1. Before taking the cut indicated by the vertical dashed line the diagram only receives contributions from hard loop momenta p µ ∼ m b , and it would thus seem appropriate to shrink all propagators to a point. In this case the diagram would contribute at order (Λ/m b ) 3 in the heavy-quark expansion, and this scaling would appear to be preserved when one takes the discontinuity of the diagram, apparently contradicting our conclusion. The loophole is that the contribution to the totalB → X s γ decay rate is not given by the discontinuity of the loop graph, which would correspond to the sum of all three possible cuts, but instead it is given by the single cut shown in the figure. In order to obtain at least some model estimate of the magnitude of the effect in (1) we adopt the vacuum insertion approximation (VIA), in which the vacuum state |0 0| is inserted between the two light-quark fields inside the four-quark operators. This is a crude approximation, which however appears to work well in the analysis of b-hadron lifetimes [17,18]. The approximation has thus been checked for local four-quark operator evaluated be-tween B-meson states. Also, it can be justified using large-N c counting rules. Applications of the VIA to nonlocal operators can be found in [8,19] In the present case, the matrix elements of the operators O 2 and T 1,2 vanish in the VIA, either due to the color-octet structure of the quark bilinears (T 1,2 ) or due to the fact that there is no external perpendicular Lorentz vector available (O 2 and T 2 ). The matrix element of O 1 can be expressed in terms of the leading light-cone distribution amplitude of the B-meson in position space [20,21]. We obtain B | O 1 |B VIA = e q f 2 B m B 4 φ B + (s) [ φ B + (t)] * , where e q now refers to the charge of the light spectator quark in the B meson. The integral over the positionspace distribution amplitude can be evaluated to yield −i 0 −∞ ds φ B + (s) = ∞ 0 dω ω φ B + (ω) = 1 λ B , where φ B + (ω) is the light-cone distribution amplitude in momentum space, and λ −1 B is the common notation for the first inverse moment of this quantity [9]. Numerical estimates of λ B are very uncertain, but typically fall in the range between 0.25 and 0.75 GeV [9,[20][21][22][23][24]. In the VIA, our estimate of the spectator-dependent, non-local power corrections then takes the final form ∆Γ VIA Γ 77 = − e q C 8g C 7γ πα s 2 1 − 1 N 2 c f 2 B m B λ 2 B m b .(2) Recalling that f B ∼ 1/ √ m B in the heavy-quark limit, we indeed recover the scaling behavior anticipated above. Numerically, with µ ∼ 1.5 GeV as a typical factorization scale and f B ≈ 0.215 GeV for the B-meson decay constant (see [25] for a recent determination using unquenched lattice QCD), we obtain ∆Γ VIA Γ 77 ≈ −0.26e q f B λ B 2 ≈ −0.05e q λ B 0.5 GeV −2 , where e q = 2/3 for decays of B − mesons, while e q = −1/3 for decays ofB 0 mesons. For the range of λ B values quoted above, the effect is between −2% and −19% times e q . Taking Γ 77 as an estimate of the total decay rate at leading power, this implies that the enhanced power corrections to the total, flavor-averagedB → X s γ decay rate are expected (in the VIA) to be between −0.3% and −3%, while these effects induce a flavor-dependent rate asymmetry Γ(B − → X s γ) − Γ(B 0 → X s γ) Γ(B → X s γ) ≈ −0.05 λ B 0.5 GeV −2 ,(3) which could amount to an effect between −2% and −19%. When considering these estimates one should keep in mind that the VIA can at best provide a very simple model of the effect of the non-local four-quark operators in (1). Conservatively, we can therefore not exclude that the type of enhanced power corrections identified in this Letter could contribute to the totalB → X s γ decay rate at the 5% level. The magnitude of the flavor-specific effects studied above could be probed by a measurement of the flavor asymmetry (3); but there are other four-quark operator contributions with flavor structurebssb (see e.g. the bottom left diagram in Figure 1), whose matrix elements vanish in the VIA but could still be significant in real QCD. Their contributions are flavor-blind and hence not tested by (3). The Babar collaboration has measured the flavordependent rate asymmetry in eq. (3), finding the value (1.2 ± 11.6 ± 1.8 ± 4.8)% , where the errors are statistical, systematic and due to the production ratioB 0 /B − , respectively [26]. The dominant error is statistical and therefore likely to decrease when more data is collected. III. CONCLUSIONS We have identified a new class of enhanced power corrections to the total inclusiveB → X s γ decay rate, which cannot be parameterized in terms of matrix elements of local operators. These effects are nevertheless "calculable" in the sense that they can be expressed in terms of subleading shape functions. At tree level, the corresponding operators are tri-local four-quark operators. While local four-quark operators contribute at order (Λ/m b ) 3 in the heavy-quark expansion of the total decay rate, the effects we have explored are enhanced by the nonlocal structure of the operators and promoted to the level of Λ/m b corrections. We have identified and estimated what we believe are the dominant corrections of this type, namely those that match the flavor quantum numbers of the external B-meson states. Our results imply that a local operator product expansion for the inclusiveB → X s γ decay rate does not exist. Even at first order in Λ/m b there are hadronic effects that can only be accounted for in terms of non-local operators. The precise impact of these power corrections will be notoriously difficult to estimate using our present command of non-perturbative QCD on the light cone. While a naive estimate using the vacuum insertion approximation suggests that the effects are at the few percent level, we conclude that they are nevertheless a source of significant hadronic uncertainty in the calculation of partial or totalB → X s γ decay rates. After the perturbative analysis of the decay rate will have been completed, the enhanced non-local power corrections will remain as the dominant source of theoretical uncertainty. A measurement of the flavor-dependent asymmetry (3) could help to corroborate our numerical estimates of such corrections. 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We propose a new mechanism for the achievment of homochirality in life without any autocatalytic production process. Our model consists of a spontaneous production together with a recycling cross inhibition in a closed system. It is shown that although the rate equations for this system predict no chiral symmetry breaking, the stochastic master equation predicts complete homochirality. This is because the fluctuation induced by the discreteness of population numbers of participating molecules plays essential roles. This fluctuation conspires with the recyling cross inhibition to realize the homochirality.Recently, the amplification of ee (but not homochirality) was realized in experiments carried out by Soai et al.,12,13and the temporal evolution of the chemical reaction was shown to be explained by a second-order autocatalytic reaction. 14, 15 Stimulated by these works, we proposed that, in addition to the nonlinear autocatalytic reaction, a recycling process induced by a back reaction gives rise to the complete homochirality in a closed system. 16 Subsequently several theoretical works related to this mechanism have been done. 17-22 *

10.1143/jpsj.77.064606

0801.2841

6f1e93d473c7850718476150cd8e1fd1788e6cff

Fluctuation Induced Homochirality 18 Jan 2008 Takeshi Sugimori Department of Physics Keio University 223-8522Yokohama Hiroyuki Hyuga Department of Physics Keio University 223-8522Yokohama Yukio Saito Department of Physics Keio University 223-8522Yokohama Fluctuation Induced Homochirality 18 Jan 2008(Received February 2, 2008)Typeset with jpsj2.cls <ver.1.2> Full Paperhomochiralityprobability distributionmaster equationrecycling cross inhibi- tiondirected random walk We propose a new mechanism for the achievment of homochirality in life without any autocatalytic production process. Our model consists of a spontaneous production together with a recycling cross inhibition in a closed system. It is shown that although the rate equations for this system predict no chiral symmetry breaking, the stochastic master equation predicts complete homochirality. This is because the fluctuation induced by the discreteness of population numbers of participating molecules plays essential roles. This fluctuation conspires with the recyling cross inhibition to realize the homochirality.Recently, the amplification of ee (but not homochirality) was realized in experiments carried out by Soai et al.,12,13and the temporal evolution of the chemical reaction was shown to be explained by a second-order autocatalytic reaction. 14, 15 Stimulated by these works, we proposed that, in addition to the nonlinear autocatalytic reaction, a recycling process induced by a back reaction gives rise to the complete homochirality in a closed system. 16 Subsequently several theoretical works related to this mechanism have been done. 17-22 * Introduction and Model System It has long been known since the discovery of Pasteur that organic molecules in life are homochiral, in other words, having a completely broken chiral symmetry. 1,2 The origin of this homochirality remains an unsolved important puzzle. [3][4][5][6] Various mechanisms for the germination of chirality imbalance have been proposed such as different intensities of circularly polarized light in a primordial era, adsorption on optically active crystals, or the parity breaking in the weak interaction. 7 Predicted asymmetries, however, have turned out to be very small, 5,7,8 and therefore their amplification is indispensable. Frank showed theoretically that an autocatalytic reaction accompanying cross inhibition can lead to the amplification of enantiomeric excess (ee) and to the eventual homochirality in an open system. 9 Following this work, numerous studies have been performed on the chiral amplification and selection in various systems. 4-6, 10, 11 In these studies of chiral amplification, the autocatalytic reaction plays an essential role either in open systems or in closed systems. 4-6, 9-11, 16-22 So far, however, any autocatalytic reaction has not been found in the process of polymerization, relevant for the formation of organic molecules in life. Granting that some pertinent autocatalytic reaction may well be discovered in future, it seems worthwhile to explore possibilities of chirality selection in non-autocatalytic way. In this paper, we demonstrate that complete chirality selection or homochirality is possible in a closed system with spontaneous production together with recycling cross inhibition but without autocatalytic reaction. Our model consists of achiral substrate molecule A and two chiral enantiomers R and S, which are produced by spontaneous productions A → R, A → S(1) with the same reaction rate k 0 . Furthermore, R and S are assumed to react back to A as R + S → 2A(2) with a reaction rate µ 0 . We call this reaction a recycling cross inhibition, which looks similar to Frank's cross inhibition but differs in that whether R and S are recycled back in the present model or eliminated out of the system in the Frank's open model. 9 In §2, the rate equation approach shows that the system has a line of fixed points and chirality selection is impossible. Since the fixed line is neutral in stability, the system is expected to be susceptible to the weakest perturbations such as fluctuation. The rate equation, however, describes only the evolution of average quantities. To include fluctuation effect, one has to consider stochastic aspects of the system evolution. This feature can be taken into account in a stocahstic master equation approach, where the system is described by a probability distribution function. [23][24][25] From stochastic analysis in §3 and §4, it will be shown that the fluctuation drives the system to homochirality. The effect of the fluctuation is attributed to the discreteness of the microscopic process, the essence of which is extracted in the system size expansion in §5. The result is summarized in §6. Rate equation approach In the rate equation approach, the reaction processes, Eq.(1) and Eq.(2), are expressed as dr dt = k 0 a − µ 0 rs,(3)ds dt = k 0 a − µ 0 rs,(4) together with c = a + r + s (5) 2/13 where a, r, s are concentrations of species A, R, S respectively and the total concentration c is assumed to be constant. The conservation of total concentration expressed by Eq.(5) implies that R and S are recycled back to A via the cross inhibition reaction (−µ 0 rs). The trajectories of the evolution are easily obtained as r − s =constant, as shown by lines with arrows in Fig. 1. The final states are obtained by solvingṙ =ṡ = k 0 a − µ 0 rs = 0 together with a + r + s = c, resulting the following hyperbola r c + k 0 cµ 0 s c + k 0 cµ 0 = k 0 cµ 0 1 + k 0 cµ 0(6) as is shown in Fig. 1. There, the rate for the cross inhibition µ 0 is chosen to be very large compared to that for the spontaneous production k 0 as cµ 0 = 5k 0 , in order to draw the fixed line of hyperbola clearly visible away from the diagonal boundary line r + s = c. We expect, however, that the cross inhibition, if it exists, should be a very rare process and cµ 0 ≪ k 0 . The conclusion of this rate equation approach is that the enantiomeric excess (ee) defined as φ = r − s r + s(7) takes any value (−1 ≤ φ ≤ 1), depending on initial conditions, thus indicating no chirality selection. There is, however, a subtle feature such that, along the fixed line (hyperbola), the system is neither stable nor unstable, namely it is neutral. Fluctuations could be decisive to destruct this neutrality and resolve the hyperbola into a few fixed points. In fact, a similar situation with a neutral fixed line appears for a closed system with a nonlinear autocatalytic process. We have adopted a stocahstic master equation approach for this sytem, and found chiral symmetry breaking in a previous paper. 25 Therefore, we adopt the same approach for the present system in the following sections. Master equation approach For the stochastic approach, the relevant system is to be described in a microscopic way. Namely, the system is confined in a fixed volume V , containing species A, R, S with population numbers N A , N R , N S , respectively. Total population number of all molecules is assumed to be constant N = cV since the system is closed, so that we have N = N A + N R + N S .(8) Chemical reaction is assumed to be stochastic where a microscopic state specified by the population number X = (N A , N R , N S ) varies according to a certain transition probabilities W (X; q) of a jump q = (q A , q R , q S ) to another state X ′ = X + q. The probability P (X, t) of a state X at time t then evolves according to the master equation ∂P (X, t) ∂t = q W (X − q; q)P (X − q, t) − q W (X; q)P (X, t).(9) The summation by q is restricted by the conservation condition Eq. (8). The transition probabilities for the present model is explicitly expressed as W (N A , N R , N S ; −1, +1, 0) = k 0 N A , W (N A , N R , N S ; −1, 0, +1) = k 0 N A , W (N A , N R , N S ; +2, −1, −1) = µN R N S(10) and those with other q's vanish. As is confirmed later, the coefficient for the microscopic cross inhibition µ is related to the macroscopic reaction rate µ 0 as µ = µ 0 V = cµ 0 N .(11) Thus the master equation in concrete form is expressed as ∂P (N A , N R , N S , t) ∂t =k 0 (N A + 1) P (N A + 1, N R − 1, N S , t) + P (N A + 1, N R , N S − 1, t) + µ(N R + 1)(N S + 1)P (N A − 2, N R + 1, N S + 1, t) − {2k 0 N A + µN R N S }P (N A , N R , N S , t).(12) Because of the conservation condition Eq.(8) the microscopic state of the system is specified by two independent variables N R and N S in a triangular region to other states are zero, 0 ≤ N R , N S , N R + N S ≤ N,(13)W (0, N, 0; q) = W (0, 0, N ; q) = 0(14) for any jump q, so that, once the system enters into this homochiral states, it remains there. In this sense, the homochiral states are regarded as a sink or a kind of black hole. All other microscopic states are connected directly or indirectly to these two homochiral states, and consequently the probabilities of the other states evolve into zero, namely P (N A , N R , N S ; t = ∞) = 0 for non-homochiral states. This can be demonstrated step-by-step as follows. Because of Eq. (14), the master equation for the homochiral state takes the form ∂P (0, N, 0, t) ∂t =k 0 P (1, N − 1, 0, t).(15) In the asymptotic limit where every temporal evolution has died out ∂P/∂t = 0, the asymptotic value for this neighboring state becomes P (1, N − 1, 0, t = ∞) = 0. Furthermore, asymptotic limit of the master equation for a general state X becomes q W (X − q; q)P (X − q, ∞) = q W (X; q)P (X, ∞)(16) so that if P (X, ∞) = 0, then all the probabilities of states X ′ connected directly to the state In the late stage, the probability disitribution develops sharp peaks at the two end points corresponding to the two homochiral states (Fig. 3(c)). By the use of this probability distribution, an expectation value of any function f (N A , N R , N S ) at a time t is easily calculated as f (N A , N R , N S ) t = N A ,N R ,N S f (N A , N R , N S )P (N A , N R , N S , t)(17) In Fig. 4, the time development of the expectation value of the population number of R species N R t is shown. In the early stage in Fig. 4(a) the two homochiral states. The slow approach is expressed well by the following form N R t = N 2 − Ae −t/τ .(18) This exponential behavior is evident in Fig. 4 |φ(t)| = N R − N S N R + N S 2 1/2 t .(19) The time development of an ee order parameter |φ| is also shown in Fig.4(a) and (b). In the very early stage k 0 t ≤ 0.2 in Fig. 4(a) |φ| is very large because the probability has a large amplitude close to the edges N R = 0 or N S = 0. As time evolves, the numbers of R and S molecules increase and the peak position of the probability distribution leaves from both edges ( Fig. 3(a)), and thus the ee value |φ| drops sharply. After the peak of the probability distribution reaches the racemic fixed point where |φ| ≈ 0, the probability spreads along the fixed line ( Fig. 3(b)), and thus |φ| value increases steadily to a final value, unity, of homochirality. By plotting the logarithm of the difference, ln[1 − |φ(t)|], as a function of k 0 t, as in Fig. 4(b), one again obtains the exponential relaxation with the same exponent 1/k 0 τ ≈ 0.0162. Eigenvalue Analysis The asymptotic relaxation of the average value and the ee order parameter turned out to be exponential with the same characteristic time, k 0 τ ≈ 60. We consider this problem in terms of eigenvalues of evolution matrix of the master equation. Since master equation is a linear equation for the probability distribution, the time evolution is written by using the evolution k 0 / µ − k 0 / Λ 3 − k 0 / Λ 4 (c) (d)matrix M dP (X, t) dt = X ′ (X|M |X ′ )P (X ′ , t),(20) where matrix elements of M are related to the transition probabilities W as (X|M |X ′ ) =    W (X ′ ; X − X ′ ) for X ′ = X, − q W (X; q) for X ′ = X.(21) By the use of eigenfunctions Ψ i and eigenvalues Λ i of the matrix M as M Ψ i = Λ i Ψ i ,(22) the time development of the probability distribution is expressed as a series P (X, t) = ∞ i=1 a i e Λ i t Ψ i .(23) Since the probability distribution satisfies the conservation as X P (X, t) = 1, all the eigenvalues must be non-positive. In the previous section, we have proven that two homochiral states corresponds to the final states, so that there are two degenerate zero eigenvalues Λ 1 = Λ 2 = 0 8/13 with eigenstates Ψ 1 = δ N A ,0 δ N R ,N δ N S ,0 and Ψ 2 = δ N A ,0 δ N R ,0 δ N S ,N , or their linear combinations. The asymptotic temporal evolution is governed then by the third largest eigenvalue Λ 3 . One can calculate eigenvalues of the matrix M numerically by using the subroutine "dgeev" in LAPACK. A few largest eigenvalues are shown in Fig. 5 k 0 /µ, asymptotically. Therefore, for a very large system with a small cross inhibition, it should take a long time of the order N/µ before the homochirality becomes observable. System size expansion For our model with spontaneous production of chiral species with recycling cross inhibition, the rate equation tells us no chirality selection whereas the stochastic master equation insists the final configuration be homochiral. The totally different conclusions are ascribed to the fluctuation due to the discreteness of microscopic processes. The fluctuation effect associated to the system size is qualitatively analyzed by the system size (precisely said, the inverse system size) expansion analysis of the master equation, developed by R. Kubo et al. 27,28 In the master Eq.(9), the probability density P (X, t) of a microscopic state X is connected to another state X + q which differs with a jump q of order unity by a transition probability W (X; q). The rate W is of macroscopic order of the system size N or the volume V as W (X; q) = V w(x; q)(24) where x is the density variable x = X/V of order unity: x = (a, r, s) = N A V , N R V , N S V .(25) Then, the probability is assumed to take the form P (X, t) = exp[V χ(x, t)], and time evolutions of the leading order contributions of the average density x t and the correlation functions σ ij (t) = V (x i − x i t )(x j − x j t )(26) are shown to be determined by moments c ij··· (x) = q q i q j · · · w(x; q)(27) 9/13 of the transition probability w as d dt x i t = c i ( x t ) d dt σ ij (t) = k ∂c i ∂ x k t σ kj (t) + σ ik (t) ∂c j ∂ x k t + c ij ( x t ).(28) The evolution equations for the higher order corrections can also be derived. 27 If the system is normal, the fluctuation correlations σ ij remains of order unity. On the other hand, if the system is unstable, as in the case of phase transitions with critical behaviors, at least one of the fluctuations σ ij is enhanced to the order of the system size. In the present model, first order moments are c r = c s = k 0 a − µ 0 rs = k 0 (c − r − s) − µ 0 rs(29) and the second order moments are c rr = c ss = k 0 a + µ 0 rs = k 0 (c − r − s) + µ 0 rs, c rs = µ 0 rs,(30) where µ 0 = V µ as defined previously in Eq. (11). Thus, the lowest order of the average concen- To detect the chiral symmetry breaking, it is more convenient to use the following symmetric and asymmetric variables x + = r + s, x − = r − s.(32) Their averages and correlation functions evolve as d dt x + = 2k 0 (c − x + ) − 1 2 µ 0 (x 2 + − x 2 − ), d dt x − = 0, d dt σ ++ = −2(2k 0 + µ 0 x + )σ ++ + 2k 0 (c − x + ) + µ 0 (x 2 + − x 2 − ), d dt σ +− = −(2k 0 + µ 0 x + )σ +− + µ 0 x − σ −− , d dt σ −− = 2k 0 (c − x + ).(33) By starting from the completely achiral initial condition, the system remains racemic with average value x − = 0, and x + approaches to the racemic fixed point value (2k 0 /µ 0 )( cµ 0 /k 0 + 1 − 1). The time the system takes to reach the racemic fixed point is of order 1/k 0 . 10/13 The fluctuation of the asymmetry variable σ −− , however, increases indefinitely in this approximation. In fact, when the average value of x + takes the racemic point value, the fluctuation increases linearly in time with a positive velocity; σ −− c = 1 2 µ 0 c x 2 + = 1 2 N µ x + c 2 > 0.(34) This increase of the fluctuation indicates that the discreteness in the microscopic process of chemical reaction evokes an intrinsic instability of the racemic fixed point (and in general, every points on the fixed line) to macroscopic level. In the numerical simulation the fluctuation of asymmetric variable σ −− (t)/c actually increases in time, as shown in Fig. 6. The initial increase shown by a continuous curve in Fig. 6(a) is linear in time k 0 t, as expected from the system size expansion analysis, shown by a dashed line. As time passes, the system size analysis fails, since the average value x + deviates from the racemic value in the actual simulation. One then has to consider higher order contributions to the average of x + . Departure from the racemic state becomes visible in a macrosopic sense when the fluctuation expressed by σ −− /c reaches the order N . The time for this to appear is of order 1/µ, as is obtained from Eq.(34) and by assuming x + ∼ c. Asymptotically, the double peak structure developes in the probability distribution. The time scale for this to fully develop is governed by the largest nonzero eigenvalue Λ 3 so that the homochirality is realized in the time of order of 1/|Λ 3 | ≈ N/µ, as is described in the previous section. Conclusion and discussion It seems to be a general consensus that an autocatalytic production process, either linear or nonlinear, is indispensable for the realization of homochirality. In the present work, we 11/13 proposed a new mechanism by using a simple model to demonstrated that the homochirality can be realized without any autocatalytic production process. Our model consists of a spontaeous production together with a recycling cross inhibition in a closed system. As was shown, the rate equations for this system predict no chiral symmetry breaking, but the stochastic master equation predicts complete homochirality. This is because the fluctuation induced by the discreteness of population numbers of participating molecules plays essential roles. This fluctuation conspires with the recyling cross inhibition to realize the homochirality. If this fluctuation mechanism could explain the homochirality in life, then this is what Pearson suggested long time ago. 29 However, the necessary time for the homochirality to set in due to the fluctuation is very large , as it is proportional to the total number of the relevant molecules (≈ N/µ), which is of macroscopic size. Taking this feature into account, we can conceive a new senario for the homochirality in macroscopic scale as follows. At first, in some small closed corner, such as in a region enclosed by a vesicle, the fluctuation induced homochirality is realized in a very long time with respect to the time scale of laboratory experiments, but not so long with respect to geological time scale. It may be conceivable that in this period some unknown autocatalytic reaction, the effect of which is too small to be detected in the laboratory experiment so far, begins to operate and generates the large scale realization of the homochiralty from the homochiral seeds produced by the fluctuation mechanism proposed here. Even though reaction systems with a recycling cross inhibition are not yet found, we hope that a simple system with only a recycling cross inhibition might be found in near future, and the establishment of homochirality be checked. Fig. 1 . 1Trajectory of the concentration r and s, prescribed by the rate equations(3)and(4), with a fixed line of hyperbola. The parameter is set as cµ 0 = 5k 0 . Fig. 2 . 2as shown in Fig. 2. Transition probabilities W connect neighboring states linked along square edges or along a diagonal, indicated in Fig. 2. The system is equivalent to the directed random walk model where a random walker jumps to the right (+1 in N R direction), to upwards (+1 in N S direction) and to the left-down diagonal ( −1 in both N R and N S directions) in the N R − N S phase space. One then notices easily that two homochiral states (N R , N S ) = (N, 0) and (0, N ) are special; there is only inflow but no outflow of the walker, namely the transition Random walk in a triangular region in N R − N S space. A walker at a site (N R , N S ) jumps along square edges or along a diagonal indicated by arrows. probabilities from the microscopic homochiral states (N A , N R , N S ) = (0, N, 0) and (0, 0, N ) XFig. 3 . 3with positive W 's must have vanishing asymptotic values P (X ′ , ∞) = 0. Since every state is interconnected, this completes the proof.This conclusion is confirmed numerically by solving the time development of the master equation Eq.(12) by using the Runge-Kutta method of the fourth order.26 Starting from a completely achiral initial condition with N A = N and no chiral ingredients N R = N S = 0, the probability distribution, started from the initial delta-peak P (N A , N R , N S , t = 0) = δ N A ,N δ N R ,0 δ N S ,0 , remains symmetric at any time, as shown inFig. 3. There, the redundant variable N A = N − N R − N S is suppressed, and the probability distribution is shown in N R and N S phase space. The probability contour is also shown at the basement. Time evolution of the probability distribution with trajectory of the probability contour at the basement, obtained by numerically integrating the master equation(9)for µ = k 0 /10. The system starts from P (N, 0, 0) = 1 and depicted times are at (a)k 0 t = 1, (b)k 0 t = 9 and (c)k 0 t = 19. The total population number is N = 50.numerical calculation, the total population number is set as N = 50 due to the limitation of the calculationa capacity and µ = k 0 /10 (or cµ 0 = 5k 0 as inFig. 1) for the sake of visibility of distribution. In the early stage at a time k 0 t = 1 inFig.3(a), the probability distribution has an approximately Gaussian shape with a central peak at the racemic fixed point N R = N S = (k 0 /µ)( N µ/k 0 + 1 − 1) ≈ 14.5 in agreement with results obtained by the rate eqautions. At the intermediate time k 0 t = 9, the probability distibution spreads along the hyperbola which is the fixed line found in the rate equation approach, as shown inFig.3(b). , the average N R t increases sharply to the racemic fixed point value 14.5 predicted by the rate equation within a time scale of k 0 t ≈ 1. This development is well described by the rate equations Eq.(3) and Eq.(4) together with Eq.(5), as is shown by a dashed curve in Fig. 4(a). The results by the rate equations completely agrees with the evolution of average value N R t until k 0 t ≈ 1. Then, the rate equations predict the saturations at the racemic value, but the numerical simulation of the master equations indicates a slow increase. The average N R t approaches ultimately to the value N/2, corresponding to the double peak profile of the final probability distribution at 6/13 N 0 Fig. 4 . 04R t ] is plotted versus k 0 t. The fitting gives values A = 10 and 1/k 0 τ = Time development of the average population N R t and the absolute value of the ee order parameter |φ|. (a) In an early stage, k 0 t ≤ 1, N R t agrees with the evolution by the rate equation, which is shown by a dashed curve. (b) Exponential relaxation in the asymptotic region. The enantiomeric excess (ee) corresponds to an order parameter in a phase transition in standard statistical mechanics. Adopting an analogous definition of an order parameter in numerical simulations for magnetic phase transitions, we define the absolute value of ee as Fig. 5 . 5A few largest eigenvalues Λ i of the time-evolution matrix plotted (a) versus total population N at a fixed cross inhibition, µ/k 0 = 0.1, and (c) versus µ/k 0 at a fixed N = 50. By plotting the inverse of eigenvalues in (b) and (d), one finds k 0 /|Λ i | is linear in N and also in k 0 /µ, asymptotically. as a function of the total population number N for a fixed cross inhibition µ = 0.1k 0 (Figs. 5(a) and (b)), or as a function of µ/k 0 for a fixed N = 50 (Figs. 5(c) and (d)). For µ = 0.1k 0 and N = 50, the third largest eigenvalue has a value Λ 3 /k 0 = −0.0163, in good agreement with the exponent 1/k 0 τ = 0.0163 obtained in the previous section from the asymptotic final relaxation of the average N R t and the ee values |φ(t)|. By plotting the inverse of eigenvalues −k 0 /Λ i as in Figs. 5(b) and (d), one notices that it is linear in the total population number N and the inverse strength of the cross inhibition trations r t and s t satisfy the rate equations Eq.(3) and Eq.(4). For simplicity, we describe these lowest order average values as r and s, hereafter. The correlation functions follow the evolution d dt σ rr (t) = −2(k 0 + µ 0 s)σ rr − 2(k 0 + µ 0 r)σ rs + k 0 (c − r − s) + µ 0 rs d dt σ rs (t) = −(k 0 + µ 0 s)σ rr − (2k 0 + µ 0 r + µ 0 s)σ rs − (k 0 + µ 0 r)σ ss + µ 0 rs d dt σ ss (t) = −2(k 0 + µ 0 r)σ ss − 2(k 0 + µ 0 s)σ rs + k 0 (c − r − s) + µ 0 rs Fig. 6 . 6Fluctuation correlation functions obtained by numerically time integrating the master equation (a) at an early stage, and (b) a whole time range. Parameter is set as µ = k 0 /10, or cµ 0 = 5k 0 . A dotted line in (a) represents the result of system size expansion. Acknowledgement Y. S. acknowledges support by a Grant-in-Aid for Scientific Research (No. 19540410) from the Japan Society for the Promotion of Science. 12/13 L Pasteur, Comptes Rendus. 26535L. Pasteur: Comptes Rendus, 26 (1848) 535. . F R Japp, Nature. 58452F. R. Japp: Nature 58 (1898) 452. L Stryer, Biochemistry (Feeman and Comp. New YorkL. Stryer: Biochemistry (Feeman and Comp., New York, 1998). M Calvin, Chemical Evolution. OxfordOxford University PressM. Calvin: Chemical Evolution (Oxford University Press, Oxford, 1969). . V I Goldanskii, V V Kuz&apos;min, Z. Phys. Chem. (Leipzig). 269216V. I. Goldanskii and V. V. Kuz'min: Z. Phys. Chem. (Leipzig) 269 (1988) 216. . I D Gridnev, Chem. Lett. 35148I. D. Gridnev: Chem. Lett. 35 (2006) 148. . B L Feringa, R A Van Delden, Angew. Chem. Int. Ed. 383419See for example a review by B. L. Feringa and R. A. van Delden: Angew. Chem. Int. Ed. 38 (1999) 3419. . M H Mills, Chem. Ind. 750M. H. Mills: Chem. Ind. (London) 51 (1932) 750. . F C Frank, Biochimi. Biophys. Acta. 11459F. C. Frank: Biochimi. Biophys. Acta 11 (1953) 459. . P G H Sandars, Orig. Life. Evol. Biosph. 33575P. G. H. Sandars: Orig. Life. Evol. Biosph. 33 (2003) 575. . A Brandenburgand, T Multamaki, Int. J. Astrobiol. 3209A. Brandenburgand T. Multamaki: Int. J. Astrobiol. 3 (2004) 209. . K Soai, T Shibata, H Morioka, K Choji, Nature. 378767K. Soai, T. Shibata, H. Morioka, and K. Choji: Nature 378 (1995) 767. . K Soai, T Shibata, I Sato, Acc. Chem. Res. 33382K. Soai, T. Shibata, and I. Sato: Acc. Chem. Res. 33 (2000) 382. . I Sato, D Omiya, K Tsukiyama, Y Ogi, K Soai, Tetrahedron: Asymmetry. 12I. Sato, D. Omiya, K. Tsukiyama, Y. Ogi, and K. Soai: Tetrahedron: Asymmetry 12 (2001) 1965. . I Sato, D Omiya, H Igarashi, K Kato, Y Ogi, K Tsukiyama, K Soai, Tetrahedron: Asymmetry. 14975I. Sato, D. Omiya, H. Igarashi, K. Kato, Y. Ogi, K. Tsukiyama, and K. Soai: Tetrahedron: Asym- metry 14 (2003) 975. . Y Saito, H Hyuga, J. Phys. Soc. Jpn. 7333Y. Saito, and H. Hyuga: J. Phys. Soc. Jpn. 73 (2004) 33. . Y Saito, H Hyuga, J. Phys. Soc. Jpn. 731685Y. Saito, and H. Hyuga: J. Phys. Soc. Jpn. 73 (2004) 1685. . Y Saito, H Hyuga, J. Phys. Soc. Jpn. 74535Y. Saito, and H. Hyuga: J. Phys. Soc. Jpn. 74 (2005) 535. . Y Saito, H Hyuga, J. Phys. Soc. Jpn. 741629Y. Saito, and H. Hyuga: J. Phys. Soc. Jpn. 74 (2005) 1629. Y Saito, H Hyuga, Progress in Chemical Physics Research. A. N. LinkeNew York65NOVAY. Saito, and H. Hyuga: in Progress in Chemical Physics Research, ed. A. N. Linke, Ch.3, (NOVA, New York, 2005) p.65. . R Shibata, Y Saito, H Hyuga, Phy. Rev. E. 7426117R. Shibata, Y. Saito, and H. Hyuga: Phy. Rev. E 74 (2006) 026117. Y Saito, H Hyuga, Topics in Current Chemistry: Amplification of Chirality. K. SoaiHeidelbergSpringerY. Saito and H. Hyuga: in Topics in Current Chemistry: Amplification of Chirality, ed. K. Soai, (Springer, Heidelberg, 2007). . G Lente, J. Phys. Chem. 1089475G. Lente: J. Phys. Chem. 108 (2004) 9475. . G Lente, J. Phys. Chem. 10911058G. Lente: J. Phys. 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Oxide heterostructures allow for detailed studies of 2D electronic transport phenomena. Herein, di erent facets of magnetotransport in selected spin-orbit-coupled systems are analyzed and characterized by their single-band and multiband behavior, respectively. Experimentally, temperature-and magnetic eld-dependent measurements in the single-band system BaPbO 3 /SrTiO 3 reveal strong interplay of weak antilocalization (WAL) and electron-electron interaction (EEI). Within a scheme which treats both, WAL and EEI, on an equal footing a strong contribution of EEI at low temperatures is found which suggests the emergence of a strongly correlated ground state. Furthermore, now considering multiband e ects as they appear, e. g., in the model system LaAlO 3 /SrTiO 3 , theoretical investigations predict a huge impact of lling on the topological Hall e ect in systems with intermingled bands. Already weak band coupling produces striking deviations from the well-known Hall conductivity that are explainable in a fully quantum mechanical treatment which builds upon the hybridization of intersecting Hofstadter bands.

10.1002/pssb.202100154

2104.07270

e9d6237afb37311a8ff954164583bd9c91f8af34

Magnetotransport of Functional Oxide Heterostructures A ected by Spin-Orbit Coupling: A Tale of Two-Dimensional Systems Robert Bartel Center for Electronic Correlations and Magnetism Experimental Physics VI Institute of Physics University of Augsburg 86135AugsburgGermany Elias Lettl Center for Electronic Correlations and Magnetism Experimental Physics VI Institute of Physics University of Augsburg 86135AugsburgGermany Patrick Seiler Center for Electronic Correlations and Magnetism Experimental Physics VI Institute of Physics University of Augsburg 86135AugsburgGermany Thilo Kopp Center for Electronic Correlations and Magnetism Experimental Physics VI Institute of Physics University of Augsburg 86135AugsburgGermany German Hammerl Center for Electronic Correlations and Magnetism Experimental Physics VI Institute of Physics University of Augsburg 86135AugsburgGermany Magnetotransport of Functional Oxide Heterostructures A ected by Spin-Orbit Coupling: A Tale of Two-Dimensional Systems (Dated: October 14, 2021)magnetotransportspin-orbit couplingoxide heterostructures2Dweak antilocalizationmultibandHofstadter bands Oxide heterostructures allow for detailed studies of 2D electronic transport phenomena. Herein, di erent facets of magnetotransport in selected spin-orbit-coupled systems are analyzed and characterized by their single-band and multiband behavior, respectively. Experimentally, temperature-and magnetic eld-dependent measurements in the single-band system BaPbO 3 /SrTiO 3 reveal strong interplay of weak antilocalization (WAL) and electron-electron interaction (EEI). Within a scheme which treats both, WAL and EEI, on an equal footing a strong contribution of EEI at low temperatures is found which suggests the emergence of a strongly correlated ground state. Furthermore, now considering multiband e ects as they appear, e. g., in the model system LaAlO 3 /SrTiO 3 , theoretical investigations predict a huge impact of lling on the topological Hall e ect in systems with intermingled bands. Already weak band coupling produces striking deviations from the well-known Hall conductivity that are explainable in a fully quantum mechanical treatment which builds upon the hybridization of intersecting Hofstadter bands. I. INTRODUCTION Perovskite-related oxides show a huge variety of intrinsic properties. [1] With oxide heterostructures, it is not only possible to combine such material characteristics but also to identify novel electronic phases emerging on the nanoscale which allows to trigger a plethora of functionalities. [2,3] At the interfaces of certain polar insulators con ned metallic electronic systems appear driven by electronic reconstruction. [4,5] In addition, inversion symmetry is systemically broken, a key ingredient for strong Rashba-type spin-orbit coupling, leading to anomalous transport properties which will be addressed in this article. Moreover, such electronic systems, when gapped, may assume nontrivial values of topological invariants causing a particular behavior of their magnetotransport. In fact, magnetotransport allows to obtain a ngerprint of the electronic state of metals, especially also of oxide heterostructures with their complex electronic properties controlled by sizable spin-orbit coupling, multiband behavior, disorder, and Coulomb interaction. This article covers two complementary spin-orbit-coupled electronic systems, both with regard to magnetotransport: a disordered and a defect-free 2D system. Correspondingly, the article is organized as follows: In Section II, we examine experimentally BaPbO 3 thin-lms grown on SrTiO 3 . The perovskiterelated oxide BaPbO 3 is a single-band metal. In this system with Rashba spin-orbit coupling disorder accounts for weak antilocalization (WAL) in the presence of electron-electron interaction (EEI). We brie y introduce these theoretical concepts of quantum corrections to transport properties before we analyze our temperature-and magnetic eld-dependent measurements. We then self-consistently extract parameters describing spin-orbit coupling and EEI-indicating a correlated * [email protected] ground state in BaPbO 3 . In a further step toward a general understanding it suggests itself to consider the spin-orbit coupling dominated magnetotransport beyond the single-band 2D systems. In Section III, we analyze the in uence of magnetic elds on the transport properties of a defect-free 2D multiband system in the fully quantum mechanical treatment of linear response theory. Our work is inspired by the fact that magnetotransport studies of LaAlO 3 /SrTiO 3 interfaces under applied hydrostatic pressure can lead to counterintuitive results if evaluated with standard semiclassical techniques. [6] However, as the semiclassical Boltzmann transport theory builds upon a single-band model its validity in case of multiband systems like LaAlO 3 /SrTiO 3 should be questioned. [7] This is especially true if one expects topological band aspects to play a fundamental role. After a general model description, we start by analyzing magnetotransport for the single-band case revisiting the results of the Hofstadter model. In a next step, we discuss multiband behavior a ected by atomic or Rashba-type spin-orbit coupling. II. MAGNETOTRANSPORT IN SINGLE-BAND SYSTEMS GOVERNED BY DISORDER Recently, we found that BaPbO 3 thin-lms grown on (001)oriented SrTiO 3 single crystals show single-band behavior and a pronounced magnetoresistance (MR) which at low magnetic elds is evidently ruled by WAL. [8] Surprisingly, temperaturedependent measurements of the sheet resistance □ ( ) account for an insulating low-temperature state, contradicting the WAL result of magnetoconductance. Such a counterintuitive behavior of thin-lm samples was observed before. [9,10] It is argued that MR and □ ( ) may originate from distinct sensitive channels leading to di erent measurement-dependent ground states. [9][10][11][12] By carefully investigating MR and □ ( ), we unveiled that the expected WAL contribution in □ ( ) is covered by a pronounced EEI contribution. However, up to now, we neglected the mutual e ect of EEI to MR as we considered it to be small. Before we examine the in uence of EEI on the WAL signal in our samples, let us discuss the generic temperature and magnetic eld dependencies on the quantum corrections of the electrical transport of a disordered 2D system. Due to weak disorder low-temperature electronic transport in 2D materials is a ected by quantum interference (QI) resulting either in insulating or metallic ground states. QI contributes signi cantly to the electrical transport only if the electrons' temperature-dependent dephasing time is large compared with, e. g., the elastic scattering time e : randomly scattered electrons will unavoidably self-interfere constructively with their time-reversal counterparts leading to WL with its insulating ground state. [13][14][15][16][17] Pronounced spin-orbit (SO) coupling described by a timescale so associated with the D'yakonov-Perel' spin relaxation ( so ≪ ) instead contributes an additional phase causing WAL which induces a metallic ground state. [15,[18][19][20] Both QI e ects, WL and WAL, are characteristically in uenced by applied time-reversal symmetry-breaking external magnetic elds which makes it possible to experimentally decide on the type of quantum corrections. A comprehensive description of the magnetic eld-dependent rst-order quantum correction to the conductivity of an ideal 2D material is given by the well-accepted Iordanskii-Lyanda-Geller-Pikus theory which relates the speci c magnetic eld dependence to the winding number of the spin expectation value around the Fermi surface. [19,21,22] In case of triple spin winding, found in, e. g., SrTiO 3 -based 2D thin-lms, [23][24][25] the Iordanskii-Lyanda-Geller-Pikus theory merges to the analytical result of the Hikami-Nagaoka-Larkin theory. [15] The rst-order quantum correction to the conductivity in applied magnetic eld triggered by QI can then be expressed as dependent resistance ( ) via MR = ( ) − (0) (0) = 1 1 + (0)Δ QI ( ) − 1, (II.4) where the 2D resistivity is identi ed with the sheet resistance □ = ⋅ with and being the measurement bar's length and width, respectively. To compare the conductivity in uenced either by magnetic elds or temperature, Equation (II.1) can be further adapted: Evaluating δ QI ( ) in the limit of zero magnetic eld the rst-order correction to the conductivity can be individually expressed for both low-temperature states associated with WL and WAL, respectively: in case of so ≫ ( so ≪ ) Equation (II.1) treats WL and simpli es to is controlled by inelastic scattering and an algebraic temperature dependence of is assumed by ( ) = + , (II.7) with being a scaling factor, modeling a saturation in dephasing at zero temperature, and being an exponent in the range between 1 and 2 combining contributions of both electron-phonon and electron-electron scattering. [27,28] With the help of Equation (II.7), the rst-order quantum corrections to the conductivity become temperature-dependent with an insulating state in case of WL Both progressions are exclusively driven by the temperature dependence of the dephasing scattering with WL and WAL being temperature-independent constants determined by WL and WAL, respectively. An insulating ground state is not necessarily induced by Anderson localization but can also be incited by EEI. [16,17,29] In 2D systems, the conductivity correction due to EEI reveals nearly the same logarithmic temperature dependence compared with WL δ EEI ( ) = 2 ℎ ln EEI , (II. 10) with being an exponent related to screening e ects and ranging between 0.35 for no screening and 1 for perfect screening, and EEI being a temperature-independent constant de ned by EEI. The temperature dependence can again be compared with magnetic eld-dependent measurements as in the presence of magnetic elds a nite Zeeman splitting (ZS) is responsible for a sizable magnetoconductivity in 2D systems: [17] Δ ZS ̃ ( ) = − 2 ℎ 2(1 − ) 3 2D ̃ ( ) , (II.11) with̃ ( ) = ( B )/( B ), the Landé factor, and 2D a function de ned by 2D ̃ ( ) = ∞ 0 d ln | | | | 1 −̃ ( ) 2 2 | | | | d 2 d 2 exp( ) − 1 , (II.12) which can be evaluated numerically. A. Sample Growth and Characterization of BaPbO 3 Thin-Films All samples discussed were grown by pulsed laser deposition (PLD). The PLD system uses a KrF excimer laser with a wavelength of 248 nm and a nominal uency of 2 J cm −2 . The used polycrystalline BaPbO 3 targets were obtained commercially with asked maximum achievable density. They are evaluated to have purities of at least 99.95 %. Prior to each sample growth the surface of the targets were carefully cleaned. BaPbO 3 thin-lms were grown on commercially available, one-side polished, (001)-oriented single-crystalline SrTiO 3 substrates with a given size of 5 mm × 5 mm × 1 mm. To obtain de ned BaPbO 3 /SrTiO 3 interfaces the substrates were either TiO 2 terminated using a hydrogen uoride (HF) bu er solution [30,31] and subsequently annealed in pure oxygen ow at about 950°C for 7 h or cleansed by lens paper as well as ultrasonic bath treatment in acetone and isopropyl. The substrates were then xed for either infrared laser heating or resistive heating on appropriate platforms using silver paste and transferred via a load-lock system and transfer chamber into permanently air-sealed PLD vacuum chambers. Depending on the pretreatment the substrates were either slowly heated to nominally 554°C during at least 60 min in case of HF-treated substrates or heated up to 800°C within a few minutes for at least 5 min in case of cleansed substrates to purify the substrate surface and then reheated to about 554°C within seconds, both in a pure oxygen background pressure of about 1 mbar. Thin-lm deposition was done using a nominal laser pulse energy of 550 mJ and 650 mJ-depending on the used PLD chamber-at a laser frequency of 5 Hz. The number of laser pulses was chosen individually resulting in desired thin-lm thicknesses. With this setup, the growth rate of BaPbO 3 was determined to be about 0.34 nm per laser pulse. After thin-lm deposition, the vacuum chamber was immediately lled with pure oxygen to at least 400 mbar, whereas the sample was cooled to about 400°C within 3 min and kept at that temperature for additional 17 min for annealing. Then the sample was allowed to freely cool-down to room temperature before the chamber was evacuated again for unloading the sample. Film thicknesses were routinely obtained by X-ray re ectivity (XRR). Conducted XRR ts resulted in averaged surface and interface roughness better than 0.6 nm and 0.7 nm, respectively. X-ray di raction (XRD) measurements indicate that all epitaxial BaPbO 3 layers are (001)-oriented. All samples were patterned into four-probe and Hall bar layouts using a standard photolithography system (mercury arc lamp) followed-up by ion-milling. To minimize contact resistances gold was sputtered onto the contact pads. All samples were electrically contacted using copper wires (0.1 mm in diameter) soldered to the puck and glued via silver paste to the samples. All electrical transport measurements were carried out using a commercial 14-T physical property measurement system (PPMS) with an electrical transport option (ETO). The applied AC currents were in the range of 0.1 µA to 1 µA with typical frequencies from 70 Hz to 128 Hz. B. Experimental Results and Discussion In this article, we account for the EEI contribution intrinsically involved in the MR data. Assuming both WAL and EEI contributing equally via Equation (II.3) and (II.11), we self-consistently evaluate MR and □ ( ) within the following iterative scheme: We start by applying Equation (II.3) to our raw MR data and extract the WAL contribution neglecting any EEI contribution during the rst iteration. Subsequently, with the help of Equation (II.9), we are able to subtract the WAL contribution to reveal the pure temperature-dependent sheet resistance due to EEI which then provides a value of the screening factor . By accounting for a pronounced Zeeman splitting the MR data can now be reevaluated again allowing for a priorly hidden EEI contribution that is described by Equation (II.11) with a presumed Landé factor = 2. We carry out this procedure successively until the screening factor settles to a constant value. To avoid oscillations which may prevent convergenceas is close and limited to 1-we average the obtained values within the last three iterations. Exemplarily the result of such a self-consistent evaluation of MR and □ ( ) in terms of WAL and EEI are shown in Figure 1 and 2. Figure 1 shows temperature-dependent MR data taken from a 15.0 nm-thick BaPbO 3 thin-lm showing an increase in MR to a maximum value at a magnetic eld of ≈ 0.85 T with a following decrease at higher magnetic elds, con rming our former results. [8] The MR data were corrected from concomitant EEI by subtracting its contribution via Equation (II.11) with = 0.91 retrieved from □ ( ) analysis. As expected, EEI contributes only slightly (see colored lines in Figure 1). The reevaluated MR data can now be perfectly tted in terms of WAL using Equation (II.3). Further, the ts result in an averaged so ≈ 0.22 T and a temperature dependence of that can be best described with = 1.99 following Equation (II.7) supporting a dephasing Figure 2). The EEI contribution for each temperature is plotted as a solid line with its corresponding color. Black solid lines show best ts (least squares method) of the EEIcorrected MR data (see "corr. ", i. e., colored dots) using Equation (II.3) resulting in an averaged value of so ≈ 0.22 T. The obtained temperature dependence of can be described by an algebraic dependence (Equation (II.7)) with = 1.99, = 0.14 mT K − , and = 7.50 mT (not shown) determining the WAL correction in the □ ( ) analysis (see Figure 2). mechanism mainly due to electron-phonon scattering. Simultaneously taken □ ( ) data are likewise a ected by EEI at low temperature, see zero-eld data in Figure 3: Upon cooling starting from room temperature □ steadily decreases, then reaches a minimum at about 11.1 K and subsequently rises again. The high-temperature progression can be well understood in terms of electron-phonon scattering as well as thermally activated dislocation scattering. [32] The low-temperature behavior is unequivocally controlled by quantum corrections. Figure 2 shows the change of the conductivity Δ = 1 ( ) − 1 ( ref ) (II.13) normalized to ref = 11.1 K. The measured data were reevaluated by subtracting the in uence of WAL following Equation (II.9) with parameters acquired from evaluations of the MR. The corrected data show a clear logarithmic increase perfectly described by EEI following Equation (II.10) that results in = 0.91. For consistency, we applied the just established self-consistent calculations of so and to the data presented in [8] comparing di erent sample thicknesses: For the sample with thickness 21.3 nm, the WAL contribution expressed by so changes in its average value from 0.10 T to 0.13 T, whereas EEI represented by remains unchanged at a value of 0.97. The 4.8 nm-thick sample shows a small increase in so from 0.23 T to 0.24 T in average, whereas changes from 0.84 to 0.89. It will be interesting to further study the thickness dependence on both the WAL and EEI contributions. An independent approach to extract the EEI contribution without being a ected by WAL is the magnetic eld dependence of □ ( ). Magnetic elds > cause the quantum corrections induced by QI (δ QI ) to become temperature independent [33] and therefore to vanish by evaluating Δ QI ( ) = δ QI ( ) − δ QI ( ref ). (II.14) Hence, in the presence of even small magnetic elds, the temperature dependence of the conductance below ref should be solely reigned by EEI. In Figure 3, the temperature-dependent progression of □ ( ) as well as Δ ( ) normalized to now ref = 6 K are plotted, both in logarithmic scale. The magnetic eld further increases □ pronouncing the insulating ground state according to the expected suppression of WAL e ects. The gradient | | (which translates directly into the value of in case of suppressed WAL) extracted from linear ts clearly increases and saturates at | | ≈ 0.915 in reasonable good agreement with our previous result ( = 0.91). III. MAGNETOTRANSPORT IN MULTIBAND SYSTEMS IN THE CLEAN LIMIT Magnetotransport studies have also been carried out on multiband oxide heterostructures. For example, for the conned electronic system at the interface of LaAlO 3 /SrTiO 3 , an EEI contribution was suggested to dominate transport at low temperatures. [32] This interpretation was challenged in a more recent WAL analysis within the framework of a semiclassical approach to multiband magnetotransport. [6] A fully quantum mechanical multiband treatment of WAL was established for degenerate, isotropic t 2g bands. [34,35] However, for various multiband systems, such as the electron system at the LaAlO 3 /SrTiO 3 interface, band hybridization at crossing points or rather lines is present in the relevant lling regime. This so far has not been addressed within a fully quantum mechanical approach to WAL. Here, as a rst step to a more realistic modeling, we develop a description of magnetotransport in the presence of band crossings within an e ective two-band model for a defect-free lattice system. We investigate explicitly the Hall conductivity in the presence of atomic and Rashba-like spin-orbit coupling. Before we reexamine the prerequisites of magnetotransport of a single-band model and the two-band case with its particular Hall conductivity, let us introduce the generic model description. We use a tight-binding representation for the Hamiltonian of a noninteracting electron system in an in nite 2D crystalline lattice where is the elementary electric charge. [39][40][41] As the coordinate operator (Equation (III.2)) is assumed to be diagonal, the e ect of a homogeneous external magnetic eld on the orbital degrees of freedom is given purely in terms of the Peierls phase. [42,43] No further parameters enter the model description. [44,45] In general, the Hamiltonian will then not commute with the lattice translation operators T , because of the real space dependence of the vector potential. For a homogeneous external magnetic eld with rational ux / per 2D unit cell, in units of the magnetic ux quantum 0 = ℎ/ , translation symmetry can be restored by introducing magnetic translation operators T M . [46][47][48] Those are a combination of a gauge transformation and a lattice translation. They do not commute with each other except if transporting a particle to the opposite corner of a parallelogram penetrated by an integer number of magnetic ux quanta. The smallest such parallelogram with a nonvanishing area is the so-called magnetic unit cell, which is a times enlarged version of the lattice unit cell, so that it is penetrated by an integer number of magnetic ux quanta. Here and in the following and are assumed to be coprime integers. The quantum numbers of the commuting magnetic translation operators are good quantum numbers to characterize the eigenstates of the Hamiltonian. They replace the lattice momenta of the translation invariant system, resulting again in a Hamiltonian in reciprocal space of the form of Equation (III.1), where , now label the states in a magnetic unit cell. From a band perspective, the enlargement of the unit cell to a magnetic one leads to a splitting of each of the initial dispersion relations without eld into magnetic Bloch bands (so-called Hofstadter bands). Each of the Hofstadter bands contains only a fraction 1/ of the states of the original bands. [49] Under applied magnetic eld the matrix elements of the current operator in the eigenbasis of the Hamiltonian, as appearing in Equation (III.4), have the same -fold degeneracy in the magnetic BZ as the eigenvalues. [49] The integral over must therefore in the magnetic case only be taken over a reduced part of the magnetic BZ. [50][51][52] A. Anisotropic Hofstadter Model Within this framework, we now consider a square lattice with one orbital per site and nearest-neighbor hopping only: H = BZ d 2 −2 x cos( x ) − 2 y cos y c † c . (III.6) The lattice spacing is set to 1 and spin polarization is assumed. We note that a rectangular lattice geometry would in the following only lead to a scaling of longitudinal conductivities and densities of states. We allow for an asymmetry in the hopping strength along the two di erent bond directions. By introducing the Peierls phase to account for a homogeneous magnetic ux through the lattice cells, one arrives at the Harper-Hofstadter Hamiltonian. [49,53] To review how band structure and topology a ect the conductivity of the anisotropic Hofstadter model, we rst choose a ux of / = 1/10. The original cosine band is then split up into = 10 separate Hofstadter bands, as long as the system is truly 2D ( x ≠ 0 ≠ y ). In case of being even the two middle sub-bands in the Hofstadter model touch. [49,54] All other bands are isolated by nite energy gaps and have a Chern number of +1. [54,55] This can be veri ed in Figure 4, as the longitudinal conductivity vanishes in those gaps, whereas the transversal conductivity is quantized in units of the conduction quantum 2 /ℎ. This holds approximately true even at nite temperatures and scattering rates, as long as temperature B and scattering-induced energy broadening ℏ are much smaller than the bandgaps. On the other hand, if the chemical potential is placed within a Hofstadter band, one calculates a nite Drude weight in case of the longitudinal conductivity and the Hall signal is shifted away from its quantized values. In the limit of a 1D system with either x = 0 or y = 0, the Peierls phase can be gauged away completely. One is left with the eld-free model with a single band with zero Hall signature. As the anisotropy between the hopping parameters in xand y-directions is in-/decreased, only the contributions to the conductivities, which are not of topological character, approach the fully an-/isotropic limit (see yellow/orange lines in Figure 4). For lling factors = / , on the other hand, where Hofstadter bands are completely lled, the conductivities are invariant as long as no single energy gap becomes too small. By reducing the magnetic ux ( Figure 5) for a xed value of the anisotropy with 0 < y / x < 1, one can see that the Hall signal is lling-wise divided into distinct regimes where it either approaches the fully anisotropic or the isotropic limit. The same holds true also for the longitudinal conductivity. The boundaries between those di erent cases are associated with the positions of the logarithmic Van Hove singularities of the eld-free model. [56] This is reasonable if one recalls that those two Van Hove singularities originate from the saddle points of the band structure and are thus at the same llings as the transitions between di erent kinds of semiclassical orbits. [57] In this speci c case, one nds closed orbits for low and high llings of the anisotropic Hofstadter model, whereas in between the logarithmic Van Hove singularities only open orbits exist (see insets in Figure 5c). The isotropic limit is a special case: the two considered Van Hove singularities merge in energy, which leads to an immediate switching from electron to hole-like closed orbits, with only a single energy level in between accommodating open orbits. [58] In the fully anisotropic limit, on the other hand, there are only open orbits, which are purely 1D and yield no Hall signal as already mentioned. The sharp topological peaks in the regions of open orbits that one nds for high magnetic elds (Figure 4b) are washed out quickly with decreasing magnetic eld by nite temperatures and scattering, as there the gaps between the Hofstadter bands become small. From semiclassical Boltzmann transport theory, one can deduce an expression for the nontopological contributions to the Hall conductivity of the considered band model at zero temperature, assuming y ≤ x : xy DC = − 2 ℎ − | x ( )| , (III.7) where | x ( )| is the absolute value of the time averaged xvalue along a semiclassical orbit at the Fermi surface for a certain band lling (compare [59,60] For a similar study about open and closed orbits in the Hofstadter model where the anisotropy is due to a diatomic basis see [56]. B. E ective Two-Band Model in a Perpendicular Magnetic Field With knowledge of the magnetotransport behavior of the single-band model from Section III A, one can now proceed to study a multiband system, where two such square lattice cosine bands are combined. Its eld-free Hamiltonian is given by H = BZ d 2 =1,2 − 2[ x cos( x ) + y cos y ](c ) † c + (c ) † c + =1,2 ( y ) , x (c ) † c , (III.8) where allows for a relative energy shift of the two bands against each other, x is the rst Pauli matrix, and Δ( y ) controls a spin-orbit-like coupling e ect (see the following text). We assume that both states in a unit cell ( = 1, 2) are centered at the same point ( 1 = 2 ). To provide a speci c example of a perovskite oxide, Hamiltonian (III.8) can accommodate each reduced set of two out of the six spin-orbital states of the e ective LaAlO 3 /SrTiO 3 band model. [7,61,62] As such, it allows us to study the complex patterns of the Hall signal for every pair of bands individually, without interference from a plethora of additional states. The interplay between the anisotropic d yz -/d zx -bands of the 3d t 2g orbitals of titanium and the isotropic d xy -band governs the main structure of the Hall signal of the e ective six-band model. From this perspective we now concentrate on the Hall conductivity emerging from coupling of an anisotropic ( = 1, 1 y = 0.25 1 x ) and an isotropic ( = 2, 2 x = 2 y = 1 x ) cosine band. Neglecting the energy shift in the e ective LaAlO 3 /SrTiO 3 band model due to spacial anisotropy at the interface, these two bands are assumed to be aligned at their bottom. This arrangement leads to a match in energy, and thus lling, of the logarithmic Van Hove singularity of the isotropic band with the upper singularity of the anisotropic band. A two-band model with slightly di erent relative band positions would be treated analogously. Two di erent coupling e ects will be considered. A constant coupling term with Δ( y ) = as it arises in the sixband model between the d xy -band and the d yz -/d zx -bands due to atomic spin-orbit coupling. Furthermore, a -dependent coupling Δ( y ) = − sin y is examined. It resembles the coupling term between the d xy -band and the d yz -/d zx -bands, introduced by the symmetry breaking at the LaAlO 3 /SrTiO 3 interface. [61,62] First, we inspect the Hall conductivity of the two uncoupled bands plotted against the lling factor, as its structure already changes nontrivially with respect to the single-band behavior studied in Section III A. The additional structural complexity is caused by the di ering densities of states of the two bands. Consequently, the conductivity of the uncoupled two-band system may only be obtained by superposition of the individual signals after a nontrivial transformation of each of them along the lling axis. By color coding the total Hall conductivity (Figure 6, purple sections belong to = 1, orange sections refer to the orbital contribution = 2), the signal is again resolvable from a single-band perspective. In addition to the asymmetry of the signal with respect to half lling, which results from the alignment of the two bands at their bottom, the most prominent new feature in the Hall conductivity is a step-like descent for llings between the logarithmic Van Hove singularities. It should not be confused with the similar looking quantized Hall conductivity resulting from gaps in the energy spectrum when plotted against the chemical potential. In Figure 6, the signal is shown versus band lling, e ectively skipping energy gaps in the dispersion speci ed by a quantized Hall conductivity. Thus, the "treads" of those steps cannot be the result of bandgaps. Instead, they are the Hall signal of the wider Hofstadter bands of the anisotropic cosine band, which has open semiclassical orbits in this range of lling, leading to a nearly suppressed transversal conductivity. The narrow energy gaps between those wider Hofstadter bands manifest themselves in Figure 6 as narrow "topological peaks" interrupting the horizontal progression of the step treads. However, as seen in the single-band case in Section III A, they are quickly washed out by scattering and temperature, remaining only visible in the vicinity of the logarithmic Van Hove singularities. The step "risers", on the other hand, can be traced back to the at Hofstadter bands of the isotropic cosine band, corresponding to closed semiclassical orbits. Typically, such a at Hofstadter band (with = 2) is placed energetically somewhere within a wider one (with = 1). When the chemical potential reaches this at Hofstadter band its much higher density of states leads to a near total suspension of the lling up of the wider band, until no empty states are left in the at band. Thus, the slope of the Hall conductivity changes abruptly compared with the step treads and the height of the riser assumes a nearly quantized value (of 2 /ℎ). The regime with the step-like behavior is then expected to be heavily a ected already by adding a weak coupling term Δ( y ) to the Hamiltonian (Figure 7), as the di erent Hofstadter bands will hybridize strongest at their intersection lines. In the case of a weak magnetic eld (Figure 7b), it is actually the only range of lling where the Hall signal of the weakly coupled bands di ers signi cantly from the one of the uncoupled bands. It is striking that a weak perturbation modi es the Hall signal qualitatively-an observation that will be explained below. The other a ected region around the coinciding logarithmic Van Hove singularities (Figure 7a), where the Hall signal switches its sign, will not be investigated closer, as it shrinks to zero width in the low magnetic eld limit. For weak coupling strengths, the deviation from the behavior of the uncoupled bands in the step-like region can be well understood by rst looking at higher magnetic elds (Figure 8). Band structure and Berry curvature are for weak coupling only distorted in the vicinity of the former band crossings. So the Hall signal is expected to stay mostly unchanged. It can only deviate signi cantly from that of uncoupled bands in the lling ranges of the step risers (e. g., 0.2 < < 0.3, in Figure 8a). For a at primary Hofstadter band intersecting a wider one, the shape of the Hall signal of the hybridized bands can be constructed based on two facts: band repulsion and the Chern numbers of the hybridized magnetic Bloch bands. By hybridization, the wider primary Hofstadter band is split apart at the energy of the at band and each part is merged with half of the at band, which is itself split along the intersection lines. Thus forming two new nonintersecting hybridized magnetic Bloch bands. For weak coupling strengths, the new bands in the regions around the former crossings are pushed above/below the energy of the primary at Hofstadter band, due to band repulsion. In contrast, in the other regions of the BZ, the band dispersions and also the Berry curvatures are nearly unchanged. This means that lling-wise the progression of the transversal conductivity only changes at the two edges of the former step riser, whereas in the middle part of that region one still nds the same linear trend as before. For strong coupling, all hybridized magnetic Bloch bands in this regime are energetically separated from each other by nite bandgaps. A Chern number of +1 can in this case easily be read o from Figure 8b for each of the new bands (see the peaks at llings of completely lled magnetic Bloch bands lined up along a descending line). This must also hold true for the weak coupling case, assuming the bands do not cross while reducing the coupling strength-albeit the hybridized bands may eventually overlap if the atter band has a nite width. Somewhere in the middle of the former step riser the energetically lower one of the two hybridized magnetic Bloch bands is completely lled. Assuming energetically nonover- lapping bands or, equivalently, that the upper hybridized band only contributes linearly up to this lling factor, the Hall signal must thus already be shifted down to the descending gray line connecting the integer topological values in Figure 8a. Otherwise, the Chern numbers of the nonintersecting hybridized bands could not be matched correctly. This leads to a broad dip replacing the step riser. It is the separation of the bands due to the hybridization that causes this sizable nite down shift of the Hall signal. Inspecting the case of a slightly weaker magnetic eld more thoroughly (Figure 7), where the assumption of totally at primary Hofstadter bands is even more accurate, one sees that such a broad dip appears at every former step riser. Thereby replacing the step-like descent by an oscillatory behavior, varying between the signal of the uncoupled bands and the "topological limit". The gaps between the wider Hofstadter bands, associated with the anisotropic cosine band, must have also been slightly enlarged by the band coupling. In particular one can now identify their narrow peaks in the whole region between the logarithmic Van Hove singularities (Figure 7a), where they were suppressed before by nite temperature and scattering. For higher temperatures, the energy broadening of B will eventually extend over the range of several magnetic Bloch bands. This then leads to an averaging out of these oscillations. Lowering the magnetic eld has the same e ect with the addition that new phenomena can arise due to a nite coupling strength, which can then also depend on the speci c form of Δ( y ). IV. CONCLUSION We discussed 2D magnetotransport in the presence of spin-orbit coupling in single-band systems with disorder as well as multiband systems in the clean limit. Experimentally, we extracted self-consistently both WAL and EEI contributions emerging as rst-order quantum corrections to the electrical transport properties of thin BaPbO 3 lms. Thus, we o er a consistent way to interpret quantum corrections on 2D lms to thoroughly identify an electronically correlated and insulating low-temperature state. Furthermore, going from a single-band system to a general multiband setup, we investigated a defect-free lattice system which reveals a striking behavior when electronic bands hybridize in the presence of a magnetic eld. We rst reanalyzed the Hall conductivity of the anisotropic Hofstadter model, where open semiclassical orbits lead to a deviation from the well-known linear behavior in the electron density of closed orbits. This fundamental knowledge of the single-band behavior of the conductivity then allowed us to fully understand an uncoupled multiband system. The additional e ects of a weak band coupling in this multiband system can be explained by the hybridization of intersecting Hofstadter bands instead of the eld-free bands. Hereafter, it would be intriguing to investigate a disordered system in a generic multiband setup to merge the aspects investigated in our complementary studies. The implementation of band hybridization into a generalized version of the Iordanskii-Lyanda-Geller-Pikus theory will be challenging but allows for a fundamental understanding of multiband quantum interference. case of so ≪ ( so ≫ ) it relates to WAL and reads δ WAL ( Figure 1 . 1Reevaluated MR data taken at di erent temperatures of a 15.0 nm-thick BaPbO 3 thin-lm grown on a (001)-oriented singlecrystalline SrTiO 3 substrate. The symmetrized raw data were selfconsistently corrected from EEI contributions (Equation (II.11)) with the presumed value of = 2 and = 0.91 retrieved from analysis of the □ ( ) measurement (see Figure 2 . 2Progression of the change in conductivity referenced to the temperature ref = 11.1 K in logarithmic scale. Blue dots show original measured data, whereas the expected progression for WAL is plotted in orange, retrieved from MR analysis (seeFigure 1)following Equation (II.9) indicating a metallic low-temperature state. In green, WAL-revised data are shown which are perfectly explained by the EEI contribution solely (Equation (II.10))-resulting in = 0.91. Figure 3 . 3Progressions of □ ( ) as well as changes in conductivity normalized to ref = 6 K while applying various perpendicular magnetic elds between 0.2 T and 1 T (in logarithmic scale). The magnetic eld suppresses WAL contributions, whereas EEI contributions are una ected. The slopes clearly show trends toward an insulating ground state as the magnetic eld increases. The gradient | | was linearly tted, representing in case of suppressed WAL. The resulting ranging between 0.91 and 0.92 is in good agreement with the prior self-consistent analysis ( = 0.91). a lattice vector. Lowercase Greek letters , label the states within a unit cell. The integral over the lattice momenta is taken over the rst Brillouin zone (BZ), the area of which we denote by BZ .The coordinate operator is assumed to be diagonal in the chosen basis { the position of the state within the unit cell. For the last equality to hold the Fourier transformation of the creation and annihilation operators must be de ned for each state individually with respect to its exact position: theory provides us with the Kubo formula for the electric conductivity in the static limit , ⟩ describes an eigenstate of the Hamiltonian in band and the corresponding eigenvalue.[36][37][38] For numerical stability has to be kept nite, which may be roughly interpreted as a nite scattering rate. The Fermi distribution ( ) actually also depends on the chemical potential and temperature. The electric current operator in the reciprocal basis can be written in terms of the gradient of the Hamilto- Figure 4 . 4Quantization of conductivities: a) longitudinal and b) transversal signals for di erent anisotropy values in the Hofstadter model plotted versus band lling with a magnetic ux per unit cell of / = 1/10 of a magnetic ux quantum. The evaluations are done setting B = 5 x ⋅ 10 −4 and ℏ = x ⋅ 10 −3 . The periodicity of the eld-free system is taken to be = 12000 lattice cells in each direction. Figure 5 . 5Hall conductivity in respect to open and closed semiclassical orbits: Hall signal for a) / = 1/30 of a ux quantum per unit cell and b) / = 1/300. All other parameters are unchanged from Figure 4. c) Density of states of the eld-free model at the Fermi energy in dependence of the lling factor. The insets show the Fermi surfaces at certain llings in the rst BZ, where the horizontal axis represents x and the vertical axis y . In all three sub gures, the positions of the logarithmic Van Hove singularities of the model with anisotropy y / x = 0.25 and the isotropic case are indicated by vertical lines. . So in case of only closed electron orbits | x | = 0 and in case of exclusively closed hole orbits at the Fermi level | x | = , whereas for open orbits, | x ( )| is bounded by the minimal and maximal absolute x -value of the open orbit. Thus, opposed to the standard textbook derivations, Equation (III.7) is not limited to the linear contributions of closed electron or hole orbits to the transversal conductivity. [60] It describes the complete lling range, even the suppression of the Hall signal for open orbits and the switching from electron to hole-like behavior at half lling. Figure 6 . 6Hall conductivity of an uncoupled two-band model (Δ( y ) = 0) broken down into distinct band contributions: second band. The two bands are aligned at their bottom ( = (−1) 0.75 1 x ). System size, temperature, and scattering rate are chosen as in Figure 4 and the magnetic ux is at / = 1/30. Vertical gray lines indicate the positions of logarithmic Van Hove singularities. By calculating the relative change in the Hall signal with lling, coming from state = 1 of the eld-free model, one obtains the impact of this state on the Hall conductivity at a certain lling factor (see color coding). Loosely speaking, dark purple sections result from the anisotropic band ( = 1) and dark orange sections are contributions from the isotropic band ( = 2). Figure 7 . 7Evolution of the Hall conductivity for a hybridized twoband model: a) Hall signal from Figure 6 (Δ( y ) = 0) compared with the two di erent types of band coupling Δ( y ) = and Δ( y ) = − sin y for small coupling constants. b) Same as a) but with a reduced magnetic eld. Figure 8 . 8Reappearance of a topologically quantized Hall conductivity for hybridized Hofstadter bands: Hall signal for the uncoupled model of Figure 6 (Δ( y ) = 0) with an increased magnetic ux of / = 1/10 compared with the two di erent types of band coupling Δ( y ) = and Δ( y ) = − sin y for a) weak and b) strong coupling. The lling of half an electron per unit cell ( = 0.25) is marked by a vertical gray line. For the lower llings, the integer topological values of the Hall signal line up (descending gray line), as there the magnetic Bloch bands all have a Chern number of +1. In the 2D case, experimental data are often presented in terms of the related MR calculated from the magnetic eld-δ QI ( ) = 2 ℎ ψ 1 2 + so + − 1 2 ψ 1 2 + + 1 2 ψ 1 2 + 2 so + − ψ 1 2 + e , (II.1) with ψ being the digamma function. [26] The introduced ef- fective magnetic elds are related to the scattering times via e/ /so = ℏ 4 e/ /so , (II.2) with being the di usion constant. 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We obtain 21 two-dimensional natural Hamiltonian systems with sextic invariants, which are polynomial of the sixth order in momenta. Following to Bertrand, Darboux, and Drach these results of the standard brute force experiments can be applied to construct a new mathematical theory.

10.1016/j.cnsns.2022.106404

2110.12860

52830bc52690b401f70f41368a416285f1a7721a

On two-dimensional Hamiltonian systems with sixth-order integrals of motion 28 Oct 2021 E O Porubov St.Petersburg State University St.PetersburgRussia A V Tsiganov [email protected] St.Petersburg State University St.PetersburgRussia On two-dimensional Hamiltonian systems with sixth-order integrals of motion 28 Oct 2021integrable systems We obtain 21 two-dimensional natural Hamiltonian systems with sextic invariants, which are polynomial of the sixth order in momenta. Following to Bertrand, Darboux, and Drach these results of the standard brute force experiments can be applied to construct a new mathematical theory. Introduction A Hamiltonian system is called natural if Hamiltonian is the sum of positive-definite kinetic energy and potential H = T (q, p) + V (q) . The search of natural Hamiltonian systems admitting integral of motion that is a polynomial of order N in the momenta is a classical problem [4,5], which is currently intensively studied, see [2,10,11,12,15,23] and references within. There is few two-dimensional natural Hamiltonian systems with invariants (constants of motion or integrals of motion), which are polynomials of the sixth order in momenta. According to [13] the Hamiltonian vector field with Hamiltonian H = p 2 1 + p 2 2 + V (q 1 , q 2 ) , admits sextic invariants for a potential of the Holt-type system V = 12q 4/3 2 + (q 2 1 + a)q −2/3 2 + bq −2 1 , Toda-type system 1 3 , superintegrable Calogero-type system V 1 = e ( √ 3q 1 −q 2 ) 2 + e q2 + e − √ 3q1 , V 2 = e ( √ 3q 1 −q 2 ) 2 + e q2 + e − √ 3qV = v(2q 2 ) + v( √ 3q 1 + q 2 ) + v(− √ 3q 1 + q 2 ) , v(z) = z −2 + a 2 2 z −2 , and a few superintegrable systems associated with the Chebychev theorem on binomial differentials. In [9] we obtained one more system with sextic invariant at V = (q −2/3 1 + a)q −2/3 2 , which is a generalization of the Fokas-Lagerström system having a cubic invariant [7,13]. On the pseudo-Euclidean space, there are more natural Hamiltonian systems with sextic invariant [14] H = p 1 p 2 + q a 1 q b 2 , where a and b are rational numbers associated with a special set of parameters of the hypergeometric function, for instance V = q 2 1 q −10/7 2 , V = q −2/3 1 q −5/6 2 , V = q −2/3 1 q −7/3 2 . Similar to the Holt and Fokas-Lagerstrom potentials we have singular potentials which are well-defined functions only on part of the plane. In [18] we construct one more Hamiltonian system with invariant of degree six on the two-dimensional space with metric depending on local coordinates H = p 2 1 2m 1 (q 1 , q 2 ) + p 2 2 2m 2 (q 1 , q 2 ) + V (q 1 , q 2 ) , i.e. a natural Hamiltonian system with position-dependent mass (effective mass). This system has been obtained using divisor arithmetic on an elliptic curve [18,19]. Below we present 20 similar natural Hamiltonian systems with effective mass and sextic invariant. Dynamical systems with position-dependent mass were first introduced in the theory of semiconductor physics, and now these systems can be found in many fields, such as classical Hamiltonian and non-Hamiltonian mechanics, quantum mechanics, relativistic mechanics, nuclear physics, molecular physics, neutrino mass oscillations, quantum information and so on, see references in recent papers [1,3,8,16,22]. Our aim is not to construct integrable systems with a sixth degree invariant or to study the applications of these systems in physics. Similar to Darboux [5] and Drach [6] we use the direct method of constructing invariants to collect a sufficient number of examples that allow us to build a mathematical theory explaining the existence of such sextic invariants. Two-dimensional geodesic motion with cubic invariant Let us consider two-dimensional metric g =    (kq 1 + q 2 )q m 1 q 1 − q 2 0 0 (kq 2 + q 1 )q m 2 q 2 − q 1    , q 1 < 0 < q 2 , and the corresponding kinetic energy T = g 11 p 2 1 + g 22 p 2 2 = (kq 1 + q 2 )q m 1 q 1 − q 2 p 2 1 + (kq 2 + q 1 )q m 2 q 2 − q 1 p 2 2 ,(1.1) derived in [18] in the framework of divisor arithmetic on an elliptic curve. Here q 1,2 are the position variables and p 1,2 are the conjugate momenta for canonical Poisson brackets {q 1 , p 1 } = {q 2 , p 2 } = 1 , {q 1 , q 2 } = {q 1 , p 2 } = {q 2 , p 1 } = {p 1 , p 2 } = 0 . At k = ±1 Hamiltonian system with geodesic Hamiltonian T (1.1) admits linear invariant and, therefore, sextic invariant may be obtained by using the Chebyshev theorem on binomial differentials [9]. Here, we do not consider this construction of integrals of motion. Proposition 1 At special values of parameters m and k • m = 1 , k = 2; • m = 3 , k = 1 2 , 3; • m = 4 , k = ±3, − 3 5 , − 1 7 , 1 5 ,1 2 ; geodesic Hamiltonian T (1.1) commutes {T, K} = 0 with the reducible cubic polynomial K = K 1 K 2 , K 1 = (q 2 1 p 1 − q 2 2 p 2 ) , where K 2 is a polynomial of the second order in momenta. These special values of parameters are related to divisor arithmetic on an elliptic curve, which allows us to get separated variables for the corresponding Hamilton-Jacobi equation T = E and prove that these Hamiltonian systems are bi-Hamiltonian. Proposition 2 At special values of parameters m and k • m = 1 , k = 2; • m = 3 , k = 1 2 , 2; • m = 4 , k = −3, 1 3 , 1 5 , 3 5 , 2; geodesic Hamiltonian T (1.1) commutes {T, L} = 0 with the reducible quartic polynomial L = K 2 1 L 2 , K 1 = (q 2 1 p 1 − q 2 2 p 2 ) , where L 2 is a polynomial of the second order in momenta. Thus, we have superintegrable geodesic motion at • m = 1 , k = 2; • m = 3 , k = 1 2 ; • m = 4 , k = −3,1 5 ; and in all these cases {L, K} = K 2 . Following the example from [18], below we consider integrable Hamiltonian systems with integrals of motion H 1 = T + V (q 1 , q 2 ) , H 2 = K 2 1 + U 1 (q 1 , q 2 ) K 2 + U 2 (q 1 , q 2 ) 2 = K 2 + U 1 K 2 2 + 2U 2 K 1 K + U 2 2 K 2 1 + 2U 1 U 2 K 2 + U 1 U 2 2 . (1.2) where V (q 1 q 2 ) and U 1,2 (q 1 q 2 ) are the solution of partial differential equations obtained from the standard equation {H 1 , H 2 } = 0 ,(1.3) which must be identically satisfied for all admissible values of p 1 and p 2 . If reducible cubic polynomial K admits a few decompositions K = K 1 K 2 =K 1K2 =K 1K2 , we obtain several integrable systems with integrals of motion (1.2) and H 1 = T +V (q 1 , q 2 ) ,Ĥ 2 = K 2 1 +Û 1 (q 1 , q 2 ) K 2 +Û 2 (q 1 , q 2 ) 2 , H 1 = T +Ṽ (q 1 , q 2 ) ,H 2 = K 2 1 +Ũ 1 (q 1 , q 2 ) K 2 +Ũ 2 (q 1 , q 2 ) 2 . (1.4) In the next section, we present formal solutions of the partial differential equations (1.3) associated with all parameters m and k in Proposition 1. Invariants of sixth order in momenta In [18] we supposed that q 1,2 are parabolic coordinates on the plane q 1 = y − x 2 + y 2 , q 2 = y + x 2 + y 2 so that the corresponding momenta are p 1 = p y 2 + √ −q 1 q 2 p x 2q 1 , p 1 = p y 2 + √ −q 1 q 2 p x 2q 2 . In these variables, kinetic energy is equal to m = 1 , T = yp 2 x 2 + x(k − 1)p x p y 2 + kyp 2 y 2 , m = 3 , T = kyx 2 p 2 x 2 + (k − 1)x 2 + 4ky 2 xp x p y 2 + (2k − 1)x 2 2 + 2ky 2 yp 2 y , m = 4 , T = x 2 (k − 1)x 2 + 4ky 2 p 2 x 4 + xy (2k − 1)x 2 + 4ky 2 p x p y + (k − 1)x 4 4 + (3k − 1)x 2 y 2 + 4ky 4 p 2 y . Below we also present potentials in these x, y-variables in two dimensional configuration space. Case m = 1 and k = 2 In this case geodesic Hamiltonian T = q 1 (2q 1 + q 2 )p 2 1 q 1 − q 2 + q 2 (q 1 + 2q 2 )p 2 2 q 2 − q 1 commutes with cubic and quartic polynomials K = q 1 q 2 (q 2 1 p 1 − q 2 2 p 2 )(p 1 − p 2 ) 2 (q 1 − q 2 ) 3 , L = (q 2 1 p 1 − q 2 2 p 2 ) 2 (q 1 p 2 1 − q 2 p 2 2 ) (q 1 − q 2 ) 3 , {K, L} = K 2 . Cubic polynomial K has the form K = K 1 K 2 =K 1K2 , where K 1 = (q 2 1 p 1 − q 2 2 p 2 ) , andK 1 = q 1 q 2 (p 1 − p 2 ) (q 1 − q 2 ) 3 . The corresponding functions in (1.2) and (1. 4) are V = a(4q 4 1 + 14q 3 1 q 2 + 19q 2 1 q 2 2 + 14q 1 q 3 2 + 4q 4 2 ) (q 1 + q 2 ) 2 + b(q 1 + q 2 ) + c 1 q 1 + 1 q 2 , U 1 = 2a(q 2 1 + q 1 q 2 + q 2 2 )(q 1 + q 2 )(q 1 − q 2 ) 2 + b(q 1 − q 2 ) 2 (q 1 + q 2 ) 2 2 , U 2 = 2aq 1 q 2 (q 1 + q 2 )(q 1 − q 2 ) − c q 1 q 2 (q 1 − q 2 ) , andV = a(q 4 1 − 4q 3 1 q 2 − 14q 2 1 q 2 2 − 4q 1 q 3 2 + q 4 2 ) (q 1 + q 2 ) 2 + b(q 2 1 + 3q 1 q 2 + q 2 2 ) (q 1 + q 2 ) 3/2 + c(q 1 + q 2 ) , U 1 = 8aq 2 2 q 2 1 (q 2 1 − q 2 2 ) 4 − 2bq 1 q 2 √ q 1 + q 2 (q 1 − q 2 ) 4 , U 2 = 2a(q 4 1 − q 4 2 ) q 1 + q 2 + b(q 1 − q 2 ) 2 2 √ q 1 + q 2 + c(q 1 − q 2 ) 2 . In x, y-variables potentials read as V = − a(x 4 − 8x 2 y 2 − 64y 4 ) 4y 2 + 2by − 2cy x 2 , (2.1) V = − a(x 4 − 8x 2 y 2 − 4y 4 ) y 2 − √ 2b(x 2 − 4y 2 ) 4y 3/2 + 2cy . (2.2) Using the method of undetermined coefficients we can prove that these Hamiltonian systems do not have quartic invariants. Indeed, substituting quartic polynomial H 3 = L + 4 k=0 k i=0 f ik (q 1 , q 2 )p k−i 1 p i 2 into the equations {T + V, H 3 } = 0 {T +V , H 3 } = 0 we obtain two inconsistent systems of partial differential equations on functions f ik (q 1 , q 2 ). It means that the corresponding Hamiltonian vector fields admit quartic invariants only at V = 0 andV = 0. Case m = 3 and k = 3 Geodesic Hamiltonian T = q 3 1 (3q 1 + q 2 )p 2 1 q 1 − q 2 + q 3 2 (q 1 + 3q 2 )p 2 2 q 2 − q 1 , commutes with cubic polynomial K = q 3/2 1 q 3/2 2 (q 1 − q 2 ) 2 (q 2 1 p 1 − q 2 2 p 2 )(q 3/2 1 p 1 − q 3/2 2 p 2 )(q 3/2 1 p 1 + q 3/2 2 p 2 ) . So, we can take K 1 = q 3/2 1 q 3/2 2 (q 2 1 p 1 − q 2 2 p 2 ) ,K 1 = q 3/2 1 q 3/2 2 (q 3/2 1 p 1 − q 3/2 2 p 2 ) . andK 1 = q 3/2 1 q 3/2 2 (q 3/2 1 p 1 + q 3/2 2 p 2 ) . Solution of the equation (1.3) for V and U 1,2 has the following form V = a(q 1 + q 2 )(q 2 1 + 3q 1 q 2 + q 2 2 ) q 3 1 q 3 2 + b(q 1 + q 2 ) + c 1 q 1 + 1 q 2 , U 1 = a(q 1 − q 2 ) 2 (q 1 + q 2 ) 2 4 + bq 3 1 q 3 2 (q 1 − q 2 ) 2 3 , U 2 = a(q 2 1 + q 1 q 2 + q 2 2 ) q 3 1 q 3 2 (q 1 − q 2 ) + b 3(q 1 − q 2 ) + c q 1 q 2 (q 1 − q 2 ) . Other decompositions give rise to potentialŝ V = a(13q 2 1 +4 √ q 3 1 q2+46q1q2+4 √ q1q 3 2 +13q 2 2 )( √ q1+ √ q2) 2 q 3 1 q 3 2 + bq1q2( √ q1+ √ q2) 2 (3q1−2 √ q1q2+3q2) 2 + c( √ q1+ √ q2) 2 q1q2 , V = a(13q 2 1 −4 √ q 3 1 q2+46q1q2−4 √ q1q 3 2 +13q 2 2 )( √ q1− √ q2) 2 q 3 1 q 3 2 + bq1q2( √ q1− √ q2) 2 (3q1+2 √ q1q2+3q2) 2 + c( √ q1− √ q2) 2 q1q2 , and functionŝ U 1 = −8a( √ q 1 − √ q 2 ) 2 ( √ q 1 + √ q 2 ) 4 + 8bq 4 1 q 4 2 ( √ q 1 − √ q 2 ) 2 3(3q 1 − 2 √ q 1 q 2 + 3q 2 ) 2 , U 1 = −8a( √ q 1 − √ q 2 ) 4 ( √ q 1 + √ q 2 ) 2 − 8bq 4 1 q 4 2 ( √ q 1 + √ q 2 ) 2 3(3q 1 + 2 √ q 1 q 2 + 3q 2 ) 2 , U 2 = − a(5q1+2 √ q1q2+5q2)(q1−6 √ q1q2+q2) ( √ q1− √ q2)q 3 1 q 3 2 − bq1q2 3( √ q1− √ q2)(3q1−2 √ q1q2+3q2) 2 + c q1q2( √ q1− √ q2) , U 2 = − a(5q1−2 √ q1q2+5q2)(q1+6 √ q1q2+q2) ( √ q1+ √ q2)q 3 1 q 3 2 + bq1q2 3( √ q1+ √ q2)(3q1+2 √ q1q2+3q2) 2 − c q1q2( √ q1+ √ q2) , In x, y-variables potentials have the form V = 2ay(x 2 − 4y 2 ) x 6 + 2by − 2cy x 2 , (2.3) V = i − 8a(iy − 5x − 8y)(iy − 5x + 8y)(iy − x) 5x 6 − b(iy − x)x 2 2(3iy + x) 2 − 2c(iy − x) x 2 , (2.4) V = i − 8a(iy + 5x − 8y)(iy + 5x + 8y)(iy + x) 5x 6 − b(iy + x)x 2 2(3iy − x) 2 − 2c(iy + x) x 2 , (2.5) where i = √ −1. In two last cases canonical transformation of coordinates y → iy and p y → −ip y , changes kinetic energy T and potentialsV ,Ṽ so that the corresponding Hamiltonianŝ H 1 = i(T +V ) , andH 1 = i(T +Ṽ ) become real functions. Case m = 3 and k = 1/2 Quadratic polynomial T = q 3 1 q1 2 + q 2 p 2 1 q 1 − q 2 + q 3 2 q 1 + q2 2 p 2 2 q 2 − q 1 commutes with cubic and quartic polynomials K = q 2 1 q 2 2 (q 2 1 p 1 − q 2 2 p 2 ) 2 (p 1 − p 2 ) (q 1 − q 2 ) 3 , L = (q 2 1 p 1 − q 2 2 p 2 ) 3 q 2 1 (q 1 + 3q 2 )p 1 + q 2 2 (3q 1 + q 2 )p 2 4(q 1 − q 2 ) 3 so that {L, K} = K 2 . Because cubic polynomial K admits two decompositions K = K 1 K 2 =K 1K2 , where K 1 = q 2 1 q 2 2 (q 2 1 p 1 − q 2 2 p 2 ) , andK 1 = p 1 − p 2 (q 1 − q 2 ) 3 , we can construct two different integrable systems with sextic invariants so that V = a(q 4 1 − 4q 3 1 q 2 − 14q 2 1 q 2 2 − 4q 1 q 3 2 + q 4 2 ) q 2 1 q 2 2 (q 1 + q 2 ) 2 + b(q 2 1 + 3q 1 q 2 + q 2 2 ) √ q 1 q 2 (q 1 + q 2 ) 3/2 + c 1 q 1 + 1 q 2 , U 1 = − 16aq 3 1 q 3 2 (q 1 − q 2 ) 2 q 1 + q 2 + 4bq 7/2 1 q 7/2 2 (q 1 − q 2 ) 2 √ q 1 + q 2 , U 2 = − 4a(q 2 1 + q 2 2 ) q 3 1 q 3 2 (q 2 1 − q 2 2 ) − b q 3/2 1 q 3/2 2 √ q 1 + q 2 (q 2 1 − q 2 2 ) − 2c q 2 1 q 2 2 (q 1 − q 2 ) , andV = a(4q 4 1 + 14q 3 1 q 2 + 19q 2 1 q 2 2 + 14q 1 q 3 2 + 4q 4 2 ) q 2 1 q 2 2 (q 1 + q 2 ) 2 + b(q 1 + q 2 ) + c 1 q 1 + 1 q 2 , U 1 = − 4a(q 1 + q 2 )(q 2 1 + q 1 q 2 + q 2 2 ) q 5 1 q 5 2 (q 1 − q 2 ) 4 − c(q 1 + q 2 ) 2 q 4 1 q 4 2 (q 1 − q 2 ) 4 , U 2 = − 4aq 1 q 2 (q 1 − q 2 ) 2 q 1 + q 2 + 2bq 2 1 q 2 2 (q 1 − q 2 ) 2 . In this case V = − a(x 4 − 8x 2 y 2 − 4y 4 ) x 4 y 2 + ib √ 2(x 2 − 4y 2 ) 4xy 3/2 − 2cy x 2 , (2.6) V = − a(x 4 − 8x 2 y 2 − 64y 4 ) 4x 4 y 2 + 2by − 2cy x 2 . (2.7) After changing parameter b → −ib we obtain real potential V (2.6). Case m = 4 and k = −3 In this case geodesic Hamiltonian T = q 4 1 (q 2 − 3q 1 )p 2 1 q 1 − q 2 + q 4 2 (q 1 − 3q 2 )p 2 2 q 2 − q 1 commutes with the reducible cubic polynomial K = K 1 K 2 =K 1K2 = q 1 q 2 (q 2 1 p 1 − q 2 2 p 2 )(q 2 1 p 1 + q 2 2 p 2 ) 2 q 1 − q 2 , where K 1 = q 2 1 p 1 − q 2 2 p 2 andK 1 = q 1 q 2 (q 2 1 p 1 + q 2 2 p 2 ) q 1 − q 2 . The corresponding quartic invariant is equal to L = (q 2 1 p 1 − q 2 2 p 2 ) 2 q 4 1 (3q 2 1 − 2q 1 q 2 + q 2 2 )p 2 1 + 2q 2 1 q 2 2 (q 2 1 + q 2 2 )p 1 p 2 + q 4 2 (q 2 1 − 2q 1 q 2 + 3q 2 2 )p 2 2 4(q 1 − q 2 ) 2 , so that {L, K} = K 2 . The corresponding two integrable systems with sextic invariants (1.2-1.4) are determined by functions V = a(19q 4 1 − 68q 3 1 q 2 + 82q 2 1 q 2 2 − 68q 1 q 3 2 + 19q 4 2 ) q 4 1 q 4 2 + bq 2 1 q 2 2 (q 1 + q 2 ) 2 + c(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 1 = − 8a(3q 2 1 − 2q 1 q 2 + 3q 2 2 )(q 1 − q 2 ) 2 q 4 1 q 4 2 − c(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 2 = − 4a(q 1 + q 2 ) 2 (q 1 − q 2 ) q 3 1 q 3 2 − bq 3 1 q 3 2 (q 1 + q 2 ) 2 (q 1 − q 2 ) , andV = a(q 4 1 − 2q 3 1 q 2 − 2q 2 1 q 2 2 − 2q 1 q 3 2 + q 4 2 ) q 4 1 q 4 2 + b(q 2 1 − 6q 1 q 2 + q 2 2 ) 4q 1 q 2 (q 1 − q 2 ) + c(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 1 = − a(q 1 + q 2 ) 2 q 2 1 q 2 2 − bq 1 q 2 q 1 − q 2 , U 2 = a(q 1 + q 2 )(q 1 − q 2 )(q 2 1 + q 2 2 ) q 4 1 q 4 2 + b(q 1 + q 2 ) 4q 1 q 2 + c 1 q 2 1 − 1 q 2 2 . In x, y-variables potentials have the form V = 16a(16x 4 + 36x 2 y 2 + 19y 4 ) x 8 + bx 4 4y 2 + 4c(x 2 + y 2 ) x 4 , (2.8) V = 4a(x 4 + 6x 2 y 2 + 4y 4 ) x 8 + b(2x 2 + y 2 ) 2x 2 x 2 + y 2 + 4c(x 2 + y 2 ) x 4 . (2.9) We can prove that the Hamiltonians H 1 (1.2) andĤ 1 (1.4) do not commute with the following polynomial of fourth order in momenta H 3 = L + 4 k=0 k i=0 f ik (q 1 , q 2 )p k−i 1 p i 2 , for any functions f ik (q 1 , q 2 ). It means that the corresponding Hamiltonian vector fields admit quartic invariants only for geodesic motion at V = 0 andV = 0. Case m = 4 and k = 3 In this case, we have T = q 4 1 (3q 1 + q 2 )p 2 1 q 1 − q 2 + q 4 2 (q 1 + 3q 2 )p 2 2 q 2 − q 1 and K = K 1 K 2 =K 1K2 = q 2 1 q 2 2 (q 1 − q 2 ) 2 (q 2 1 p 1 − q 2 2 p 2 ) 2 (q 2 1 p 1 + q 2 2 p 2 ) , where K 1 = q 2 1 q 2 2 (q 2 1 p 1 − q 2 2 p 2 ) andK 1 = (q 2 1 p 1 + q 2 2 p 2 ) . The first decomposition yields the following solutions of (1.3) V = a(q 4 1 + 4q 3 1 q 2 + 4q 2 1 q 2 2 + 4q 1 q 3 2 + q 4 2 ) q 4 1 q 4 2 + b(q 2 1 + 4q 1 q 2 + q 2 2 ) q 2 1 q 2 2 + c 1 q 1 + 1 q 2 , U 1 = − a(3q 2 1 + 2q 1 q 2 + 3q 2 2 )(q 1 − q 2 ) 2 4 − bq 2 2 q 2 1 (q 1 − q 2 ) 2 , U 2 = a(q 1 + q 2 )(q 2 1 + q 2 2 ) q 4 1 q 4 2 (q 1 − q 2 ) + b(q 1 + q 2 ) q 2 1 q 2 2 (q 1 − q 2 ) + c q 1 q 2 (q 1 − q 2 ) . The second decomposition leads to the solutionŝ V = a(q 2 1 + 6q 1 q 2 + q 2 2 )(q 1 − q 2 ) 2 q 4 1 q 4 2 + bq 2 1 q 2 2 (q 1 + q 2 ) 2 + c(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 1 = − a(q 1 − q 2 ) 4 q 4 1 q 4 2 + 2bq 2 1 q 2 2 (q 1 + q 2 ) 2 , U 2 = − 2a(3q 2 1 + 2q 1 q 2 + 3q 2 2 ) q 2 1 q 2 2 + 2c . In x, y-variables potentials have the form V = − 2a(x 4 − 8y 4 ) x 8 − 2b(x 2 − 2y 2 ) x 4 − 2cy x 2 , (2.10) V = − 16a(x 4 − y 4 ) x 8 + bx 4 4y 2 + 4c(x 2 + y 2 ) x 4 . (2.11) 2.6 Case m = 4 and k = −3/5 Geodesic Hamiltonian T = q 4 1 q 2 − 3q1 5 p 2 1 q 1 − q 2 + q 4 2 q 1 − 3q2 5 p 2 2 q 2 − q 1 commutes with reducible polynomial K = (q 2 1 p 1 − q 2 2 p 2 )(q 2 1 p 1 + q 2 2 p 2 ) q 2 1 (5q 2 1 − 10q 1 q 2 + q 2 2 )p 1 + q 2 2 (q 2 1 − 10q 1 q 2 + 5q 2 2 )p 2 q 1 q 2 (q 1 − q 2 ) , from which we can extract three different linear polynomials in momenta K 1 = (q 2 1 p 1 − q 2 2 p 2 ) ,K 1 = (q 2 1 p 1 + q 2 2 p 2 ) q 1 q 2 (q 1 − q 2 ) andK 1 = q 2 1 (5q 2 1 − 10q 1 q 2 + q 2 2 )p 1 + q 2 2 (q 2 1 − 10q 1 q 2 + 5q 2 2 )p 2 . The corresponding potentials in (1.2-1.4) are equal to V = a(19q 2 1 − 26q 1 q 2 + 19q 2 2 ) q 2 1 q 2 2 + bq 2 1 q 2 2 (q 1 + q 2 ) 2 , U 1 = − 40a(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 2 = 5a(5q 2 1 − 6q 1 q 2 + 5q 2 2 )(q 1 + q 2 ) 2 2q 3 1 q 3 2 (q 1 − q 2 ) − 10bq 1 q 2 (q 1 − q 2 ) (q 1 + q 2 ) 2 and V = a(31q 4 1 − 68q 3 1 q 2 + 58q 2 1 q 2 2 − 68q 1 q 3 2 + 31q 4 2 ) q 1 q 2 (q 1 − q 2 ) 3 − b √ q 1 q 2 (7q 2 1 − 18q 1 q 2 + 7q 2 2 ) (q 1 − q 2 ) 5/2 + cq 1 q 2 q 1 − q 2 , U 1 = − 40a(q 1 + q 2 ) 2 q 3 1 q 3 2 (q 1 − q 2 ) 3 + 40b q 3/2 1 q 3/2 2 (q 1 − q 2 ) 5/2 , U 2 = 5a(q 1 + q 2 )(3q 1 − q 2 ) 2 (q 1 − 3q 2 ) 2 q 1 q 2 (q 1 − q 2 ) 2 + 5b √ q 1 q 2 )(3q 1 − q 2 )(q 1 − 3q 2 )(q 1 + q 2 ) (q 1 − q 2 ) 3/2 −5cq 1 q 2 (q 1 + q 2 ) . Third decomposition K =K 1K2 yields the third integrable systemH 1 = T +Ṽ with potential V = a(5q 2 1 − 6q 1 q 2 + 5q 2 2 ) q 1 q 2 (q 1 − q 2 ) ,Ũ 1 = 0 ,Ũ 2 = 5a(q 1 + q 2 ) q 1 q 2 . Thus, in this case, we have three integrable systems with sextic invariants and potentials V = 4a(16x 2 + 19y 2 ) x 4 + bx 4 4y 2 , (2.12) V = 2a(16x 4 + 48x 2 y 2 + 31y 4 ) x 2 (x 2 + y 2 ) 3/2 + √ 2bx(8x 2 + 7y 2 ) 2(x 2 + y 2 ) 5/4 − cx 2 2 x 2 + y 2 , (2.13) V = 2a(4x 2 + 5y 2 ) x 2 x 2 + y 2 . (2.14) 2.7 Case m = 4 and k = −1/7 As above geodesic Hamiltonian T = q 4 1 q 2 − q1 7 p 2 1 q 1 − q 2 + q 4 2 q 1 − q2 7 )p 2 2 q 2 − q 1 commutes with reducible cubic polynomial K = (q 2 1 p 1 − q 2 2 p 2 ) q 2 1 (q 1 − 3q 2 )p 1 + q 2 2 (3q 1 − q 2 )p 2 q 1 q 2 (q 1 − q 2 ) 3 × q 2 1 (3q 3 1 − 27q 2 1 q 2 + 33q 1 q 2 2 − q 3 2 )p 1 + q 2 2 (q 3 1 − 33q 2 1 q 2 + 27q 1 q 2 2 − 3q 3 2 )p 2 from which we can extract three different linear polynomials K 1 = q 2 1 p 1 − q 2 2 p 2 ,K 1 = q 2 1 (q 1 − 3q 2 )p 1 + q 2 2 (3q 1 − q 2 )p 2 andK 1 = q 2 1 (3q 3 1 − 27q 2 1 q 2 + 33q 1 q 2 2 − q 3 2 )p 1 + q 2 2 (q 3 1 − 33q 2 1 q 2 + 27q 1 q 2 2 − 3q 3 2 )p 2 . The corresponding potentials in (1.2-1.4) are equal to V = a(11q 2 1 − 42q 1 q 2 + 11q 2 2 ) q 2 1 q 2 2 + bq 2 1 q 2 2 (q 1 + q 2 ) 2 , U 1 = 56a(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 2 = 14a(3q 1 − q 2 )(q 1 − 3q 2 )(q 1 + q 2 ) 2 q 3 1 q 3 2 (q 1 − q 2 ) − 28bq 1 q 2 (q 2 1 − 6q 1 q 2 + q 2 2 ) (q 1 + q 2 ) 2 (q 1 − q 2 ) , andV = a(11q 4 1 − 84q 3 1 q 2 + 66q 2 1 q 2 2 − 84q 1 q 3 2 + 11q 4 2 ) q 1 q 2 (q 2 1 − 6q 1 q 2 + q 2 2 ) 3/2 − 2bq 1 q 2 q 2 1 − 6q 1 q 2 + q 2 2 , U 1 = − 112a(q 1 − q 2 ) 2 (q 1 + q 2 ) 2 q 1 q 2 q 2 1 − 6q 1 q 2 + q 2 2 , U 2 = 14a(q 1 + q 2 )(q 2 1 − 14q 1 q 2 + q 2 2 ) 2 (q 2 1 − 6q 1 q 2 + q 2 2 ) 3/2 (q 1 − q 2 )q 2 1 q 2 2 + 28b(q 1 + q 2 ) q 2 1 − 6q 1 q 2 + q 2 2 (q 1 − q 2 ) . Third decomposition K =K 1K2 gives rise to the third integrable system so that V = a(3q 1 − q 2 )(q 1 − 3q 2 ) q 1 q 2 q 2 1 − 6q 1 q 2 + q 2 2 ,Ũ 1 = 0 ,Ũ 2 = 14a(q 1 + q 2 ) q 2 1 q 2 2 (q 1 − q 2 ) q 2 1 − 6q 1 q 2 + q 2 2 . In x, y-variables these potentials have the form V = 4a(16x 2 + 11y 2 ) x 4 + bx 4 4y 2 , ,(2.15) V = − 2a(16x 4 + 32x 2 y 2 + 11y 4 ) x 2 (2x 2 + y 2 ) 3/2 + bx 2 2x 2 + y 2 , (2.16) V = − 2a(4x 2 + 3y 2 ) x 2 2x 2 + y 2 . (2.17) Case m = 4 and k = 1/5 In this case geodesic Hamiltonian T = q 4 1 q1 5 + q 2 p 2 1 q 1 − q 2 + q 4 2 q 1 + q2 5 p 2 2 q 2 − q 1 commutes with the cubic and quartic polynomials K = q 1 q 2 (q 2 1 p 1 − q 2 2 p 2 ) q 2 1 (q 1 + 3q 2 )p 1 − q 2 2 (3q 1 + q 2 )p 2 2 (q 1 − q 2 ) 3 , and L = (q 2 1 p 1 −q 2 2 p 2 ) 2 q 4 1 (q 3 1 +9q 2 1 q 2 +21q 1 q 2 2 +q 3 2 )p 2 1 −2q 2 1 q 2 2 (q 1 −q 2 )(q 2 1 +4q 1 q 2 +q 2 2 )p 1 p 2 −q 4 2 (q 3 1 +21q 2 1 q 2 +9q 1 q 2 2 +q 3 2 )p 2 2 4(q 1 −q 2 ) 3 . so that {L, K} = K 2 . This cubic polynomial can be decomposed as K = K 1 K 2 or K =K 1K2 , where K 1 = (q 2 1 p 1 − q 2 2 p 2 ) andK 1 = q 2 1 (q 1 + 3q 2 )p 1 − q 2 2 (3q 1 + q 2 )p 2 . Integrals of motion (1.2-1.4) involve the following functions V = a(11q 4 1 + 156q 3 1 q 2 + 546q 2 1 q 2 2 + 156q 1 q 3 2 + 11q 4 2 ) q 4 1 q 4 2 + b(q 2 1 + 6q 1 q 2 + q 2 2 ) q 2 1 q 2 2 + cq 2 1 q 2 2 (q 1 + q 2 ) 2 , U 1 = 40a(q 1 − q 2 ) 2 (q 2 1 + 10q 1 q 2 + q 2 2 ) q 4 1 q 4 2 + 5b(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 2 = 20a(q 2 1 + 6q 1 q 2 + q 2 2 )(q 2 + q 1 ) 2 q 3 1 q 3 2 (q 1 − q 2 ) − 5cq 3 1 q 3 andV = a(q 4 1 + 6q 3 1 q 2 + 6q 2 1 q 2 2 + 6q 1 q 3 2 + q 4 2 ) q 4 1 q 4 2 + b(3q 2 1 + 14q 1 q 2 + 3q 2 2 ) q 1 q 2 q 2 1 + 6q 1 q 2 + q 2 2 + c(q 2 1 + 6q 1 q 2 + q 2 2 ) q 2 1 q 2 2 , U 1 = − 5a(q 1 − q 2 ) 2 (q 2 1 + 6q 1 q 2 + q 2 2 )(q 1 + q 2 ) 2 q 4 1 q 4 2 − 20b(q 1 − q 2 ) 2 q 2 1 + 6q 1 q 2 + q 2 2 q 1 q 2 , U 2 = 5a(q 1 + q 2 ) q 2 1 q 2 2 (q 1 − q 2 ) − 5b(q 1 + q 2 ) 2 q 2 1 + 6q 1 q 2 + q 2 2 (q 1 − q 2 ) − 5c(q 1 + q 2 ) 2q 1 q 2 (q 1 − q 2 ) . In x, y-variables potentials are equal to V = 16a(16x 4 − 28x 2 y 2 + 11y 4 ) x 8 − 4b(x 2 − y 2 ) x 4 + cx 4 4y 2 , (2.18) V = − 4a(x 4 + 2x 2 y 2 − 4y 4 ) x 8 + 2b(2x 2 − 3y 2 ) x 2 y 2 − x 2 − 4c(x 2 − y 2 ) x 4 . (2.19) We can prove that Hamiltonians H 1 (1.2) andĤ 1 (1.4) do not commute with polynomial of fourth order in momenta H 3 = L + 4 k=0 k i=0 f ik (q 1 , q 2 )p k−i 1 p i 2 for any f ik (q 1 , q 2 ). As above it means that the corresponding Hamiltonian vector fields admit quartic invariants only for geodesic motion. Case m = 4 and k = 1/2 In this case geodesic Hamiltonian T = q 4 1 q1 2 + q 2 p 2 1 q 1 − q 2 + q 4 2 q 1 + q2 2 p 2 2 q 2 − q 1 commute with cubic polynomial K = K 1 K 2 =K 1K2 = q 2 1 q 2 2 (q 2 1 p 1 − q 2 2 p 2 ) 2 q 2 1 (2q 1 + 3q 2 )p 1 − q 2 2 (3q 1 + 2q 2 )p 2 (q 1 − q 2 ) 3 , where K 1 = q 2 1 (2q 1 + 3q 2 )p 1 − q 2 2 (3q 1 + 2q 2 )p 2 , andK 1 = (q 2 1 p 1 − q 2 2 p 2 ) . As a result, we obtain two integrable systems (1. 2-1.4) with V = a(q 2 1 + 3q 1 q 2 + q 2 2 )(3q 2 1 + 8q 1 q 2 + 3q 2 2 ) q 4 1 q 4 2 + b(q 2 1 + 3q 1 q 2 + q 2 2 ) q 2 1 q 2 2 + cq 2 1 q 2 2 (q 1 + q 2 ) 2 , U 1 = − 2a(q 1 − q 2 ) 2 (q 2 1 + 3q 1 q 2 + q 2 2 ) 2 q 4 1 q 4 2 − 2cq 2 1 q 2 2 (q 1 − q 2 ) 2 (q 1 + q 2 ) 2 , U 2 = 2a(q 2 1 + 4q 1 q 2 + q 2 2 ) q 2 1 q 2 2 (q 1 − q 2 ) + 2b q 1 − q 2 , and V = a(17q 4 1 + 108q 3 1 q 2 + 198q 2 1 q 2 2 + 108q 1 q 3 2 + 17q 4 2 ) q 4 1 q 4 2 + b(7q 2 1 + 18q 1 q 2 + 7q 2 2 ) q 2 1 q 2 2 + c(q 1 + q 2 ) q 1 q 2 , U 1 = 16a(q 1 − q 2 ) 2 (q 2 1 + 4q 1 q 2 + q 2 2 ) q 4 1 q 4 2 + 8b(q 1 − q 2 ) 2 q 2 1 q 2 2 , U 2 = 8a(q 1 + 3q 2 )(3q 1 + q 2 )(q 1 + q 2 ) q 2 1 q 2 2 (q 1 − q 2 ) + 4b(q 1 + q 2 ) q 1 − q 2 + 2cq 2 q 1 q 1 − q 2 . In x, y-variables potentials are equal to V = 2a(x 2 − 4y 2 )(x 2 − 6y 2 ) x 8 − b(x 2 − 4y 2 ) x 4 − cx 4 4y 2 , (2.20) V = 16a(x 4 − 10x 2 y 2 + 17y 4 ) x 8 − 4b(x 2 − 7y 2 ) x 4 − 2cy x 2 . (2.21) Thus, we present 21 integrable systems with sextic invariants associated with metric (1.1) depending on coordinates. Other nontrivial families of integrable systems with quartic invariants can be obtained similarly. Indeed, we can consider integrals of motion H 1 = T + V (q 1 , q 2 ) , H 2 = K 2 1 + U 1 (q 1 , q 2 ) L 2 + U 2 (q 1 , q 2 ) (2.22) where V and U 1,2 are solutions of the equation (1.3). For instance, at m = 1 and k = 2 if we substitute Hamiltonian H 1 = q 1 (2q 1 + q 2 )p 2 1 q 1 − q 2 + q 2 (q 1 + 2q 2 )p 2 2 u 2 − u 1 + V (q 1 , q 2 ) and H 2 = (q 2 1 p 1 − q 2 2 p 2 ) 2 + U 1 (q 1 , q 2 ) (q 1 p 2 1 − q 2 p 2 2 ) (q 1 − q 2 ) 3 + U 2 (q 1 , q 2 ) into {H 1 , H 2 } = 0 and solve the resulting equation, then we obtain the following solution V = a(q 1 + q 2 ) 2q 2 1 + q 1 q 2 + 2q 2 2 + b 2q 2 1 + 3q 1 q 2 + 2q 2 2 + c(q 1 + q 2 ) + d(q 1 + q 2 ) q 1 q 2 + e(q 1 + q 2 ) 3 q 3 1 q 3 2 and integrable Hamiltonian H 1 = y(p 2 x + 2p 2 y ) 2 + xp x p y 2 + 2ay(3x 2 + 8y 2 ) + b(x 2 + 8y 2 ) + 2cy − 2dy x 2 + 8ey 3 x 6 depending on five parameters a, b, c, d, and e. The corresponding Hamiltonian vector fields has the quartic invariant H 2 . Conclusion In 1910 Darboux studied two-dimensional systems in the Euclidean plane with quadratic invariants [5] and his calculations became one of the foundations of the modern theory of separation of variables in orthogonal curvilinear coordinates. In 1935 Drach studied two-dimensional systems in the pseudo-Euclidean plane with cubic invariants [6]. Seven of the ten Drach potentials are superintegrable with two quadratic additional integrals [17] and, therefore, cubic Drach invariants exist due to the Abel and Riemann-Roch theorems [20,21]. In [14] we studied three families of superintegrable two-dimensional systems in the pseudo-Euclidean planes, for which one additional first integral is quadratic, and the second one can be arbitrarily polynomials of a high degree in momenta. Some of these systems were obtained using degenerate deformations of the Poisson brackets but we do not know of a mathematical justification for the existence of these systems till now. In this note, we start with two-dimensional systems with a position-dependent mass having cubic invariants. Some of these systems are also superintegrable. In contrast with the Drach systems, these cubic invariants can be used to construct systems with sextic invariants. From the technical point of view, it is a consequence of dependence of metrics on the coordinates. In [18] we obtain a triad of bivectors Π ± and π, which form a Poisson bivector Π ′ = Π − a + π + aΠ + compatible with canonical Poisson bivector Π, so that X = ΠdH 1 = Π ′ dH 2 for a system with sextic invariant at m = 4 and k = 3. In the future, we plan to study similar triads of bivectors Π ± and π and the corresponding bi-Hamiltonian vector fields X for other integrable cases obtained in this note. We hope that this mathematical experiment allows us to get new results for natural Hamiltonian systems with other metrics. The work was supported by the Russian Science Foundation (project 21-11-00141). (q 1 − q 2 )(q 2 + q 1 ) 2 , On Hamiltonians with positiondependent mass from Kaluza-Klein compactifications. A Ballesteros, I Gutiérrez-Sagredo, Pedro Naranjo, P , Physics Letters A, v. 3817Ballesteros A., Gutiérrez-Sagredo I., Pedro Naranjo P., On Hamiltonians with position- dependent mass from Kaluza-Klein compactifications, Physics Letters A, v. 381 (7), pp. 701-706, 2017. A survey on polynomial in momenta integrals for billiard problems. M Bialy, A E Mironov, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. v. 376, n.2131Bialy M., Mironov A.E., A survey on polynomial in momenta integrals for billiard problems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, v. 376, n.2131, 20170418, 2018. Information-theoretic measures for a position-dependent mass system in an infinite potential well. Da Costa, G Bruno, S Ignacio, Physica A: Statistical Mechanics and its Applications. v. 541, 123698da Costa, Bruno G., Ignacio S., Information-theoretic measures for a position-dependent mass system in an infinite potential well, Physica A: Statistical Mechanics and its Appli- cations, v. 541, 123698, 2020. Leçons sur la théorie générale des surfaces. G Darboux, Chelsea Publishing 1894Darboux G., Leçons sur la théorie générale des surfaces, v. 3, Chelsea Publishing 1894. G Darboux, Leçons sur les Systems Orthogonaux et les Coordinées Curvilignes. ParisGauthier-VillarsDarboux G., Leçons sur les Systems Orthogonaux et les Coordinées Curvilignes, Paris: Gauthier-Villars, 1910. Sur l'intégration logique et la transformation deséquations de la dyanamiqueá deux Forces conservatives. Intégrales cubiques. J Drach, Comptes Rendus. Drach J., Sur l'intégration logique et la transformation deséquations de la dyanamiqueá deux Forces conservatives. Intégrales cubiques, Comptes Rendus, (Paris), v.200, pp.22-26, 1935. Quadratic and cubic invariants in classical mechanics. S Fokas, P A Lagerström, J. Math. Anal. Appl., v. 74Fokas S., Lagerström P.A., Quadratic and cubic invariants in classical mechanics, J. Math. Anal. Appl., v.74, pp. 325-341, 1980. Integrability of nonholonomic Heisenberg type systems. Grigoryev Yu, . A Sozonov, A P Tsiganov, A V , SIGMA, v. 1214Grigoryev Yu.A, Sozonov A.P., Tsiganov A.V., Integrability of nonholonomic Heisenberg type systems, SIGMA, v.12, 112, 14 pp, 2016. On superintegrable systems separable in Cartesian coordinates Physics Letters A, v. Grigoriev A Yu, A V Tsiganov, 382Grigoriev Yu.A., Tsiganov A. V., On superintegrable systems separable in Cartesian coor- dinates Physics Letters A, v.382, n.32, pp.2092-2096, 2018. Two-dimensional geodesic flows having first integrals of higher degree. K Kiyohara, Math. Ann., v.320, n.3Kiyohara K, Two-dimensional geodesic flows having first integrals of higher degree, Math. Ann., v.320, n.3, pp.487-505, 2001. Polynomial integrals of geodesic flows on a two-imensional torus. V V Kozlov, N V Denisova, Mat. Sb., v.185, n.1Kozlov, V.V., Denisova, N.V., Polynomial integrals of geodesic flows on a two-imensional torus, Mat. Sb., v.185, n.1, pp.49-64, 1994. Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric. B S Kruglikov, V S Matveev, Phys. Rev. D, v. 85124057Kruglikov, B.S., Matveev, V.S., Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric, Phys. Rev. D, v.85, 124057, 2012. Direct methods for the search of the second invariant. J Hietarinta, Phys. 2ReportHietarinta J., Direct methods for the search of the second invariant, Phys.Report, v.147, n.2, pp. 87-154, 1987. On algebraic construction of certain integrable and super-integrable systems. A J Maciejewski, M Przybylska, A V Tsiganov, Physica D. 240Maciejewski A.J., Przybylska M., Tsiganov A.V., On algebraic construction of certain integrable and super-integrable systems, Physica D, v. 240, p.1426-1448, 2011. General N -th order integrals of the motion in the Euclidean plane. S Post, P Winternitz, J. Phys. A: Math. Theor., v. 4824Post S., Winternitz P., General N -th order integrals of the motion in the Euclidean plane, J. Phys. A: Math. Theor., v.48, 405201 (24pp), 2015. Position-dependent finite symmetric mass harmonic like oscillator: Classical and quantum mechanical study, Open Physics. B Rath, P Mallick, P Mohapatra, J Asad, H Shanak, R Jarrar, 19Rath B., Mallick P., Mohapatra P., Asad J., Shanak H., Jarrar R., Position-dependent finite symmetric mass harmonic like oscillator: Classical and quantum mechanical study, Open Physics, v.19, no.1, pp. 266-276, 2021. On the Drach superintegrable systems. A V Tsiganov, J. Phys.A., v. 33Tsiganov A. V., On the Drach superintegrable systems, J. Phys.A., v.33, p.7407-7423, 2000. New bi-Hamiltonian systems on the plane. A V Tsiganov, Journal of Mathematical Physics. v.58, 062901Tsiganov A. V., New bi-Hamiltonian systems on the plane, Journal of Mathematical Physics, v.58, 062901, 2017. Elliptic curve arithmetic and superintegrable systems. A V Tsiganov, Physica Scripta. v.94, n.8, 085207Tsiganov A.V., Elliptic curve arithmetic and superintegrable systems, Physica Scripta, v.94, n.8, 085207, 2019. Discretization and superintegrability all rolled into one. A V Tsiganov, Nonlinearity. 33Tsiganov A.V., Discretization and superintegrability all rolled into one, Nonlinearity, v.33, pp.4924-4939, 2020. Superintegrable systems and Riemann-Roch theorem. A V Tsiganov, Journal of Mathematical Physics. Tsiganov A.V., Superintegrable systems and Riemann-Roch theorem, Journal of Mathe- matical Physics, v.61, 012701, 2020. On a time-dependent nonholonomic oscillator, Russ. A V Tsiganov, J. Math. Phys., v. 27Tsiganov A.V., On a time-dependent nonholonomic oscillator, Russ. J. Math. Phys., v.27, pp.399-409, 2020. G Valent, arXiv:2110.02703Superintegrable geodesic flows on the hyperbolic plane. PreprintValent G., Superintegrable geodesic flows on the hyperbolic plane, Preprint arXiv: 2110.02703, 2021.

Context. If relativistic particle acceleration takes place in colliding-wind binaries, hard X-rays and γ-rays are expected through inverse Compton emission, but to date these have never been unambiguously detected. Aims. To detect this emission, observations of η Carinae were performed with INTEGRAL, leveraging its high spatial resolution. Methods. Deep hard X-ray images of the region of η Car were constructed in several energy bands.Results. The hard X-ray emission previously detected by BS around η Car originates from at least 3 different point sources. The emission of η Car itself can be isolated for the first time, and its spectrum unambiguously analyzed. The X-ray emission of η Car in the 22-100 keV energy range is very hard (Γ 1 ± 0.4) and its luminosity is 7 × 10 33 erg s −1 .Conclusions. The observed emission is in agreement with the predictions of inverse Compton models, and corresponds to about 0.1% of the energy available in the wind collision. η Car is expected to be detected in the GeV energy range.

10.1051/0004-6361:20078981

0712.1491

6835282598db131a7da86305971017ff1f210e5c

Hard X-ray emission from η Carinae February 3, 2008 J.-C Leyder [email protected] Institut d'Astrophysique et de Géophysique Université de Liège Allée du 6-Août 17, Bâtiment B5cB-4000LiègeBelgium INTEGRAL Science Data Centre Université de Genève Chemin d'Écogia 16CH-1290VersoixSwitzerland R Walter INTEGRAL Science Data Centre Université de Genève Chemin d'Écogia 16CH-1290VersoixSwitzerland Observatoire de Genève Université de Genève Chemin des Maillettes 51CH-1290SauvernySwitzerland G Rauw Institut d'Astrophysique et de Géophysique Université de Liège Allée du 6-Août 17, Bâtiment B5cB-4000LiègeBelgium Hard X-ray emission from η Carinae February 3, 2008Received 2 November 2007 / Accepted 16 November 2007Astronomy & Astrophysics manuscript no. Article-Final L   Egamma rays: observations -X-rays: binaries -X-rays: individuals: η Car -X-rays: individuals: 1E 10481-5937 - X-rays: individuals: IGR J10447-6027 Context. If relativistic particle acceleration takes place in colliding-wind binaries, hard X-rays and γ-rays are expected through inverse Compton emission, but to date these have never been unambiguously detected. Aims. To detect this emission, observations of η Carinae were performed with INTEGRAL, leveraging its high spatial resolution. Methods. Deep hard X-ray images of the region of η Car were constructed in several energy bands.Results. The hard X-ray emission previously detected by BS around η Car originates from at least 3 different point sources. The emission of η Car itself can be isolated for the first time, and its spectrum unambiguously analyzed. The X-ray emission of η Car in the 22-100 keV energy range is very hard (Γ 1 ± 0.4) and its luminosity is 7 × 10 33 erg s −1 .Conclusions. The observed emission is in agreement with the predictions of inverse Compton models, and corresponds to about 0.1% of the energy available in the wind collision. η Car is expected to be detected in the GeV energy range. Introduction η Carinae is one of the most peculiar objects in our Galaxy (see Davidson & Humphreys 1997). Once the second brightest object in the sky (during its eruption in 1843), it decreased to a V magnitude m V of 8 by the end of the XIXth century, before slowly and irregularly increasing again, up to its current value of m V ∼ 5 (see e.g. Viotti 1995). The large quantities of matter that were ejected during these dramatic luminosity variations are now forming an extended nebula (the so-called homunculus), while η Car is still ejecting matter through energetic stellar winds. Observations lead to an estimated mass-loss rate of 10 −4 -10 −3 M yr −1 (Andriesse et al. 1978;Hillier et al. 2001;Pittard & Corcoran 2002;van Boekel et al. 2003). Optical spectra of η Car reveal long periods of "high spectroscopic state" characterized by an emission line spectrum, followed by shorter periods of "low spectroscopic state"also called "spectroscopic events" -typically lasting a few months, during which the high-excitation emission lines fade away (Rodgers & Searle 1967;Viotti 1969;Zanella et al. 1984;Altamore et al. 1994). A period of ∼2023 days (5.53 yr) was inferred from periodic changes in the optical (Damineli 1996;Damineli et al. 2000) and IR (Whitelock et al. 1994(Whitelock et al. , 2004 domains. There is now strong evidence that η Car is a binary system; some examples are the radial velocity variations occurring with the period of 5.53 yr, or the "spectroscopic event" (believed to occur near periastron passage). Many X-ray observations have also been performed, which have extended our understanding of the physical nature of η Car. The structured X-ray emission can be divided into two components, as shown by the Einstein satellite (Chlebowski et al. 1984;Rebecchi et al. 2001;Viotti et al. 2002b), and further confirmed by Chandra observations (Seward et al. 2001): -a soft (kT SX ∼ 0.5 keV) thermal X-ray component (η SX), which is spatially extended (up to about 20 ) and inhomogenous, dominates the spectrum mostly below 1.5 keV, and is probably associated to the interaction of the stellar wind with the interstellar matter; -a hard (kT HX ∼ 4.7 keV) thermal X-ray component (η HX), which is point-like, centered on the stellar system, dominates the spectrum in the 2-10 keV range, and is likely linked to the wind collision between the two massive stars that form the binary system. The X-ray light curves also exhibit strong variations, with the same periodicity of 2024±2 days (Ishibashi et al. 1999;Corcoran 2005). The X-ray spectrum suggests a colliding-wind binary (CWB) scenario (Ishibashi et al. 1999;Pittard & Corcoran 2002;Corcoran 2005): the dense stellar wind coming from the massive luminous blue variable (LBV) primary star (η Car A) collides with the higher-velocity, lower-density wind from the hotter and luminous, probably late-type nitrogen-rich O or WR type (Iping et al. 2005), star companion (η Car B) in a highly eccentric orbit. Finally, at hard X-rays, BeppoSAX detected η Car only once exhibiting a hard X-ray tail (i.e. at a flux higher than an extrapolation of the hard thermal X-ray component). In this Letter, new high-resolution hard X-ray observations performed by INTEGRAL are presented, which allow to clearly detect the emission from η Car for the first time. Such a hard X-ray detection is important, as it proves that non-thermal particle acceler-arXiv:0712.1491v1 [astro-ph] 10 Dec 2007 ation is at work in the wind collision, and as it suggests likely gamma-ray emission in the MeV and GeV energy ranges. High-energy observations and data analysis Previous BeppoSAX observations The BeppoSAX satellite carried out 4 observations of η Car between 1996 and 2000, covering its 5.53 yr cycle (Φ = 1 corresponds to the 1998 minimum): -December 1996 (Φ = 0.83), December 1999 (Φ = 1.37), and June 2000 (Φ = 1.46) in the high spectroscopic state; -March 1998 (Φ = 1.05) in the low spectroscopic state. The first hard X-ray detection at energies between 10 to 20 keV in the vicinity of η Car was obtained with the phoswich detector system (PDS) instrument on-board BeppoSAX in December 1996, when the 13-20 keV flux (0.15−0.17 count s −1 ) was clearly in excess of the 4.7 keV thermal fit found from the 2-10 keV energy range (Rebecchi et al. 2001;Viotti et al. 2002a). This flux excess is found again in the other two observations in the high spectroscopic state (December 1999 and June 2000), but not in the low spectroscopic state observation (March 1998) where only an upper limit on the 13-20 keV flux could be extracted (Viotti et al. 2004). Moreover, the June 2000 observation was longer (nominal exposure time of 100 ks), and revealed a high-energy tail extending up to 50 keV. This emission, detected with the non-imaging PDS instrument, was attributed to η Car, based mostly on the fact that η Car is the strongest and hardest source seen in the simultaneous medium energy concentrator spectrometer (MECS) 1.5-10 keV imaging observations (Viotti et al. 2002a(Viotti et al. , 2004. However, the INTEGRAL observations presented in Sect. 2.2 indicate the presence of at least two additional hard X-ray sources in the PDS field of view (FOV; see Fig. 1), but which were not within the MECS FOV. Therefore, a significant fraction of the hard X-ray flux detected by PDS originates from sources nearby η Car, and could affect the PDS spectral analysis performed by Viotti et al. (2004). INTEGRAL observations The ESA INTEGRAL γ-ray satellite carries (in addition to an optical monitoring camera) 3 co-aligned instruments dedicated to the observation of the high-energy sky, from 3 keV up to 10 MeV (Winkler et al. 2003). The INTEGRAL soft gamma-ray imager (ISGRI; Lebrun et al. 2003), the most sensitive detector between 15 and 200 keV, offers the first unambiguous detection of η Car at hard X-rays. All available public INTEGRAL data located within 10 • of η Car were selected, resulting in 1131 pointings, for a total elapsed observing time of 3.3 Ms and a deadtime-corrected good exposure of 2.3 Ms. There are 3 major periods during which the source was frequently observed, listed in Table 1. Together, they represent about 85% of the data used; the remaining observations come from Galactic plane scans, and are therefore well spaced over time. The effective exposure time (corresponding to an onaxis observation) amounts to 1.1 Ms (in the 22-35 keV energy range). ISGRI pointing sky images were produced using OSA 1 , version 6.0, with standard parameters. Good time intervals were built using a strong constraint on the attitude stability (∆ < 3 ). The image cleaning step used an input catalogue of 24 sources with fixed source positions, and was applied independently of the source strength (thus allowing for negative source models) in order to avoid introducing any bias in the process. A broad band (22-100 keV) mosaic image of the field was produced, along with narrower bands in 3 energy ranges : 22-35 keV, 35-50 keV, 50-100 keV. The most external parts of each individual image were skipped, as they are more noisy but do not add much signal. The final mosaic images were built in equatorial coordinates with a tangential projection, using an oversampling factor of two with respect to the individual sky images. The photometric integrity and accurate astrometry are obtained by calculating the intersection between input and output pixels, and weighting the count rates with the overlapping area. The ISGRI 22-100 keV image is shown in Fig. 1. Several flux excesses are clearly detected; Table 2 summarizes their best-fit position, error circle, intensity and significance. Besides η Car, one source has an error circle that includes the anomalous X-ray pulsar (AXP) 1E 1048.1-5937, and another one (named IGR J10447-6027) is coincident with the South of the giant dust pillar nebula of the Carina region, and in particular with a massive young stellar object (YSO; IRAS 10423-6011), probably corresponding to an embedded B0 star (Rathborne et al. 2004). The emission observed by INTEGRAL for the latter source could be interpreted either as evidence for a new high-mass X-ray binary, or as a signature of accretion and/or particle acceleration in the YSO. A significant fraction of the flux detected with PDS is likely to originate not only from η Car but also from these other two sources, in proportions that depend on both the energy range and the time. Table 2. The most significant source is coincident with the most powerful X-ray source in the field (η Car), although it should be noted that another close X-ray source (WR 25) lies just outside of the error box. However, the shape of the point-spread function (PSF) in the best fit (5.4 by 6.3 ) is consistent with a point source. Moreover, the positions extracted from the images (in all individual energy bands, as well as in the 22-100 keV image) indicate that only η Car is consistently inside the error box, while WR 25 always lies close, but outside. The angular separation between η Car and WR 25 is larger than 7 , as opposed to the typical error box of 3 . Therefore, the hard X-ray emission observed with INTEGRAL can be associated with η Car. Based on the 22-100 keV images, Table 3 lists (for the three major observing periods, as well as for the whole data set) the effective exposure, the count rate and the detection significance of η Car (or the 3σ-upper limit in the case of the first period where the object is not detected). As expected from the highly uneven effective exposure durations of the 3 different major observing periods, the source is not detected during the first period, well detected during the second one, and slightly below the detection level during the third period. The average source flux observed by ISGRI, extracted from this image by fixing the position of the object and by assuming a PSF of 6 , is 0.15 ± 0.02 count s −1 in the 22-100 keV energy range. This corresponds to a flux of 1.11 × 10 −11 erg cm −2 s −1 when assuming a photon index of 1. Based on a distance of 2.3 kpc (Smith 2006), the hard X-ray luminosity is 7 × 10 33 erg s −1 (22-100 keV). The ISGRI unfolded spectrum (extracted under the same conditions) is shown in Fig. 2, together with the archival MECS spectrum from June 2000. They are fitted with a wabs*mekal model (kT 5.1 keV and N H 4.3 × 10 22 cm −2 , both in agreement with Viotti et al. 2004). This mekal model provides an excellent fit to the MECS data, and to the first ISGRI bin (i.e. up to 30 keV), but is unable to reproduce the spectrum at higher energy. Given the limited number of points in the ISGRI data, the spectral shape of the high-energy emission is poorly constrained. A simple powerlaw model fit to the hard X-ray data gives a photon index Γ around 1±0.4 (with harder values reached when used in combination with the mekal component to fit the low-energy part of the ISGRI spectrum). The photon index is much harder than found by Viotti et al. (2004), but the data are not simultaneous, and the 2 other sources within the PDS FOV could have been responsible for a signifi-cant contribution. The ISGRI spectrum extends at least up to 100 keV, while the PDS X-ray tail does not go beyond 50 keV. At energies between 20 and 50 keV, the average spectrum obtained from ISGRI observations is much weaker than the PDS spectrum from June 2000, while at energies above 50 keV, the ISGRI flux is stronger. Hence, the ISGRI photon index is much harder than an extension of the hard X-ray thermal component observed below 10 keV. Based on ISGRI observations listed in Table 3, the tiny difference in observing time between periods 1 and 3 does not justify the total absence of source detection in period 1. Therefore, it seems that some variability is observed, in agreement with that reported by Viotti et al. (2004) following the high and low spectroscopic states. However, given the presence of 3 sources within the PDS FOV, it is unclear whether this variation can be trusted in the BeppoSAX data. In particular, the flux observed between 20 and 30 keV with ISGRI is roughly 10 times lower than the flux observed with PDS, and this difference could perhaps be attributed to the 2 additional sources. It should also be noted that the AXP is variable, and exhibited a new X-ray burst detected by Swift in March 2007, with a flux up to 6.3 × 10 −11 erg s −1 cm −2 (in the 1-10 keV energy range; Campana & Israel 2007); this is 8 times brighter than observed in the averaged ISGRI mosaics. Discussion The high-energy flux excess observed at the position of η Car reveals unambiguously for the first time the presence of nonthermal emission at hard X-rays in a CWB. Although possibilities, such as an unknown background object or a highly obscured super-giant X-ray binary, may be envisioned, it is unlikely that such an object would not have been detected in X-rays, especially in the frequently observed Carina region. Inverse Compton (IC) scattering of low-frequency photons by high-energy electrons accelerated in the wind collision zone of CWBs (see e.g. Benaglia & Romero 2003) is proposed as the emission mechanism responsible for the hard X-ray detection. As explained in Sect. 1, η Car is an eccentric CWB, thus with a long orbital period and a large orbital separation. This implies that the winds from the two members of the binary system reach their terminal velocity before colliding 2 , leading to strong shocks with high temperatures. This also means that the density of UV stellar photons is relatively low in the shock zone between the winds 3 , as opposed to CWBs with short orbital periods. The question a priori is therefore whether the level of IC emission is sufficient to allow a detection against the thermal emission from the shocked winds. The total power contained in stellar wind interactions can be evaluated as: L = 0.5ΞṀv 2 (Pittard & Stevens 2002). For η Car, the parameters from Pittard & Corcoran (2002) can be adopted :Ṁ 1 = 2.5 × 10 −4 M /yr, v ∞,1 = 500 km s −1 ,Ṁ 2 = 1 × 10 −5 M /yr, and v ∞,2 = 3000 km s −1 ; corresponding to η = (Ṁ 2 v ∞,2 )/(Ṁ 1 v ∞,1 ) = 0.2 (hence Ξ 1 0.05 and Ξ 2 0.35). These values yield a total power L 1 + L 2 of ∼ 10 37 erg s −1 , which is potentially available for thermal and non-thermal emission. The luminosity detected by ISGRI represents only 0.1% of the total power involved in the stellar wind interactions. η Car shares some properties with WR 140 (e.g. long period, high eccentricity). The non thermal radio detection of WR 140 allowed Pittard & Dougherty (2006) to estimate the expected (but not yet observed) hard X-ray IC emission as 0.5% of the kinetic power involved in the wind collision, close to the value observed in η Car. However, WR 140 is not detected with ISGRI (De Becker et al. 2007). Assuming that all the synchrotron emission produced in the shock region could escape without suffering a significant absorption, an upper limit of the magnetic field in the shock region can be estimated: B 2 < 8πU ph L radio /L IC , where U ph ∼ 0.1 is derived from the luminosity and orbital size of η Car (see e.g. Kashi & Soker 2007). The observed radio flux at 3.5 cm is typically around 5% of the IC emission observed by INTEGRAL (Kashi & Soker 2007). Although it is likely that a substantial fraction of the synchrotron flux is actually absorbed in the wind interaction zone, it is remarkable that the derived value of ∼ 0.3G is comparable with the values inferred for other CWB systems (see e.g. Benaglia & Romero 2003). The IC emission of η Car is expected to vary along the orbit, ranging from a weak emission with a low-energy cutoff around periastron, to a detectable flux and a cutoff in the GeV region when the soft photon density decreases (Reimer et al. 2006). The spectral slope integrated over the shock region is believed to vary between 1.5 and 2, depending on the geometry of the system. The closest EGRET source is 3EG 1048-5840 (Hartman et al. 1999), located 1.1 • away from η Car, and which was associated with PSR J1048-5832 (Kaspi et al. 2000). Even when using this EGRET spectrum as an upper limit for the average GeV emission of η Car, the spectral slope observed with ISGRI is significantly harder than a powerlaw extrapolation between 2 Given the high eccentricity of η Car (e ∼ 0.9), the separation at periastron could mean the primary wind does not reach its terminal velocity. However, as shown by Table 3, the observations are dominated by the phases when η Car is far from periastron. 3 Unless one of the two stars has a very weak wind, thus implying that the collision region would be displaced very close towards that star. This seems however unlikely, given what is already known about their parameters from the analysis of the X-ray lightcurve of η Car (see below). the ISGRI measurements and the EGRET upper limits, thus indicating that the high-energy spectrum of η Car should gradually steepen with energy in the 100 MeV-1 GeV region. This is in reasonable agreement with the predictions of Reimer et al. (2006), and makes η Car a source likely to be detected by Agile, GLAST, and perhaps HESS-II. Conclusions The first unambiguous detection of η Car in the hard X-rays unveils a luminosity of 7 × 10 33 erg s −1 (22-100 keV), i.e., 0.1% of the kinetic energy available in the wind collision. This is the first observation of inverse Compton emission and particle acceleration in a colliding-wind binary. The absence of non-thermal radio emission allows to constrain the magnetic field in the particle acceleration region to be below 0.3 G. η Car is expected to be detected in the GeV range, with Agile and GLAST. A detection (or an upper limit) with HESS-II will also be very useful to constrain the emission spectrum. Two other nearby hard X-ray sources have also been detected, the AXP 1E 1048.1-5937 and a new source (IGR J10447-6027), which coincides with a YSO in the South of the Carina giant dust pillar nebula. Send offprint requests to: J.-C. Leyder FNRS Research Fellow FNRS Research Associate Fig. 1 . 1ISGRI significance image (22-100 keV; effective exposure time of 1.1 Ms; significance goes from 2σ to 5σ) around η Car, showing the positions of η Car and WR 25 (in magenta). The enclosing green circle symbolizes the PDS FOV. The white and red circles represent the best-fit positions listed in Fig. 2 . 2Unfolded spectrum of η Car, with MECS (2-10 keV) and ISGRI (22-100 keV) data, both shown with red crosses. The data are fitted with a wabs*mekal model (in green; kT 5.1 keV). Table 1 . 1Main INTEGRAL public data available for η Car.Period Rev. Time [MJD] Phase Φ of η Car 1 76-88 52 787-52 827 1.99-2.01 2 192-209 53 134-53 188 2.16-2.19 3 322-330 53 523-53 550 2.35-2.37 Table 2 . 2Sources detected within the PDS FOV around η Car (the flux and significance values were extracted from the 22-100 keV energy band, with the best-fit position, and with the PSF size left free to vary; the error circle corresponds to a 90% probability).Source Intensity [cnt s −1 ] Position (J2000) Error circle radius [ ] Significance η Car 0.16 ± 0.02 RA = 10 h 45 m 02, Dec = −59 • 43 38 2.8 7.9 1E 1048.1-5937 0.09 ± 0.02 RA = 10 h 50 m 38, Dec = −59 • 50 40 5.1 4.5 IGR J10447-6027 0.12 ± 0.02 RA = 10 h 44 m 47, Dec = −60 • 27 15 3.4 5.8 Table 3 . 3INTEGRAL observations of η Car (22-100 keV fluxes extracted by fixing the position of η Car and the PSF size to 6 ).Period Eff. exp. [ks] Count rate [cnt s −1 ] Significance 1 122 < 0.19 - 2 717 0.16 ± 0.03 6.2 3 180 0.18 ± 0.05 3.3 All data 1113 0.15 ± 0.02 7.3 The Offline Scientific Analysis (OSA) software is available from the ISDC website : http://isdc.unige.ch Winkler, C., Courvoisier, T. J.-L., Di Cocco, G., et al. 2003, A&A, 411, L1 Zanella, R., Wolf, B., & Stahl, O. 1984 Acknowledgements. This research has made use of the SIMBAD database (CDS), of public data from INTEGRAL (ESA) & BeppoSAX (ASI). JCL and GR acknowledge support through the XMM-INTEGRAL PRODEX project. . A Altamore, J.-P Maillard, R Viotti, A&A. 292208Altamore, A., Maillard, J.-P., & Viotti, R. 1994, A&A, 292, 208 . C D Andriesse, B D Donn, R Viotti, MNRAS. 185771Andriesse, C. D., Donn, B. D., & Viotti, R. 1978, MNRAS, 185, 771 . P Benaglia, G E Romero, A&A. 3991121Benaglia, P. & Romero, G. E. 2003, A&A, 399, 1121 S Campana, G L Israel, The Astronomer's Telegram. 10431Campana, S. & Israel, G. L. 2007, The Astronomer's Telegram, 1043, 1 . 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K Ishibashi, M F Corcoran, K Davidson, ApJ. 524983Ishibashi, K., Corcoran, M. F., Davidson, K., et al. 1999, ApJ, 524, 983 . A Kashi, N Soker, MNRAS. 3781609Kashi, A. & Soker, N. 2007, MNRAS, 378, 1609 . V M Kaspi, J R Lackey, J Mattox, ApJ. 528445Kaspi, V. M., Lackey, J. R., Mattox, J., et al. 2000, ApJ, 528, 445 . F Lebrun, J P Leray, P Lavocat, A&A. 411141Lebrun, F., Leray, J. P., Lavocat, P., et al. 2003, A&A, 411, L141 . J M Pittard, M F Corcoran, A&A. 383636Pittard, J. M. & Corcoran, M. F. 2002, A&A, 383, 636 . J M Pittard, S M Dougherty, MNRAS. 372801Pittard, J. M. & Dougherty, S. M. 2006, MNRAS, 372, 801 . J M Pittard, I R Stevens, A&A. 38820Pittard, J. M. & Stevens, I. R. 2002, A&A, 388, L20 . J M Rathborne, K J Brooks, M G Burton, M Cohen, S Bontemps, A&A. 418563Rathborne, J. M., Brooks, K. J., Burton, M. G., Cohen, M., & Bontemps, S. 2004, A&A, 418, 563 S Rebecchi, R F Viotti, P Grandi, ASP Conf. Ser. 234: X-ray Astronomy. 107Rebecchi, S., Viotti, R. F., Grandi, P., et al. 2001, in ASP Conf. Ser. 234: X-ray Astronomy 2000, ed. R. Giacconi, S. Serio, & L. Stella, 107 . A Reimer, M Pohl, O Reimer, ApJ. 6441118Reimer, A., Pohl, M., & Reimer, O. 2006, ApJ, 644, 1118 . A W Rodgers, L Searle, MNRAS. 13599Rodgers, A. W. & Searle, L. 1967, MNRAS, 135, 99 . F D Seward, Y M Butt, M Karovska, ApJ. 553832Seward, F. D., Butt, Y. M., Karovska, M., et al. 2001, ApJ, 553, 832 . N Smith, ApJ. 6441151Smith, N. 2006, ApJ, 644, 1151 . R Van Boekel, P Kervella, M Schöller, Ap&SS. 410323A&Avan Boekel, R., Kervella, P., Schöller, M., et al. 2003, A&A, 410, L37 Viotti, R. 1969, Ap&SS, 5, 323 R Viotti, Revista Mexicana de Astronomia y Astrofisica Conference Series. V. Niemela, N. Morrell, & A. Feinstein1Viotti, R. 1995, in Revista Mexicana de Astronomia y Astrofisica Conference Series, ed. V. Niemela, N. Morrell, & A. Feinstein, 1 . R F Viotti, L A Antonelli, M F Corcoran, A&A. 385874Viotti, R. F., Antonelli, L. A., Corcoran, M. F., et al. 2002a, A&A, 385, 874 . R F Viotti, L A Antonelli, S Rebecchi, C Rossi, Journal of Astrophysics and Astronomy. 2319Viotti, R. F., Antonelli, L. A., Rebecchi, S., & Rossi, C. 2002b, Journal of Astrophysics and Astronomy, 23, 19 . R F Viotti, L A Antonelli, C Rossi, S Rebecchi, A&A. 420527Viotti, R. F., Antonelli, L. A., Rossi, C., & Rebecchi, S. 2004, A&A, 420, 527 . P A Whitelock, M W Feast, C Koen, G Roberts, B S Carter, MNRAS. 270364Whitelock, P. A., Feast, M. W., Koen, C., Roberts, G., & Carter, B. S. 1994, MNRAS, 270, 364 . P A Whitelock, M W Feast, F Marang, E Breedt, MNRAS. 352447Whitelock, P. A., Feast, M. W., Marang, F., & Breedt, E. 2004, MNRAS, 352, 447

We address the problem of phase shift operator acting as time evolution operator in Pegg-Barnett formalism. It is argued that standard shift operator is inconsistent with the behaviour of the state vector under cyclic evolution. We consider a generally deformed oscillator algebra at q root of unity, as it yields the same Pegg-Barnett phase operator and show that shift operator within this algebra meets our requirement. 03.65.-w In recent years, Pegg-Barnett (PB) formalism has attracted wide attention as a theory for quantum phase [1],[2]. Alongside, the subject of quantum algebras and their realizations in terms of q-deformed oscillators has also been studied with great interest [3]-[5]. The problem of quantum phase has also been persued from q-deformation theoretic point of view. There are certain justifications for this approach. A feature of a q-deformed theory or framework is that one can identify an inherent scale in it, of magnitude ∼ |1 − q|, where q is called deformation parameter. As q → 1, one retreives the undeformed or "classical" theory. Now PB formalism can also be looked upon as inherently q-deformed [6] in the abovesaid sense and in this case q = exp(i2π/s + 1), where (s + 1) is the dimension of the Hilbert space. Secondly, the phase observable which is hermitian phase operator in PB theory, can be consistently defined only in a finite dimensional Hilbert space (FDHS). These two features form the motivation to study phase using a q-oscillator with q = exp(i2π/(s + 1)) [7]. Firstly, q being root of unity, naturally truncates the q-oscillator to a FDHS. Secondly, infinite dimensional limit (s → ∞) also corresponds to the deformation free (q → 1) limit. However, the problem of negative norm in this representation was recognised later and there now exist representations of q-oscillator [8] or generally deformed oscillator [9] with positive norm for q as root of unity and for which the Pegg-Barnett phase operator can be consistently defined.Let us first recapitulate relevant key points of PB formalism. Here the phase operator Φ and the number operator N are not canonically conjugate, but satisfy a complicated commutator * e-mail: raman%[email protected] eigenstates of Φ which form an orthonormal set of phase states, are related to the number states by Fourier transformwhere θ m = θ 0 + 2πm s+1 , {m = 0, 1, 2, ..., s}. θ 0 is the arbitrary phase window which defines the phase angle interval 2π modulo, θ 0 ≤ θ m < θ 0 + 2π. Apart from the hermitian phase operator Φ, the unitary phase operator e iΦ is also of significance in PB theory. It acts as shift operator on number statesThus the action of e iΦ is cyclic and it steps down the number states by unity. Its adjoint acts as step up operator. Thus one can write a realization of unitary phase operator as e iΦ = |0 1| + |1 2| + · · · + |s − 1 s| + e i(s+1)θ0 |s 0|.The operator dual to e iΦ is the operator q N , which acts as shift operator on the phase statesNote that the apparent duality between the two kinds of shift operators seems incomplete due to the extra phase factor in Eq. (4) or the lack of corresponding factor in Eq.(7). This is due to the arbitrariness in the choice of phase window in PB formalism, while there is no such choice in the ground state eigenvalue of number operator, which is necessarily zero. Thus the realization of q −N in terms of phase states is q −N = |θ 0 θ 1 | + |θ 1 θ 2 | + · · · + |θ s−1 θ s | + |θ s θ 0 |.Now the unitary phase shift operator q −N can be thought as time evolution operator, which operated once on the phase state advances the phase by 2π/(s + 1). Thus if 1

quant-ph/0003114

770ed75c248fc2ad0898d262693c3f8a72881957

Phase shift operator and cyclic evolution in finite dimensional Hilbert space 24 Mar 2000 Ramandeep S Johal Department of Physics Panjab University 160 014ChandigarhIndia Phase shift operator and cyclic evolution in finite dimensional Hilbert space 24 Mar 2000arXiv:quant-ph/0003114v1 (March 31, 2022) We address the problem of phase shift operator acting as time evolution operator in Pegg-Barnett formalism. It is argued that standard shift operator is inconsistent with the behaviour of the state vector under cyclic evolution. We consider a generally deformed oscillator algebra at q root of unity, as it yields the same Pegg-Barnett phase operator and show that shift operator within this algebra meets our requirement. 03.65.-w In recent years, Pegg-Barnett (PB) formalism has attracted wide attention as a theory for quantum phase [1],[2]. Alongside, the subject of quantum algebras and their realizations in terms of q-deformed oscillators has also been studied with great interest [3]-[5]. The problem of quantum phase has also been persued from q-deformation theoretic point of view. There are certain justifications for this approach. A feature of a q-deformed theory or framework is that one can identify an inherent scale in it, of magnitude ∼ |1 − q|, where q is called deformation parameter. As q → 1, one retreives the undeformed or "classical" theory. Now PB formalism can also be looked upon as inherently q-deformed [6] in the abovesaid sense and in this case q = exp(i2π/s + 1), where (s + 1) is the dimension of the Hilbert space. Secondly, the phase observable which is hermitian phase operator in PB theory, can be consistently defined only in a finite dimensional Hilbert space (FDHS). These two features form the motivation to study phase using a q-oscillator with q = exp(i2π/(s + 1)) [7]. Firstly, q being root of unity, naturally truncates the q-oscillator to a FDHS. Secondly, infinite dimensional limit (s → ∞) also corresponds to the deformation free (q → 1) limit. However, the problem of negative norm in this representation was recognised later and there now exist representations of q-oscillator [8] or generally deformed oscillator [9] with positive norm for q as root of unity and for which the Pegg-Barnett phase operator can be consistently defined.Let us first recapitulate relevant key points of PB formalism. Here the phase operator Φ and the number operator N are not canonically conjugate, but satisfy a complicated commutator * e-mail: raman%[email protected] eigenstates of Φ which form an orthonormal set of phase states, are related to the number states by Fourier transformwhere θ m = θ 0 + 2πm s+1 , {m = 0, 1, 2, ..., s}. θ 0 is the arbitrary phase window which defines the phase angle interval 2π modulo, θ 0 ≤ θ m < θ 0 + 2π. Apart from the hermitian phase operator Φ, the unitary phase operator e iΦ is also of significance in PB theory. It acts as shift operator on number statesThus the action of e iΦ is cyclic and it steps down the number states by unity. Its adjoint acts as step up operator. Thus one can write a realization of unitary phase operator as e iΦ = |0 1| + |1 2| + · · · + |s − 1 s| + e i(s+1)θ0 |s 0|.The operator dual to e iΦ is the operator q N , which acts as shift operator on the phase statesNote that the apparent duality between the two kinds of shift operators seems incomplete due to the extra phase factor in Eq. (4) or the lack of corresponding factor in Eq.(7). This is due to the arbitrariness in the choice of phase window in PB formalism, while there is no such choice in the ground state eigenvalue of number operator, which is necessarily zero. Thus the realization of q −N in terms of phase states is q −N = |θ 0 θ 1 | + |θ 1 θ 2 | + · · · + |θ s−1 θ s | + |θ s θ 0 |.Now the unitary phase shift operator q −N can be thought as time evolution operator, which operated once on the phase state advances the phase by 2π/(s + 1). Thus if 1 We address the problem of phase shift operator acting as time evolution operator in Pegg-Barnett formalism. It is argued that standard shift operator is inconsistent with the behaviour of the state vector under cyclic evolution. We consider a generally deformed oscillator algebra at q root of unity, as it yields the same Pegg-Barnett phase operator and show that shift operator within this algebra meets our requirement. 03.65.-w In recent years, Pegg-Barnett (PB) formalism has attracted wide attention as a theory for quantum phase [1], [2]. Alongside, the subject of quantum algebras and their realizations in terms of q-deformed oscillators has also been studied with great interest [3]- [5]. The problem of quantum phase has also been persued from q-deformation theoretic point of view. There are certain justifications for this approach. A feature of a q-deformed theory or framework is that one can identify an inherent scale in it, of magnitude ∼ |1 − q|, where q is called deformation parameter. As q → 1, one retreives the undeformed or "classical" theory. Now PB formalism can also be looked upon as inherently q-deformed [6] in the abovesaid sense and in this case q = exp(i2π/s + 1), where (s + 1) is the dimension of the Hilbert space. Secondly, the phase observable which is hermitian phase operator in PB theory, can be consistently defined only in a finite dimensional Hilbert space (FDHS). These two features form the motivation to study phase using a q-oscillator with q = exp(i2π/(s + 1)) [7]. Firstly, q being root of unity, naturally truncates the q-oscillator to a FDHS. Secondly, infinite dimensional limit (s → ∞) also corresponds to the deformation free (q → 1) limit. However, the problem of negative norm in this representation was recognised later and there now exist representations of q-oscillator [8] or generally deformed oscillator [9] with positive norm for q as root of unity and for which the Pegg-Barnett phase operator can be consistently defined. Let us first recapitulate relevant key points of PB formalism. Here the phase operator Φ and the number operator N are not canonically conjugate, but satisfy a complicated commutator * e-mail: raman%[email protected] [Φ, N ] = 2πh s + 1 l n,n ′ =−l (n ′ − n)|n ′ n| exp[2πi(n − n ′ )/(s + 1)] − 1 .(1) The eigenstates of Φ which form an orthonormal set of phase states, are related to the number states by Fourier transform |θ m = 1 √ s + 1 s n=0 exp(inθ m )|n ,(2) where θ m = θ 0 + 2πm s+1 , {m = 0, 1, 2, ..., s}. θ 0 is the arbitrary phase window which defines the phase angle interval 2π modulo, θ 0 ≤ θ m < θ 0 + 2π. Apart from the hermitian phase operator Φ, the unitary phase operator e iΦ is also of significance in PB theory. It acts as shift operator on number states e iΦ |n = |n − 1 , n = 0 (3) e iΦ |0 = e i(s+1)θ0 |s .(4) Thus the action of e iΦ is cyclic and it steps down the number states by unity. Its adjoint acts as step up operator. Thus one can write a realization of unitary phase operator as e iΦ = |0 1| + |1 2| + · · · + |s − 1 s| + e i(s+1)θ0 |s 0|.(5) The operator dual to e iΦ is the operator q N , which acts as shift operator on the phase states q −N |θ m = |θ m−1 , m = 0 (6) q −N |θ 0 = |θ s .(7) Note that the apparent duality between the two kinds of shift operators seems incomplete due to the extra phase factor in Eq. (4) or the lack of corresponding factor in Eq. (7). This is due to the arbitrariness in the choice of phase window in PB formalism, while there is no such choice in the ground state eigenvalue of number operator, which is necessarily zero. Thus the realization of q −N in terms of phase states is q −N = |θ 0 θ 1 | + |θ 1 θ 2 | + · · · + |θ s−1 θ s | + |θ s θ 0 |.(8) Now the unitary phase shift operator q −N can be thought as time evolution operator, which operated once on the phase state advances the phase by 2π/(s + 1). Thus if we operate it (s + 1) times on a phase state, we complete one cycle and return to the same phase state. On the other hand, we have the results of Ref. [10], where it was shown that for cyclic evolution of harmonic oscillator in a general state n c n |n , in FDHS, the state vector can change sign which depends on the dimesnionality of the space; if (s + 1) is even sign changes, otherwise not. However, if we take q −N as equivalent to time evolution operator, we note that according to realization of Eq. (8), the state vector always returns exactly to initial state, irrespective of the dimensionality of the space. The purpose of this paper is to make the action of phase shift operator consistent with that of time evolution operator in the context of cyclic evolution in FDHS. We take as our model the recently proposed generally deformed oscillator [9], which has certain advantages over other approaches from algebraic point of view, namely, i) The creation and annihilation operators in PB theory do not form a closed algebra by themselves, and they do not go over to corresponding relations in the s → ∞ limit ii) we can algebraically define PB phase operator in the approach of [9], iii) for q as root of unity, positive norm is also assured. Briefly, in the approach of [9], new creation and annihilation operators are defined A † = F (q N )e −iΦ , A = e iΦ F (q N ), q N = q N +η .(9) The action of these operators on generalized number states is A † |n + η = F (q n+η+1 )|n + η + 1 , n = s (10) A † |s + η = e −i(s+1)θ0 F (q n )|η (11) A|n + η = F (q n+η )|n + η − 1 , n = 0 (12) A|η = F (q η )e i(s+1)θ0 |s + η (13) q N |n + η = q n+η |n + η . The parameter η is chosen such that i) the above defines a cyclic representation, ii) the function F is hermitian and non-negative, iii) in s → ∞ limit, A † and A go over to the creation and annihilation operators of the ordinary oscillator. Also the condition for cyclic representation (F (q η ) = 0) in Eqs. (11) and (13)) also ensures that one can consistently define unitary phase operator by inverting A † and A in Eq. (9). Note that this approach exactly recovers the PB phase operator. However a significant fact that was missed in [9] is that in the above representation, q ±N can also act as phase shift operator on the phase states. As one can easily see, its action gives q −N |θ m = q −η |θ m−1 and q −N |θ 0 = q −η |θ s , which is just same as Eqs. (6) and (7). However, as we show below, the significance of this operator lies in its being consistent with the results of cyclic evolution in FDHS [10]. As a solution for restoring the duality in e iΦ and q −N , we propose to modify the Eq. (2) as follows: |θ m = 1 √ s + 1 s n=0 exp(i(n + η)θ m )|n + η ,(15) so that now we have q −N |θ m = |θ m−1 , m = 0 (16) q −N |θ 0 = e −i2πη |θ s .(17) The action of e iΦ on the phase states remains as such, i.e. e iΦ |θ m = θ m |θ m . Moreover, the action of e iΦ on the (new) number states remains same as before. Thus from Eq. (15) |n + η = 1 √ s + 1 s m=0 exp(−i(n + η)θ m )|θ m ,(18) we can write e iΦ |n + η = |n + η − 1 , n = 0 (19) e iΦ |η = e i(s+1)θ0 |s + η .(20) Thus the duality between e iΦ and q −N is exactly obeyed, so that parameter η plays the role equivalent to θ 0 . We can as well write the following realization for modified unitary operator q −N = |θ 0 θ 1 | + |θ 1 θ 2 | + · · · + |θ s−1 θ s | + e −i2πη |θ s θ 0 |.(21) Therefore, operating the above unitary operator (s + 1) times, we get q −N s+1 |θ m = e −i2πη |θ m .(22) Next, we are interested to know if under such cyclic evolution, the state vector changes sign or not. Thus if η is an integer, no change in sign occurs, while for η as half-odd integer, there is change in sign. Now the usual time evolution operator is e −iHt/h , where for the case of harmonic oscillator in FDHS [10], the hamiltonian H has the following energy spectrum E n =hω n + 1 2 + (s + 1) 2 δ n,s .(23) Thus under evolution through one time period, t = 2π/ω, the state vector |n is multiplied by the phase factor exp (−i2π{n + 1/2 + (s + 1)δ n,s /2}). On the other hand, if we consider time evolution through unitary shift operator q −N , this means that state vector is multiplied by the factor exp (−i2π{n + η}). Thus we note that for a harmonic oscillator in FDHS, for n = s, we have η = 1/2, whereas for n = s, η = 1/2 + (s + 1)/2. So (s + 1) as even number is equivalent to η as half-odd integer, which from previous discussion, implies change in sign under cyclic evolution, whereas (s + 1) odd is equivalent to η as integer and consequently no change in sign of the state vector under one cycle. Also, the case of infinite dimensional harmonic oscillator requires that E n = (n + 1/2)hω, which is consistent with η = 1/2. Finally, it is interesting to note that states |n + η can be obtained from usual number states |n , by applying a continuous unitary transformation i.e. when η is not an integer (e −iηΦ |n = |n + η ). As was pointed out in [1], such continuous unitary transformations are useful to construct the phase-moment generating functions. Concluding, we have argued that phase shift operator in standard PB formalism is inconsistent with cyclic evolution of harmonic oscillator in finite dimesional Hilbert space. To treat this, we have shown that phase shift operator of a generally deformed oscillator algebra at q root of unity, and which yields the same PB phase operator, can simulate the behaviour of time evolution operator for cyclic evolution. This also restores the duality in the actions of phase-and number-shift operators. The author would like to acknowledge the kind hospitality of H.S. Mani and Sumathi Rao at Mehta Research Institute, Allahabad, where this work was initiated and S. Abe, for careful reading of the manuscript. . D T Pegg, S M Barnett, Europhys. Lett. 61665Phys. Rev. AD.T. Pegg and S.M. Barnett, Europhys. Lett. 6 (1988) 483; Phys. Rev. A 39 (1989) 1665. . V Buzeck, A D Wilson-Gordon, P L Knight, Phys. Rev. A. 453289and references thereinV. Buzeck, A.D. Wilson-Gordon and P.L. Knight, Phys. Rev. A 45 (1992) 3289; and references therein. . A J Macfarlane, J. Phys. A: Math. Gen. 224581A.J. Macfarlane, J. Phys. A: Math. Gen. 22 (1989) 4581; . L C Biedenharn, L873, L.C. Biedenharn, ibid., L873. . C-P Sun, H-C Fu, J. Phys. A: Math. Gen. 22983C-P Sun and H-C Fu, J. Phys. A: Math. Gen. 22 (1989) L983. . P P Kulish, E V Damaskinsky, J. Phys. A: Math. Gen. 23415P.P. Kulish and E.V. Damaskinsky, J. Phys. A: Math. Gen. 23 (1990) L415. . S Abe, Phys. Lett. A. 200239S. Abe, Phys. Lett. A 200 (1995) 239. . D Ellinas, Phys. Rev. A. 453358D. Ellinas, Phys. Rev. A 45 (1992) 3358. . K Fujikawa, L C Kwek, C H Oh, Mod. Phys. Lett. A. 102543K. Fujikawa, L.C. Kwek and C.H. Oh, Mod. Phys. Lett. A 10 (1995) 2543. . H C Fu, R Sasaki, J. Phys. A: Math. Gen. 294049H.C. Fu and R. Sasaki, J. Phys. A: Math. Gen. 29 (1996) 4049. . A K Pati, S V Lawande, Phys. Rev. A. 515012A.K. Pati and S.V. Lawande, Phys. Rev. A 51 (1995) 5012.

In the era of social media and networking platforms, Twitter has been doomed for abuse and harassment toward users specifically women. Monitoring the contents including sexism and sexual harassment in traditional media is easier than monitoring on the online social media platforms like Twitter, because of the large amount of user generated content in these media. So, the research about the automated detection of content containing sexual or racist harassment is an important issue and could be the basis for removing that content or flagging it for human evaluation. Previous studies have been focused on collecting data about sexism and racism in very broad terms. However, there is no much study focusing on different types of online harassment alone attracting natural language processing techniques. In this work, we present an attentionbased approach for the detection of harassment in tweets and the detection of different types of harassment as well. Our approach is based on the Recurrent Neural Networks and particularly we are using a deep, classication specific attention mechanism. Moreover, we present a comparison between different variations of this attention-based approach.

10.1007/978-3-030-43887-6_26

1909.13104

2fde20910fba6df7d9b2b043edaac0c6c1765ff9

Attention-based method for categorizing different types of online harassment language Christos Karatsalos Athens University of Economics and Business AthensGreece Yannis Panagiotakis [email protected] Athens University of Economics and Business AthensGreece Attention-based method for categorizing different types of online harassment language Text classification · Twitter · Hate Speech · Deep Learning · Attention Mechanism In the era of social media and networking platforms, Twitter has been doomed for abuse and harassment toward users specifically women. Monitoring the contents including sexism and sexual harassment in traditional media is easier than monitoring on the online social media platforms like Twitter, because of the large amount of user generated content in these media. So, the research about the automated detection of content containing sexual or racist harassment is an important issue and could be the basis for removing that content or flagging it for human evaluation. Previous studies have been focused on collecting data about sexism and racism in very broad terms. However, there is no much study focusing on different types of online harassment alone attracting natural language processing techniques. In this work, we present an attentionbased approach for the detection of harassment in tweets and the detection of different types of harassment as well. Our approach is based on the Recurrent Neural Networks and particularly we are using a deep, classication specific attention mechanism. Moreover, we present a comparison between different variations of this attention-based approach. Introduction In the era of social media and networking platforms, Twitter has been doomed for abuse and harassment toward users specifically women. In fact, online harassment becomes very common in Twitter and there have been a lot of critics that Twitter has become the platform for many racists, misogynists and hate groups which can express themselves openly. Online harassment is usually in the form of verbal or graphical formats and is considered harassment, because it is neither invited nor has the consent of the receipt. Monitoring the contents including sexism and sexual harassment in traditional media is easier than monitoring on the online social media platforms like Twitter. The main reason is because of the large amount of user generated content in these media. So, the research about the automated detection of content containing sexual harassment is an important issue and could be the basis for removing that content or flagging it for human evaluation. The basic goal of this automatic classification is that it will significantly improve the process of detecting these types of hate speech on social media by reducing the time and effort required by human beings. Previous studies have been focused on collecting data about sexism and racism in very broad terms or have proposed two categories of sexism as benevolent or hostile sexism [1], which undermines other types of online harassment. However, there is no much study focusing on different types online harassment alone attracting natural language processing techniques. In this paper we present our work, which is a part of the SociaL Media And Harassment Competition of the ECML PKDD 2019 Conference. The topic of the competition is the classification of different types of harassment and it is divided in two tasks. The first one is the classification of the tweets in harassment and non-harassment categories, while the second one is the classification in specific harassment categories like indirect harassment, physical and sexual harassment as well. We are using the dataset of the competition, which includes text from tweets having the aforementioned categories. Our approach is based on the Recurrent Neural Networks and particularly we are using a deep, classication specific attention mechanism. Moreover, we present a comparison between different variations of this attention-based approach like multi-attention and single attention models. The next Section includes a short description of the related work, while the third Section includes a description of the dataset. After that, we describe our methodology. Finally, we describe the experiments and we present the results and our conclusion. Related Work Waseem et al. [2] were the first who collected hateful tweets and categorized them into being sexist, racist or neither. However, they did not provide specific definitions for each category. Jha and Mamidi [1] focused on just sexist tweets and proposed two categories of hostile and benevolent sexism. However, these categories were general as they ignored other types of sexism happening in social media. Sharifirad S. and Matwin S. [3] proposed complimentary categories of sexist language inspired from social science work. They categorized the sexist tweets into the categories of indirect harassment, information threat, sexual harassment and physical harassment. In the next year the same authors proposed [4] a more comprehensive categorization of online harassment in social media e.g. twitter into the following categories, indirect harassment, information threat, sexual harassment, physical harassment and not sexist. For the detection of hate speech in social media like twitter, many approaches have been proposed. Jha and Mamidi [1] tested support vector machine, bidirectional RNN encoder-decoder and FastText on hostile and benevolent sexist tweets. They also used SentiWordNet and subjectivity lexicon on the extracted phrases to show the polarity of the tweets. Sharifirad et al. [5] trained, tested and evaluated different classification methods on the SemEval2018 dataset and chose the classifier with the highest accuracy for testing on each category of sexist tweets to know the mental state and the affectual state of the user who tweets in each category. To overcome the limitations of small data sets on sexist speech detection, Sharifirad S. et al. [6] have applied text augmentation and text generation with certain success. They have generated new tweets by replacing words in order to increase the size of our training set. Moreover, in the presented text augmentation approach, the number of tweets in each class remains the same, but their words are augmented with words extracted from their ConceptNet relations and their description extracted from Wikidata. Zhang et al. [7] combined convolutional and gated recurrent networks to detect hate speech in tweets. Others have proposed different methods, which are not based on deep learning. Burnap and Williams [8] used Support Vector Machines, Random Forests and a metaclassifier to distinguish between hateful and non-hateful messages. A survey of recent research in the field is presented in [9]. For the problem of the hate speech detection a few approaches have been proposed that are based on the Attention mechanism. Pavlopoulos et al. [10] have proposed a novel, classification-specific attention mechanism that improves the performance of the RNN further for the detection of abusive content in the web. Xie et al. [11] for emotion intensity prediction, which is a similar problem to ours, have proposed a novel attention mechanism for CNN model that associates attention-based weights for every convolution window. Park and Fung [14] transformed the classication into a 2-step problem, where abusive text rst is distinguished from the non-abusive, and then the class of abuse (Sexism or Racism) is determined. However, while the first part of the two step classication performs quite well, it falls short in detecting the particular class the abusive text belongs to. Pitsilis et al. [15] have proposed a detection scheme that is an ensemble of RNN classiers, which incorporates various features associated with user related information, such as the users tendency towards racism or sexism Dataset description The dataset from Twitter that we are using in our work, consists of a train set, a validation set and a test set. It was published for the "First workshop on categorizing different types of online harassment languages in social media". The whole dataset is divided into two categories, which are harassment and non-harassment tweets. Moreover, considering the type of the harassment, the tweets are divided into three sub-categories which are indirect harassment, sexual and physical harassment. We can see in Table 1 the class distribution of our dataset. One important issue here is that the categories of indirect and physical harassment seem to be more rare in the train set than in the validation and test To tackle this issue, as we describe in the next section, we are performing data augmentation techniques. However, the dataset is imbalanced and this has a significant impact in our results. Proposed methodology Data augmentation As described before one crucial issue that we are trying to tackle in this work is that the given dataset is imbalanced. Particularly, there are only a few instances from indirect and physical harassment categories respectively in the train set, while there are much more in the validation and test sets for these categories. To tackle this issue we applying a back-translation method [16], where we translate indirect and physical harassment tweets of the train set from english to german, french and greek. After that, we translate them back to english in order to achieve data augmentation. These "noisy" data that have been translated back, increase the number of indirect and physical harassment tweets and boost significantly the performance of our models. Another way to enrich our models is the use of pre-trained word embeddings from 2B Twitter data [17] having 27B tokens, for the initialization of the embedding layer. Text processing Before training our models we are processing the given tweets using a tweet pre-processor 1 . The scope here is the cleaning and tokenization of the dataset. RNN Model and Attention Mechanism We are presenting an attention-based approach for the problem of the harassment detection in tweets. In this section, we describe the basic approach of our work. We are using RNN models because of their ability to deal with sequence information. The RNN model is a chain of GRU cells [18] that transforms the tokens w 1 , w 2 , ..., w k of each tweet to the hidden states h 1 , h 2 , ..., h k , followed by an LR Layer that uses h k to classify the tweet as harassment or non-harassment (similarly for the other categories). Given the vocabulary V and a matrix E ∈ R d×|V | containing d-dimensional word embeddings, an initial h 0 and a tweet w =< w 1 , .., w k >, the RNN computes h 1 , h 2 , ..., h k , with h t ∈ R m , as follows: h t = tanh(W h x t + U h (r t h t−1 ) + b h ) h t = (1 − z t ) h t−1 + z t h t z t = σ(W z x t + U z h t−1 + b z ) r t = σ(W r x t + U r h t−1 + b r ) where h t ∈ R m is the proposed hidden state at position t, obtained using the word embedding x t of token w t and the previous hidden state h t−1 , represents the element-wise multiplication, r t ∈ R m is the reset gate, z t ∈ R m is the update gate, σ is the sigmoid function. Also W h , W z , W r ∈ R m×d and U h , U z , U r ∈ R m×m , b h , b z , b r ∈ R m . After the computation of state h k the LR Layer estimates the probability that tweet w should be considered as harassment, with W p ∈ R 1×m , b p ∈ R: P RN N (harassment|w) = σ(W p h k + b p ). We would like to add an attention mechanism similar to the one presented in [10], so that the LR Layer will consider the weighted sum h sum of all the hidden states instead of h k : h sum = k t=1 α t h t P attentionRN N = σ(W p h sum + b p ) Alternatively, we could pass h sum through an MLP with k layers and then the LR layer will estimate the corresponding probability. More formally, where h * is the state that comes out from the MLP. The weights α t are produced by an attention mechanism presented in [10] (see Fig. 2), which is an MLP with l layers. This attention mechanism differs from most previous ones [19,20], because it is used in a classification setting, where there is no previously generated output sub-sequence to drive the attention. It assigns larger weights α t to hidden states h t corresponding to positions, where there is more evidence that the tweet should be harassment (or any other specific type of harassment) or not. In our work we are using four attention mechanisms instead of one that is presented in [10]. Particularly, we are using one attention mechanism per category. Another element that differentiates our approach from Pavlopoulos et al. [10] is that we are using a projection layer for the word embeddings (see Fig. 1). In the next subsection we describe the Model Architecture of our approach. Model Architecture The Embedding Layer is initialized using pre-trained word embeddings of dimension 200 from Twitter data that have been described in a previous sub-section. After the Embedding Layer, we are applying a Spatial Dropout Layer, which drops a certain percentage of dimensions from each word vector in the training sample. The role of Dropout is to improve generalization performance by preventing activations from becoming strongly correlated [13]. Spatial Dropout, which has been proposed in [12], is an alternative way to use dropout with convolutional neural networks as it is able to dropout entire feature maps from the convolutional layer which are then not used during pooling. After that, the word embeddings are passing through a one-layer MLP, which has tanh as activation function and 128 hidden units, in order to project them in the vector space of our problem considering that they have been pre-trained using text that has a different subject. In the next step the embeddings are fed in a unidirectional GRU having 1 Stacked Layer and size 128. We prefer GRU than LSTM, because it is more efficient computationally. Also the basic advantage of LSTM which is the ability to keep in memory large text documents, does not hold here, because tweets supposed to be not too large text documents. The output states of the GRU are passing through four self-attentions like the one described above [10], because we are using one attention per category (see Fig. 2). Finally, a one-layer MLP having 128 nodes and ReLU as activation function computes the final score for each category. At this final stage we have avoided using a softmax function to decide the harassment type considering that the tweet is a harassment, otherwise we had to train our models taking into account only the harassment tweets and this might have been a problem as the dataset is not large enough. Experiments Training Models In this subsection we are giving the details of the training process of our models. Moreover, we are describing the different models that we compare in our experiments. Batch size which pertains to the amount of training samples to consider at a time for updating our network weights, is set to 32, because our dataset is not large and small batches might help to generalize better. Also, we set other hyperparameters as: epochs = 20, patience = 10. As early stopping criterion we choose the average AUC, because our dataset is imbalanced. The training process is based on the optimization of the loss function mentioned below and it is carried out with the Adam optimizer [21], which is known for yielding quicker convergence. We set the learning rate equal to 0.001: L = 1 2 BCE(harassment) + 1 2 ( 1 5 BCE(sexualH) + 2 5 BCE(indirectH) + 2 5 BCE(physicalH)) where BCE is the binary cross-entropy loss function, BCE = − 1 n n i=1 [y i log(y i ) + (1 − y i )log(1 − y i ))] i denotes the ith training sample, y is the binary representation of true harassment label, and y is the predicted probability. In the loss function we have applied equal weight to both tasks. However, in the second task (type of harassment classification) we have applied higher weight in the categories that it is harder to predict due to the problem of the class imbalance between the training, validation and test sets respectively. Evaluation and Results Each model produces four scores and each score is the probability that a tweet includes harassment language, indirect, physical and sexual harassment language respectively. For any tweet, we first check the score of the harassment language and if it is less than a specified threshold, then the harassment label is zero, so the other three labels are zero as well. If it is greater than or equal to that threshold, then the harassment label is one and the type of harassment is the one among these three having that has the greatest score (highest probability). We set this threshold equal to 0.33. We compare eight different models in our experiments. Four of them have a Projected Layer (see Fig. 1), while the others do not have, and this is the only difference between these two groups of our models. So, we actually include four models in our experiments (having a projected layer or not). Firstly, Last-StateRNN is the classic RNN model, where the last state passes through an MLP and then the LR Layer estimates the corresponding probability. In contrast, in the AvgRNN model we consider the average vector of all states that come out of the cells. The AttentionRNN model is the one that it has been presented in [10]. Moreover, we introduce the MultiAttentionRNN model for the harassment language detection, which instead of one attention, it includes four attentions, one for each category. We have evaluated our models considering the F1 Score, which is the harmonic mean of precision and recall. We have run ten times the experiment for each model and considered the average F1 Score. The results are mentioned in Table 2. Considering F1 Macro the models that include the multi-attention mechanism outperform the others and particularly the one with the Projected Layer has the highest performance. In three out of four pairs of models, the ones with the Projected Layer achieved better performance, so in most cases the addition of the Projected Layer had a significant enhancement. Conclusion -Future work We present an attention-based approach for the detection of harassment language in tweets and the detection of different types of harassment as well. Our approach is based on the Recurrent Neural Networks and particularly we are using a deep, classication specific attention mechanism. Moreover, we present a comparison between different variations of this attention-based approach and a few baseline methods. According to the results of our experiments and considering the F1 Score, the multi-attention method having a projected layer, achieved the highest performance. Also, we tackled the problem of the imbalance between the training, validation and test sets performing the technique of back-translation. In the future, we would like to perform more experiments with this dataset applying different models using BERT [22]. Also, we would like to apply the models presented in this work, in other datasets about hate speech in social media. Fig. 1 . 1Projection Layer sets. Fig. 2 . 2Attention mechanism, MLP with l Layers Table 1 . 1Class distribution of the dataset.Dataset Tweets Harassment Harassment(%) Indirect (%) Sexual (%) Physical (%) train 6374 2713 42.56 0.86 40.50 1.19 validation 2125 632 29.74 3.34 24.76 1.69 test 2123 611 28.78 9.28 14.69 4.71 Table 2 . 2The results considering F1 Score.Model sexual f1 indirect f1 physical f1 harassment f1 f1 macro attentionRNN 0.674975 0.296320 0.087764 0.709539 0.442150 MultiAttentionRNN 0.693460 0.325338 0.145369 0.700354 0.466130 MultiProjectedAttentionRNN 0.714094 0.355600 0.126848 0.686694 0.470809 ProjectedAttentionRNN 0.692316 0.315336 0.019372 0.694082 0.430276 AvgRNN 0.637822 0.175182 0.125596 0.688122 0.40668 LastStateRNN 0.699117 0.258402 0.117258 0.710071 0.446212 ProjectedAvgRNN 0.655676 0.270162 0.155946 0.675745 0.439382 ProjectedLastStateRNN 0.696184 0.334655 0.072691 0.707994 0.452881 https://pypi.org/project/tweet-preprocessor/ P attentionRN N = σ(W p h * + b p ) When does a compliment become sexist: Analysis and classification of ambivalent sexism using twitter data. 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In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular coefficients, which does not necessarily have the maximum principle. Our method is probabilistic. The time reversal of symmetric Markov processes and the theory of Dirichlet forms play a crucial role in our approach.

10.1214/08-aop427

0907.4301

9cfc326be56a80d7801e28b74f8dfd983bd148ad

TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS Jul 2009. 2009 Zhen-Qing Chen University of Washington University of Manchester Tusheng Zhang University of Washington University of Manchester TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 373Jul 2009. 200910.1214/08-AOP427 In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular coefficients, which does not necessarily have the maximum principle. Our method is probabilistic. The time reversal of symmetric Markov processes and the theory of Dirichlet forms play a crucial role in our approach. 1. Introduction. The pioneering work by Kakutani [18] in 1944 on representing the solution to the classical Dirichlet boundary value problem ∆u = 0, in D, u = f, on ∂D, using Brownian motion started a new era in the very fruitful interplay between probability theory and analysis. Here D is a bounded connected open subset of R n . Since then, in place of Laplacian ∆, there are two classes of second-order elliptic differential operators that have been studied in connection with probabilistic approach. One is the nondivergence form operator L = 1 2 n i,j=1 a ij (x) ∂ 2 ∂x i ∂x j + n i=1 b i (x) ∂ ∂x i + q. (1.1) The other is the divergence form operator where A(x) = (a ij (x)) is an n × n symmetric bounded positive definite matrix. For nondivergence form operator L in (1.1), one can run stochastic differential equation L = 1 2 n i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j + n i=1 b i (x) ∂ ∂x i + q,(1.dX t = σ(X t ) dB t + b(X t ) dt, where σ is a symmetric n × n matrix such that σ 2 = A, b = (b 1 , . . . , b n ) and B is a Brownian motion on R n . The infinitesimal generator of X is L 0 = L − q. Under some suitable conditions, the solution for Lu = 0 in D with u = f on ∂D can be solved by u(x) = E x exp τ D 0 q(X s ) ds f (X τ D ) for x ∈ D; see [12]. Here τ D = inf{t ≥ 0 : X t / ∈ D} is the first exit time from D by X. When L is the divergence form operator of the form (1.2), one has to run symmetric diffusion associated with 1 2 ∇(A∇). Observe that X is in general not a semimartingale when A is just measurable. Nevertheless one can still use the symmetric diffusion X to solve the Dirichlet boundary value problem Lu = 0 in D with u = f on ∂D through a combination of Girsanov and Feynman-Kac transforms (see [8]). The Dirichlet form theory plays the role of Itô's calculus in the divergence form operator case. In this paper, we study the Dirichlet boundary value problems for secondorder elliptic operators of the following form: L = 1 2 ∇ · (A∇) + b · ∇ − div( b·) + q (1.3) = 1 2 n i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j + n i=1 b i (x) ∂ ∂x i − " div( b·)" + q(x) in a bounded domain D ⊂ R n . Here A = (a ij ) : R n → R n × R n is a Borel measurable, symmetric matrix-valued function that is uniformly elliptic and bounded, that is, there is a constant λ ≥ 1 such that λ −1 I n×n ≤ A(x) ≤ λI n×n for every x ∈ R n ; (1.4) b = (b 1 , . . . , b n ) and b = ( b 1 , . . . , b n ) are Borel measurable R n -valued functions on R n and q is a Borel measurable function on R n such that 1 D (|b| 2 + | b| 2 + |q|) ∈ K n . where |µ| denotes the total variational measure of µ. Kato class K n can also be defined for n = 1, 2; see [9] for details. A function q is said to be in K n if µ(dx) := q(x) dx is in K n . Clearly L ∞ (R n ) ⊂ K n , and it is easy to see by using Hölder's inequality that L p (R n ) ⊂ K n for p > n/2. By taking b = b = 0 = q off D, we may and do assume in the sequel that (1.5) holds without the restriction of D by 1 D . Note that "div( b·)" in (1.3) is just a formal writing because the divergence really does not exist for the merely measurable vector field b. It should be interpreted in the distributional sense. It is exactly due to the nondifferentiability of b, all the previous known methods in solving the elliptic boundary value problems such as those in [8] and [12] ceased to work. The lower-order term div( b·) cannot be handled by Girsanov transform or Feynman-Kac transform. We will show in this paper that this term in fact can be tackled by the time-reversal of Girsanov transform from the first exit time τ D from D by the symmetric diffusion X associated with 1 2 ∇(A∇). This is the novelty of this paper. Note that time reversal of a Girsanov transform from a deterministic time was first studied in [20] for diffusions, and very recently in [4] in the context of general m-symmetric Markov processes. We point out that time reversal from a deterministic time in [4,11,20] are defined under the stationary measure P m . Doing time reversal from a random time τ D involves many delicate technical issues for an effective analysis. We are able to circumvent these difficulties through a certain h-transform. See (1.12) below for details. Let (Q, D(Q)) be the bilinear form associated with the operator L, where D(Q) = {u ∈ L 2 (R n ) : ∇u ∈ L 2 (R n )} = W 1,2 (R n ) and for u, v ∈ W 1,2 (R n ) Q(u, v) = 1 2 n i,j=1 R n a ij (x) ∂u ∂x i ∂v ∂x j dx − n i=1 R n b i (x) ∂u ∂x i v(x) dx (1.6) − n i=1 R n b i (x) ∂v ∂x i u(x) dx − R n q(x)u(x)v(x) dx. For an open subset D ⊂ R n , denote by C ∞ c (D) the space of smooth functions on D with compact support. The L 2 -domain D(L) of L is defined to be {u ∈ W 1,2 (R n ): there is g ∈ L 2 (R n ) so that Q(u, v) = (−g, v) L 2 (R n ) for every v ∈ C ∞ c (R n )} and we denote g by Lu. Clearly, it follows from this definition that Q(u, v) = (−Lu, v) L 2 (R n ) for u ∈ D(L) and v ∈ D(Q). It is well known that the differential operator L enjoys the maximum principle if − div( b) + q ≤ 0 in R n in the following distributional sense: n i=1 R n b i (x) ∂φ ∂x i dx + R n q(x)φ(x) dx ≤ 0 (1.7) for all nonnegative function φ in C ∞ c (R n ). Trüdinger [26], Theorem 3.2 and Corollary 5.5, has proved the following: Theorem 1.1. Assume the Markovian condition (1.7) holds. For every f ∈ W 1,2 (D), there exists a unique weak solution u ∈ W 1,2 (D) such that Lu = 0 in D with u − f ∈ W 1,2 0 (D). (1.8) Here W 1,2 0 (D) is the completion of C ∞ c (D) under the Sobolev norm u 1,2 := D (u(x) 2 + |∇u(x)| 2 ) dx 1/2 in W 1,2 (D) . Moreover u is locally Hölder continuous in D. Recall that Lu = 0 in D is understood in the following distributional sense: Q(u, φ) = 0 for every φ ∈ C ∞ c (D) . We stress that condition (1.7) plays a key role in Trüdinger's approach because of the critical use of the maximum principle there. The aim of the present paper is twofold. The first is to give a probabilistic representation for the weak solution of the Dirichlet boundary value problem (1.8). This is highly nontrivial because there is no longer a Markov process associated with the operator L due to the appearance of the lowerorder term div( b·), nor can that lower-order term be handled via Girsanov transform or Feynman-Kac transform. Our idea is to use the symmetric diffusion process X associated with the divergence form operator 1 2 ∇(A∇), the symmetric part of L, and treat L as its lower-order perturbation via a combination of Girsanov and Feynman-Kac transforms and a time-reversal of Girsanov transform at the first exit time τ D from D by X. Based on the new probabilistic representation, our second aim is to establish the existence and uniqueness of the weak solution to problem (1.8) without the Markovian assumption (1.7). To this end, we introduce a kind of h-transformation which transforms the solution of the problem (1.8) to the solution of a Dirichlet boundary value problem for operators which do not involve the adjoint vector field like b. The time reversal and the theory of Dirichlet forms play an essential role throughout this paper. The remaining of the paper is organized as follows. Let X be the symmetric diffusion with infinitesimal generator 1 2 ∇(A∇). It is well known (cf. [25]) that X is a conservative Feller process on R n that has Hölder continuous transition density function which admits a two-sided Aronson's Gaussian type estimate. In general, X is a not a semimartingale but it admits the following Fukushima's decomposition (cf. [14]): X t = X 0 + M t + N t , t ≥ 0, (1.9) where M = (M, . . . , M n ) is a martingale additive functional (MAF) of X with quadratic co-variation M i , M j t = t 0 a ij (X s ) ds and N = (N 1 , . . . , N n ) is a continuous additive functional (CAF) of X locally of zero quadratic variations. Note that, since X has a continuous density function and each a ij is bounded on R d , by [13], Theorem 2, M and N can be refined to be CAFs of X in the strict sense without exceptional set and (1.9) holds under P x for every x ∈ R n . Without loss of generality, we work on the canonical continuous path space C([0, ∞), R n ) of X and, for t > 0, denote by r t the reverse operator of X from time t. In Section 2, we prove that, under the Markovian condition (1.7), the solution u of the problem (1.8) admits the following probabilistic representation: u(x) = E x [Z τ D f (X τ D )] for x ∈ D, (1.10) where τ D := inf{t ≥ 0 : X t / ∈ D} is the first exit time from D by the symmetric diffusion X and Z t = exp t 0 (A −1 b)(X s ) dM s + t 0 (A −1 b)(X s ) dM s • r t (1.11) − 1 2 t 0 (b − b)A −1 (b − b) * (X s ) ds + τ D 0 q(X s ) ds . All the vectors in this paper are row vectors and we use b * to denote the transpose of a vector b. For two vectors α and β in R n , we use α · β or α, β to denote their inner product. Note that Z τ D is well defined under P x for quasi-every x ∈ R n . This is because t → t 0 (A −1 b)(X s ) dM s is a square integrable martingale of X in the strict sense having finite energy and so it follows from [3] and [23] that there is a continuous additive functional L of X having zero energy (which in general may admit an exceptional set) such that for quasi-every (q.e.) x ∈ R d , P x -a.s. t 0 (A −1 b)(X s ) dM s • r t = − t 0 (A −1 b)(X s ) dM s + L t , t ≥ 0. To prove the probabilistic representation (1.10), we introduce a sequence of approximating operators L k with b in the definition of L replaced by smooth b k . We first show that the solution u k of the Dirichlet boundary value problem for the operator L k has the representation (1.10) with b replaced by smooth b k . To complete the proof, we then show that both the approximating solutions and the corresponding probabilistic expressions converge in an appropriate sense. The purpose of Section 3 is to show that the weak solution u to the Dirichlet boundary value problem (1.8) is continuous up to the boundary of the domain D. The crucial step is to show that for every L p -integrable R n -valued function f in a ball B R with radius R > 0 and p > n, there exists a function v ∈ W 1,p 0 (B R ) such that the following identity holds: t 0 f (X s ) dM s • r t = − t 0 f (X s ) dM s + N v t (1.12) for t < inf{s : X s / ∈ B R }, where N v is the zero-energy part of the Fukushima's decomposition for v(X t ) − v(X 0 ) (see Section 3). By Sobolev embedding theorem, if we extend v to R n by taking v = 0 on B c R , then v is continuous on R n . It then follows from [13], Theorem 1, that N v can be refined to be a CAF of X in the strict sense. Consequently, t → ( t 0 f (X s ) dM s ) • r t can be refined to be a CAF of X in the strict sense by using the expression on the right-hand side of (1.12). In particular, Z τ D is then well defined under P x for every x ∈ D. Equation (1.12) allows us to get rid of the reverse operator r t in the expression of the solution u making the analysis possible. In Section 4, we establish a general existence and uniqueness result for the weak solution of the Dirichlet boundary value problem (1.8) for the operator L without the Markovian assumption (1.7). Let Z be defined by (1.11). We establish as Theorem 4.4 the following important gauge theorem under suitable condition of D, A, b, b and q, which is of independent interest if E x 0 [Z τ D ] < ∞ for some x 0 ∈ D, then x → E x [Z τ D ] is bounded between two positive constants. We then show that if E x 0 [Z τ D ] < ∞ for some x 0 ∈ D, then for every f ∈ C(∂D), the equation Lu = 0 in D has a unique weak solution that is continuous on D such that u = f on ∂D. Moreover, this solution can be expresses as u(x) = E x [Z τ D f (X τ D )] for every x ∈ D. Our strategy is, by using (1.12), to reduce the Dirichlet boundary value problem Lu = 0 in D with u = f on ∂D to a corresponding Dirichlet boundary value problem for an operator that does not have the lower-order term div( b·). In Section 5, we consider the special case of L in (1.3) where b = b = −A∇ρ where ρ ∈ W 1,2 (R n ) with ∇ρ ∈ L p (R n ; dx) for some p > n. By Sobolev embedding theorem, ρ is continuous on R n . In this case, the quadratic form (Q, W 1,2 (R n )) in (1.6) takes the following form: Q(u, v) = 1 2 n i,j=1 R n a ij (x) ∂u ∂x i ∂v ∂x j dx + 1 2 n i,j=1 R n a ij (x) ∂(uv) ∂x i ∂ρ ∂x j dx + R n u(x)v(x)q(x) dx for u, v ∈ W 1,2 (R n ). For this case, we can establish a stronger result without additional condition on the diffusion matrix A. Let X be the symmetric diffusion with infinitesimal generator 1 2 ∇(A∇), and recall the Fukushima's decomposition in the strict sense (cf. [13], Theorem 1): ρ(X t ) − ρ(X 0 ) = M ρ t + N ρ t , t ≥ 0, where M ρ is the MAF of X in the strict sense having finite energy and N ρ is the continuous additive functional of X in the strict sense having zero energy. Define Z t = exp N ρ t + t 0 q(X s ) ds , t ≥ 0. We prove in Theorem 5.1 that if D is a bounded Lipschitz domain and if E x 0 [Z τ D ] < ∞ for some x 0 ∈ D, then x → E x [Z τ D ] is bounded between two positive constants. Moreover, assuming that E x 0 [Z τ D ] < ∞ for some x 0 ∈ D, we show that for every f ∈ C(∂D), u(x) := E x [Z τ D f (X τ D )], x ∈ D, gives the unique weak solution of Lu = 0 in D that is continuous on D with u = f on ∂D. In this paper, we use ":=" as a way of definition. A statement is said to hold quasi-everywhere (q.e.) on some set A ⊂ R n if there is an exceptional set N of zero capacity so that the statement holds on A \ N . For the general theory of Dirichlet forms and Markov processes and their terminology, we refer readers to [14] and [21] . 2. Probabilistic representation. In this section, we will give a probabilistic representation of the weak solutions of Dirichlet boundary value problems. Consider the following regular Dirichlet form on R n : E(u, v) = 1 2 n i,j=1 R n a ij (x) ∂u ∂x i ∂v ∂x j dx, (2.1) D(E) = W 1,2 (R n ). 8 Z.-Q. CHEN AND T. ZHANG It is well known that there is a symmetric conservative diffusion process X = {X t , θ t , r t , P x , x ∈ R n } associated with it. Since X has Hölder continuous transition density function with respect to the Lebesgue measure on R n (cf. [25]), X can be modified to start from every point in R n . Without loss of generality, we may and do assume that X is defined on the canonical sample space Ω = C([0, ∞) → R n ) on which the time-shift operators {θ t , t ≥ 0} and time-reversal operators {r t , t > 0} are well defined: for t > 0, θ t (ω)(s) = ω(t + s) for s ≥ 0 (2.2) and r t (ω)(s) := ω(t − s), if 0 ≤ s ≤ t, ω(0), if s ≥ t. (2.3) Let {F t , t ≥ 0} be the minimal augmented filtration generated by the diffusion process X. For every u ∈ D(E), the following Fukushima's decomposition holds: for q.e. x ∈ R n , u(X t ) − u(X 0 ) = M u t + N u t , P x -a.s., (2.4) where M u a continuous MAF of X having finite energy and N u t is a CAF of X having zero energy. Note that the MAF M u and the CAF N u typically admit an exceptional set N of zero capacity in their definition. However since X has a Hölder continuous transition density function, by [13], Theorem 1, both M u , N u and the above Fukushima decomposition (2.4) can be strengthened to admit no exceptional set [so in particular, (2.4) holds for every x ∈ R n ] if u is continuous and the energy measure for M u µ u (dx) := n i,j=1 a i,j (x) ∂u(x) ∂x i ∂u(x) ∂x j dx is a smooth measure in the strict sense. The latter means that there is an increasing sequence of finely open sets {D k , k ≥ 1} so that ∞ k=1 D k = R n , 1 D k µ u is a finite Borel measure and G 1 (1 D k µ u ) is bounded for every k ≥ 1. In the sequel we call an additive functional strict if it admits no exceptional set and call the refined decomposition of (2.4) without exceptional set a strict Fukushima decomposition. In fact by [13], Theorem 2, both M u and N u can be taken to be strict AFs of X and the strict Fukushima decomposition holds for every continuous function u that is locally in F such that µ u is a smooth measure in the strict sense. Applying the above to coordinate functions f j (x) := x j for j = 1, . . . , n, we have X t = x + M t + N t , P x -a.s. (2.5) for every x ∈ R n , where M t = (M 1 t , . . . , M n t ) is a continuous local MAF of X in the strict sense with M i , M j t = t 0 a ij (X s ) ds TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 9 and N t is a CAF of X locally of zero energy in the strict sense. In particular, there is a Brownian motion B = (B 1 , . . . , B n ), which is a martingale AF of X in the strict sense, such that M = t 0 σ(X s ) dB s , t ≥ 0, where σ(x) is the positive definite symmetric square root of the matrix A(x). Note that M is a MAF of X in the strict sense. Here is the first representation result. Lemma 2.1. Assume condition (1.7) holds and that b is C 1 -smooth. Then for every f ∈ W 1,2 (D) ∩ C(D), the unique weak solution u to (1.8) admits the following representation: for x ∈ D u(x) = E x f (X τ D ) exp τ D 0 (A −1 b)(X s ) dM s + τ D 0 (A −1 b)(X s ) dM s • r τ D (2.6) − 1 2 τ D 0 (b − b)A −1 (b − b) * (X s ) ds + τ D 0 q(X s ) ds . Moreover, u ∈ C(D). Proof. First we note that by [20], (46), we for a.e. x ∈ D, P x -a.s. have τ D 0 (A −1 b)(X s ) dM s • r τ D (2.7) = − τ D 0 (A −1 b)(X s ) dM s − τ D 0 div( b)(X s ) ds. Note that since b ∈ C 1 (R n ), both t → t 0 (A −1 b)(X s ) dM s and t → t 0 div( b)× (X s ) ds are continuous AFs of X in the strict sense. Therefore ( τ D 0 (A −1 b) × (X s ) dM s ) • r τ D can be refined using the right-hand side of (2.7) so that it is well defined under P x for every x ∈ D. In particular, the right-hand side of (2.6) is well defined for every x ∈ D. Under our assumptions, the operator L can be written as L = 1 2 n i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j + n i=1 (b i (x) − b i (x)) ∂ ∂x i − div b(x) + q(x). Define a family of measures {Q x , x ∈ R n } on F ∞ by dQ x dP x Ft = H t , (2.8) where H t = exp t 0 A −1 (b − b)(X s ) dM s − 1 2 t 0 (b − b)A −1 (b − b) * (X s ) ds . (2.9) It is known (cf. [8]) that under measure {Q x , x ∈ R n }, X is a diffusion process on R n having infinitesimal generator L 0 = 1 2 n i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j + n i=1 (b i (x) − b i (x)) ∂ ∂x i . By Theorem 5.11 in [8], when f ∈ W 1,2 (D) ∩ C(D), the unique weak solution u to (1.8) is continuous on D and has the following probabilistic representation u(x) = E Q x f (X τ D ) exp τ D 0 (− div( b)(X s ) + q(X s )) ds , x ∈ D, where E Q x stands for the expectation with respect to the measure Q x . Since {H t∧τ D , t ≥ 0} is a uniformly integrable martingale under P x for every x ∈ D, due to the fact that |b − b| 2 ∈ K n (see [6], page 746), we have u(x) = E x f (X τ D ) exp τ D 0 (A −1 (b − b))(X s ) dM s − 1 2 τ D 0 (b − b)A −1 (b − b) * (X s ) ds × exp τ D 0 (− div( b)(X s ) + q(X s )) ds . This together with (2.7) implies (2.6). Put J(x) =    c 0 exp − 1 1 − |x| 2 , if |x| < 1, 0, if |x| ≥ 1, where c 0 > 0 is a normalizing constant so that R n J(x) dx = 1. For any positive integer k ≥ 1, we set J k (x) := k n J(kx) for x ∈ R n and b k (x) = J k * b(x) := R n b(y)J k (x − y) dy, (2.10) q k (x) = J k * q(x) := R n q(y)J k (x − y) dy. Recall that we assume that b, b and q are set to be zero off D, and |b| 2 + | b| 2 + |q| ∈ K n . Lemma 2.2. (i) b k → b in L 2 loc (R n ) and q k → q in L 1 loc (R n ) as k → ∞. (ii) For every nonnegative function φ in C ∞ c (R n ) and k ≥ 1, n i=1 R n b ki (x) ∂φ ∂x i dx + R n q k (x)φ(x) dx ≤ 0. (iii) Assume n ≥ 3. Then lim r→0 sup k≥1 x∈R n {y∈R n : |x−y|≤r} | b k (y)| 2 + |q k (y)| |x − y| n−2 dy = 0. (iv) Assume n ≥ 3. Then lim k→∞ sup x∈R n D | b k (y) − b(y)| 2 + |q k (y) − q(y)| |x − y| n−2 dy = 0. Proof. (i) follows easily from the fact that b ∈ L 2 loc (R n ) and q ∈ L 1 loc (R n ). For a nonnegative function φ in C ∞ c (R n ) and z ∈ R n , put φ z (x) = φ(x + z). Then it is easy to see that for k ≥ 1, n i=1 R n b ki (x) ∂φ ∂x i dx + R n q k (x)φ(x) dx = R n J k (z) n i=1 R n b i (x) ∂φ z ∂x i dx + R n q(x)φ z (x) dx dz and so (ii) follows from (1.7). Since | b| 2 ∈ K n , (iii) is a consequence of the following inequality: sup x∈R n k≥1 |x−y|≤r | b k | 2 (y) |x − y| n−2 dy ≤ |x−y|≤r R n J k (z)| b| 2 (y − z) dz |x − y| n−2 dy = R n J k (z) |x−y|≤r | b| 2 (y − z) |x − y| n−2 dy dz = R n J k (z) |x−z−y|≤r | b| 2 (y) |x − z − y| n−2 dy dz ≤ sup x∈R n |x−y|≤r | b| 2 (y) |x − y| n−2 dy, where we used the fact R n J k (z) dz = 1. The proof for q is similar. To prove (iv) we observe from the proof of (iii) that for any r > 0, sup x∈R n R n | b k − b| 2 (y) |x − y| n−2 dy ≤ 2 sup x∈R n |x−y|≤r | b| 2 (y) |x − y| n−2 dy + 1 r n−2 R n | b k − b| 2 (y) dy. Using (i), this implies that lim k→∞ sup x∈R n R n | b k − b| 2 (y) |x − y| n−2 dy ≤ 2 sup x∈R n |x−y|≤r | b| 2 (y) |x − y| n−2 dy for any r > 0. Letting r → 0 we get (iv) for b k − b. The proof for |q k − q| goes in a similar way. For integer k ≥ 1, define L k := 1 2 n i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j (2.11) + (b(x) − b k (x)) · ∇ − div b k (x) + q k (x), where b k , q k are the functions defined by (2.10). Denote by Q k (·, ·) the quadratic form associated with L k . Now we can drop the assumption of b being C 1 from Lemma 2.1 by using the smooth approximation b k for b. Theorem 2.3. Suppose that condition (1.7) holds and f ∈ W 1,2 (D) ∩ C(D) . Then the unique weak solution u of (1.8) has the following probabilistic representation: for q.e. x ∈ D, u(x) = E x f (X τ D ) exp τ D 0 (A −1 b)(X s ) dM s + τ D 0 (A −1 b)(X s ) dM s • r τ D (2.12) − 1 2 τ D 0 (b − b)A −1 (b − b) * (X s ) ds + τ D 0 q(X s ) ds . TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 13 Proof. For simplicity, we assume n ≥ 3. [The case of n = 1 and n = 2 can be handled similarly with a corresponding version for Lemma 2.2(iii) and (iv).] Recall the differential operator L k defined by (2.11). Let u k denote the unique weak solution of the following Dirichlet boundary value problem: L k u k = 0 in D with u k − f ∈ W 1,2 0 (D). Since b k is smooth, it follows from Lemmas 2.2(ii) and 2.1 that u k (x) = E x f (X τ D ) exp τ D 0 (A −1 b)(X s ) dM s + τ D 0 (A −1 b k )(X s ) dM s • r τ D − 1 2 τ D 0 (b − b k )A −1 (b − b k ) * (X s ) ds + τ D 0 q k (X s ) ds . Let v denote the right-hand side of (2.12). We will show that lim k→∞ u k (x) = v(x). To this end, put Z k := exp τ D 0 (A −1 b)(X s ) dM s + τ D 0 (A −1 b k )(X s ) dM s • r τ D − 1 2 τ D 0 (b − b k )A −1 (b − b k ) * (X s ) ds + τ D 0 q k (X s ) ds and Z := exp τ D 0 (A −1 b)(X s ) dM s + τ D 0 (A −1 b)(X s ) dM s • r τ D − 1 2 τ D 0 (b − b)A −1 (b − b) * (X s ) ds + τ D 0 q(X s ) ds . We first prove that Z k → Z in probability as k → ∞. It is clear that τ D 0 (b − b k )A −1 (b − b k ) * (X s ) ds + τ D 0 q k (X s ) ds converges in probability under P x for every x ∈ D to τ D 0 (b − b)A −1 (b − b) * (X s ) ds + τ D 0 q(X s ) ds. Thus, it is sufficient to show that τ D 0 (A −1 b k )(X s ) dM s • r τ D → τ D 0 (A −1 b)(X s ) dM s • r τ D (2.13) 14 Z.-Q. CHEN AND T. ZHANG in probability under P x as k → ∞ for q.e. x ∈ D. Define M k t := t 0 (A −1 b k )(X s ) dM s , t ≥ 0 and M t := t 0 (A −1 b)(X s ) dM s , t ≥ 0, which are MAFs of X in the strict sense of finite energy (recall that we as- sumed b = b = 0 off D) . It follows from [3] and [23] that there are continuous processes N k t and N t of zero energy such that M k t • r t = − M k t + N k t and M t • r t = − M t + N t . Moreover, since M k → M as k → ∞ with respect to the energy norm in the martingale space, for every subsequence {n k }, there is a sub-subsequence {n k j } so that N n k j t converges to N t uniformly on compact intervals P x -a.s. for q.e. x ∈ D (cf. [14]). Thus, for any T > 0, on {ω : τ D (ω) ≤ T }, it holds that τ D 0 (A −1 b k )(X s ) dM s • r τ D − τ D 0 (A −1 b)(X s ) dM s • r τ D ≤ sup 0≤t≤T | M k t • r t − M t • r t | ≤ sup 0≤t≤T |M k t − M t | + sup 0≤u≤T |N k t − N t |. This proves (2.13). So, to show that u n → v, it suffices to prove that the family {Z n , n ≥ 1} is uniformly integrable under P x for q.e. x ∈ D. In view of Lemmas 2.1, (2.7) and 2.2(ii), we have Z k ≤ L k := exp τ D 0 (A −1 (b − b k ))(X s ) dM s − 1 2 τ D 0 (b − b k )A −1 (b − b k ) * (X s ) ds . Define L in the same way as L k but with b in place of b k . Then Ω |L k − L| dP x = {L k >L} (L k − L) dP x + {L k ≤L} (L − L k ) dP x = Ω L k dP x − Ω L dP x + 2 {L k ≤L} (L − L k ) dP x = 2 {L k ≤L} (L − L k ) dP x → 0 as k → ∞. TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 15 Here we have used the fact that E x [L k ] = 1 = E x [L] , which is a consequence of the Kato class assumption on |b| 2 + | b| 2 (cf. [2]). This particularly implies that {L k , k ≥ 1} is uniformly integrable under P x for every x ∈ D, so is {Z k , k ≥ 1}. To show that the weak solution u of (1.8) is equal to v, by the uniqueness, it suffices to show that v is a weak solution to (1.8). By Theorem 3.2 (and its proof) of Trüdinger [26], there is a constant C > 0, independent of k, such that u k 1,2 ≤ C f 1,2 for every k ≥ 1. By taking a subsequence if necessary, we may assume that u n converges weakly to some v 1 in W 1,2 (D) and that its Cesaro mean {k −1 k j=1 u j , k ≥ 1} converges to some v 2 in (W 1,2 (D), · 1,2 ). Clearly v 1 = v 2 = v. Moreover, since u k − f ∈ W 1,2 0 (D), we have v − f ∈ W 1,2 0 (D). As Q k (u k , φ) = 0 for any φ ∈ C ∞ c (D), if we can show that for any φ ∈ C ∞ c (D), Q(v, φ) = lim k→∞ Q k (u k , φ), then v ∈ W 1,2 (D) is a weak solution for Lv = 0 in D with v − f ∈ W 1,2 0 (D). Observe that Q k (u k , φ) = 1 2 n i,j=1 R n a ij (x) ∂u k ∂x i ∂φ ∂x j dx − n i=1 R n b i (x) ∂u k ∂x i φ dx − n i=1 R n b ki (x) ∂φ ∂x i u k (x) dx − R n q k (x)u k (x)φ dx. Obviously lim k→∞ 1 2 n i,j=1 R n a ij (x) ∂u k ∂x i ∂φ ∂x j dx = 1 2 n i,j=1 R n a ij (x) ∂v ∂x i ∂φ ∂x j dx and lim k→∞ n i=1 R n b i (x) ∂u k (x) ∂x i φ(x) dx = n i=1 R n b i (x) ∂v(x) ∂x i φ(x) dx. On the other hand, for φ ∈ C ∞ c (D), by Lemma 2.2 we have lim k→∞ n i=1 R n b ki (x) ∂φ(x) ∂x i u k (x) dx = n i=1 R n b i (x) ∂φ(x) ∂x i v(x) dx and lim k→∞ R n q k (x)u k (x)φ(x) dx = R n q(x)v(x)φ(x) dx. This proves that 3. Continuity at the boundary. In this section, we study the regularity of the solution of the Dirichlet boundary value problems (1.8) at the boundary of the domain. First, we prepare two useful lemmas. The next result is due to Meyers [22], Theorem 1. Q(v, φ) = lim k→∞ Q k (u k , φ) = 0 for every φ ∈ C ∞ c( Lemma 3.1. For every x 0 ∈ R n , R > 0 and p > n, there is a constant ε ∈ (0, 1), depending only on n, R and p, such that if (1 − ε)I n×n ≤ A(x) ≤ I n×n for a.e. x ∈ B R := B(x 0 , R), (3.1) then 1 2 ∇(A∇u) = div f in B R has a unique weak solution in W 1,p 0 (B R ) for every f = (f 1 , . . . , f n ) ∈ L 2 (B R ; dx). Moreover, there is a constant c > 0 in- dependent of f such that ∇u L p (B R ;dx) ≤ c f L p (B R ;dx) . Observe that since u ∈ W 1,p 0 (D) ⊂ R n with p > n, by the classical Sobolev embedding theorem (see, e.g., [16], Theorem 7.10) u ∈ C(D) if we take u = 0 on D c . Recall that D is a bounded domain in R n . Select x 0 ∈ R n and R > 0 so that B(x 0 , R) ⊃ D. For simplicity, denote B(x 0 , R) by B R . Let X B R denote the symmetric diffusion in B R associated with the infinitesimal generator 1 2 ∇(A∇). The time-reversal operator for X B R will still be denoted as r t . For a MAF M of X B R of finite energy, let Γ( M ) t = − 1 2 ( M t + M t • r t ) for t < τ B R . According to the representation theorem of martingale additive functionals in [14], there is a measurable function F : B R → R n such that M t = t 0 F (X s ) dM s for t < τ B R . We have the following result. Lemma 3.2. Suppose that p > n and the diffusion matrix A satisfies the condition (3.1). Then for every L p -integrable R n -valued function F : B R → R n , there exists a function v ∈ W 1,p 0 (B R ) ⊂ W 1,2 0 (B R ) with ∇v L p (B R ;dx) ≤ c F L p (B R ;dx) such that Γ( M ) t = Γ(M v ) t = N v t for t < τ B R . (3.2) Moreover, if we extend v to R n by taking v = 0 on B c R , then v ∈ C(R n ). Proof. By [10], Corollary 3.2, the following orthogonal decomposition holds with respect to the inner product induced by the energy norm. M t = M v t + K t for t < τ B R ,(3.3) where v is an element in W 1,2 0 (B R ), K is a MAF of X B R satisfying Γ(K) = 0 and µ K,M φ (B R ) = 0 for all φ ∈ D(E). Here µ K,M φ is the signed Revuz measure of CAF K, M φ of X of finite variation. Hence, Γ( M ) t = Γ(M v ) t = N v t for t < τ B R . By the representation of MAFs, we have M v t = t 0 ∇v(X s ) dM s and K t = t 0 G(X s ) dM s for t < τ B R , for some measurable vector field G = (G 1 (x), . . . , G n (x)) : B R → R n . Since µ K,M φ (B R ) = 0 for all φ ∈ W 1,2 0 (B R ), we have D n i,j=1 a ij (x) ∂φ ∂x i G j (x) dx = 0 for all φ ∈ C 1 c (B R ) . This says that div(AG) = 0 in B R . Note that F (x) = ∇v(x) + G(x). Multiplying both sides of the above equation by the matrix function A(x), we see that v ∈ W 1,2 0 (B R ) solves the equation div(A∇v) = div(AF ) in B R . By Lemma 3.1, v ∈ W 1,p 0 (B R ) with v L p (B R ;dx) ≤ c AF L p (B R ;dx) ≤ c 1 F L p (B R ;dx) . If we take v = 0 on B c R , then v ∈ W 1,p (R n ) and so by the Sobolev embedding theorem [16], Theorem 7.10, v ∈ C(R n ). This completes the proof of the lemma. Henceforth, we select and fix a ball B R ⊃ D. Proof. It is enough to prove the theorem for nonnegative function f . Let {Ω, F, X t , Q x , x ∈ R n } denote the diffusion process defined as in (2.8). As in the proof of Theorem 2.3, put M t = t∧τ B R 0 (A −1 b)(X s ) dM s . By Lemma 3.2, there exits a bounded, function v ∈ W 1,p 0 (B R ) ⊂ W 1,2 0 (B R ) such that M t • r t = − M t + N v t for t < τ B R . Note that M is a MAF of X in the strict sense and N v is a CAF of X in the strict sense of zero energy in view of [13], Theorem 1, since v ∈ W 1,p 0 (B R ) and so it is continuous on R n if we extend v to take value 0 on B c R . Therefore M t • r t can be refined to be a CAF of X in the strict sense. It follows that u can be expressed as u(x) = E Q x [f (X τ D ) exp(A τ D )], where A t = N v t + t 0 q(X s ) ds. Under condition (1.7), the CAF A t is negative and decreasing in t ∈ [0, τ B R ). Hence for x ∈ D, |u(x) − E Q x [f (X τ D )]| ≤ f ∞ E Q x [| exp(A τ D ) − 1|] ≤ f ∞ E Q x [1 − exp(A τ D ) ]. We know from [8] that for y ∈ ∂D that is regular for ( 1 2 ∆, D), lim x→y,x∈D E Q x [f (X τ D )] = f (y) and lim x→y,x∈D Q x (τ D > t) = 0 for every t > 0. Thus, it suffices to show that lim t↓0 lim x→y,x∈D E Q x [exp(A τ D ∧t )] = 1. For this, note that by Jensen's inequality, 1 ≥ E Q x [exp(A τ D ∧t )] ≥ exp(E Q x [A τ D ∧t ]) . On the other hand, by the Cauchy-Schwarz inequality, (E Q x [A τ D ∧t ]) 2 ≤ E x [H 2 τ D ∧t ]E x [A 2 τ D ∧t ], where H is the martingale given by (2.9). We know from [8] that sup x∈D sup s∈[0,1] E x [H 2 s ] < ∞. Observe that A t = N v t + t 0 q(X s ) ds = v(X t ) − v(X 0 ) − t 0 ∇v(X s ) dM s + t 0 q(X s ) ds. TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 19 Since v is bounded and continuous, it is known (see, e.g., [8]) that lim x→y,x∈D E x [(v(X τ D ∧t ) 2 ] = v(x) 2 and lim x→y,x∈D E x [v(X τ D ∧t ] = v(x). Thus, lim x→y,x∈D E x [A 2 τ D ∧t ] ≤ lim x→y,x∈D cE x (v(X τ D ∧t ) − v(X 0 )) 2 + τ D ∧t 0 |∇v(X s )| 2 ds + τ D ∧t 0 |q(X s )| ds 2 ≤ c lim x→y,x∈D E x τ D ∧t 0 |∇v(X s )| 2 ds + 2 τ D ∧t 0 |q(X s )| τ D ∧t s |q(X r )| dr ds ≤ c lim x→y,x∈D E x τ D ∧t 0 |∇v(X s )| 2 ds + 2 sup z∈D G D |q|(z)| τ D ∧t 0 |q(X s )| ds = 0. In the second to the last inequality we used the Markov property of X, while in the last equality we used the fact that |∇v| 2 and q are in the Kato class and the dominated convergence theorem. This proves that lim t↓0 lim x→y,x∈D E Q x × [A τ D ∧t ] = 0. Consequently lim t↓0 lim x→y,x∈D E Q x [exp(A τ D ∧t )] = 1. This proves the theorem. Markovian assumption (1.7). In this section, we will drop the Markovian condition (1.7) and give a general result on the existence and uniqueness of the weak solutions of the Dirichlet boundary value problem (1.8). Without Let h = (h 1 (x), . . . , h n (x)) : R n → R n be a measurable function such that h ∈ L p (R n → R n ) for some p > n. Let µ be a signed measure in K n . Consider G = 1 2 n i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j + n i=1 h i (x) ∂ ∂x i + µ. The quadratic from (C, D(C)) associated with G is given by D(C) = W 1,2 (R n ) and for u, v ∈ W 1,2 (R n ), C(u, v) = (−Gu, v) = 1 2 n i,j=1 R n a ij (x) ∂u ∂x i ∂v ∂x j dx − n i=1 R n h i (x) ∂u ∂x i v(x) dx (4.1) − R n u(x)v(x)µ(dx). We will regard (C, W 1,2 (R n )) as a lower-order perturbation of the symmetric Dirichlet form (E, W 1,2 (R n )) associated with the infinitesimal generator 1 2 ∇(A∇). Recall that X is the symmetric diffusion process with infinitesimal generator 1 2 ∇(A∇), or equivalently, with (E, W 1,2 (R n )). Let A µ t be the CAF of X whose Revuz measure is µ. It is proved in [8] (see also [20]) that the semigroup {T t , t ≥ 0} associated with (C, W 1,2 (R n )) is given by T t g(x) = E x g(X t ) exp t 0 (A −1 h)(X s ) dM s − 1 2 t 0 hA −1 h * (X s ) ds + A µ t (4.2) = E P * x [g(X t )e A µ t ], where E P * x stands for the expectation with respect to the diffusion measure {P * x , x ∈ R n }, defined by dP * x dP x Ft = H t , (4.3) where H t = exp t 0 (A −1 h)(X s ) dM s − 1 2 t 0 hA −1 h * (X s ) ds . (4.4) Let ν be a positive Radon measure of finite energy with respect to the symmetric Dirichlet form (E, W 1,2 (R n )). Note that due to the L p -integrability of h and the Kato class condition on µ, there exists a constant α 0 ≥ 1 such that for every α > α 0 , C α (u, v) := C(u, v) + α(u, v) is a positive definite quadratic form satisfying c −1 α C α (u, u) ≤ E 1 (u, u) ≤ c α C α (u, u) for every u ∈ W 1,2 (R n ). By Lax-Milgram theorem, for any α > α 0 , there is a unique function in D(E), denoted by U α ν, such that C α (U α ν, v) = R n v(x)ν(dx) (4.5) for all v ∈ D(E). U α ν is called the α-potential of ν associated with the quadratic form (C α , D(C)). We have the following useful representation for U α ν. U α ν(x) = E x ∞ 0 R (α) t dA ν t = E P * x ∞ 0 exp(−αt + A µ t ) dA ν t , where R (α) t = exp t 0 (A −1 h)(X s ) dM s − 1 2 t 0 hA −1 h * (X s ) ds − αt + A µ t . (4.6) Proof. By the same proof of [14], Lemma 2.2.5, we see that there exists an E-nest consisting of an increasing sequence {F k , k ≥ 1} of compact sets of R n such that U α (1 F k ν) ∞ < ∞ for each k ≥ 1. By restricting to F k if necessary, we may assume that g(x) := U α ν(x) is bounded in the sequel. By (4.5), it follows that for every v ∈ W 1,2 (R n ), E(g, v) = 1 2 n i,j=1 R n a ij (x) ∂g ∂x i ∂v ∂x j dx = n i=1 R n h i (x) ∂g ∂x i v(x) dx + R n g(x)v(x)µ(dx) (4.7) + R n v(x)ν(dx) − α R n g(x)v(x) dx. By [14], Theorem 5.4.2, (4.7) implies that P x -a.s. g(X t ) = g(X 0 ) + t 0 ∇g(X s ) dM s − n i=1 t 0 h i (X s ) ∂g ∂x i (X s ) ds − t 0 g(X s ) dA µ s + α t 0 g(X s ) ds − A ν t . Note that R (α) t satisfies dR (α) t = R (α) t (A −1 h)(X t ) dM t + R (α) t dA µ t − αR (α) t dt. By Itô's formula, it follows that g(X t )R (α) t = g(X 0 ) + K t − t 0 R (α) s A ν s , (4.8) where K t := t 0 g(X s )R (α) s (A −1 h)(X s ) dM s + t 0 R (α) s ∇g(X s ) dM s is a local martingale. Let {τ k , k ≥ 1} be an increasing sequence of stopping times approaching to ∞ such that {K t∧τ k , t ≥ 0} is a martingale for every k ≥ 1. It follows from (4.8) that g(x) = E x t∧τ k 0 R (α) s dA ν s + E x [g(X t∧τn )R (α) t∧τ k ]. Note that since |µ| ∈ K n , it follows from Khasminskii's inequality, Cauchy-Schwarz inequality and [8], Theorem 3.1, {R (α) s , 0 ≤ s ≤ T } is uniformly integrable under P x for every x ∈ R n and T > 0. Letting k → ∞, by dominated and monotone convergence theorems we get g(x) = E x t 0 R (α) s dA ν s + E x [g(X t )R (α) t ]. (4.9) Since the Revuz measure µ of A µ is in the Kato class, for sufficiently large α, we have lim t→∞ E x [R (α) t ] = 0. As g is assumed to be bounded, letting t → ∞ in (4.9) we obtain that g(x) = E x ∞ 0 R (α) s A ν s , which completes the proof. Assume from now on that D is a bounded Lipschitz domain. In particular, the boundary of D is regular with respect to ( 1 2 ∆, D). Let R be the multiplicative functional of X defined as in (4.6) with α = 0; that is, R t = exp t 0 (A −1 h)(X s ) dM s − 1 2 t 0 hA −1 h * (X s ) ds + A µ t . The following is a gauge theorem involving the Girsanov as well as Feynman-Kac functional, which extends the corresponding result in [15] where the diffusion matrix A is the identity matrix and the Kato class measure µ is absolutely continuous with respect to the Lebesgue measure [i.e., µ(dx) = q(x) dx]. Theorem 4.2. Let D be a bounded Lipschitz domain in R n , h ∈ L p (R n → R n ) for some p > n and µ a signed measure in K n . Define u(x) := E x [R τ D ]. Then if u(x) < ∞ for some x ∈ D, then u is bounded between two positive constants in D. Proof. Let G D denote the Green function of the symmetric diffusion X in D and µ + , µ − be the positive, negative part of µ, respectively. Since µ is in the Kato class, we have by Jensen's inequality inf x∈D u(x) ≥ inf x∈D exp E x τ D 0 (A −1 h)(X s ) dM s − 1 2 τ D 0 hA −1 h * (X s ) ds + A µ τ D = inf x∈D exp E x A µ τ D − 1 2 τ D 0 hA −1 h * (X s ) ds ≥ exp(− G D µ − ∞ − c G D |h| 2 ∞ ). Let Y be the diffusion process with infinitesimal generator 1 2 n i,j=1 ∂ ∂x i (a ij (x)× ∂ ∂x j ) + n i=1 h i (x) ∂ ∂x i . It is known (see [8]) that Y can be obtained from X through Girsanov transform by the exponential martingale H in (4.4). That is, if for every x ∈ R n , we let P * x be the measure defined by dP * x dP x Ft = H t for every t ≥ 0, (4.10) then the diffusion process X under P * x has the same distribution as Y starting from x. Note that R τ D = H τ D exp(A µ τ D ). For every k ≥ 1, we have by (62) of Sharpe [24], E P * x [k ∧ exp(A µ τ D )] = E x τ D 0 k ∧ exp(A µ t ) d(−H s ) + (k ∧ exp(A µ τ D ))H τ D = E x [(k ∧ exp(A µ τ D ))H τ D ]. Passing k → ∞, we have E P * x [exp(A µ τ D )] = E x [exp(A µ τ D )H τ D ] = E x [R τ D ]. (4.11) Since h ∈ L p (D; dx) for some p > n, by Ancona [1], the Green function of Y in D is comparable to that of X in D. Hence µ is in the Kato class of Y in D. By [5], Theorem 2.2, we have x → E P * x [exp(A µ τ D )] is either bounded or identically infinite on D. It then follows from (4.11) that u(x) is either bounded on D or identically infinite, which proves the theorem. Now consider the Dirichlet boundary value problem Gu = 0 in D with u = f on ∂D, (4.12) where f ∈ C(∂D). Recall that {P * x , x ∈ D} are the probability measures defined by (4.10). Recall that R is the multiplicative functional of X defined as in (4.6) with α = 0. Assume that E x 0 [R τ D ] < ∞ for some x 0 ∈ D. Then there exists a unique, continuous weak solution to the Dirichlet boundary value problem (4.12), which is given by u(x) = E x [R τ D f (X τ D )] = E P * x [e A µ τ D f (X τ D )], x ∈ D. (4.13) Proof. We first show that u defined by (4.13) is a weak solution to the Dirichlet boundary value problem (4.12). Put v 1 (x) = E P * x [f (X τ D )] and v 2 (x) = E P * x τ D 0 v 1 (X s )e A µ s dA µ s . Then u 1 (x) = v 1 (x) + v 2 (x). By [8], Theorem 4.5, we know that v 1 is locally in W 1,2 (R n ) and C(v 1 , φ) = − D v 1 (x)φ(x)µ(dx) for every φ ∈ W 1,2 0 (D). Here the quadratic form (C, W 1,2 (D)) is defined by (4.1). We will show now that v 2 ∈ W 1,2 0 (D). Let G β g(x) = E P * x τ D 0 e −βt e A µ t g(X s ) ds . Denote by G β the adjoint operator of G β in L 2 (D; dx). Applying Lemma 4.1 to the process X killed upon leaving D, we have v 2 (x) − βG 1 β v 2 (x) = E P * x τ D 0 e −βt e A µ t v 1 (X t ) dA µ t =Ū β (v 1 µ), whereŪ β (v 1 µ) is defined as in the proof of Lemma 4.1 but with the killed process X D in place of X. Since µ belongs to the Kato class of X (and of Y by the proof of Theorem 4.2) in D, we have β(v 2 − βG β v 2 , v 2 ) = β D E P * x τ D 0 e −βt e A µ t v 1 (X t ) dA µ t v 2 (x) dx = β DŪ β (v 1 µ)v 2 (x) dx = D β G β v 2 (x)v 1 (x)µ(dx) ≤ C 1 D |∇v 1 | 2 dx + D |v 1 | 2 dx + C 2 D |β G β v 2 | 2 dx + C(β G β v 2 , β G β v 2 ) TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 25 ≤ C 1 D |∇v 1 | 2 dx + D |v 1 | 2 dx + C 2 β β − α 0 D |β G β v 2 | 2 dx + 1 2 β(v 2 − βG β v 2 , v 2 ), where in the last inequality [19], Lemma 3.1(i), is used. Thus, sup β>0 β(v 2 − βG β v 2 , v 2 ) < ∞, which implies that v 2 ∈ W 1,2 0 (D). Moreover for φ ∈ W 1,2 0 (D), C(v 2 , φ) = lim β→∞ β(v 2 − βG β v 2 , φ) = lim β→∞ D β G β φ(x)v 1 (x)µ(dx) = D v 1 (x)φ(x)µ(dx). The last equation follows from the fact that β G β φ(x) converges to φ in the Dirichlet space (E 1 , W 1,2 0 (D)). Thus, for φ ∈ W 1,2 0 (D), C(u, φ) = C(v 1 , φ) + C(v 2 , φ) = 0. This means that u is a weak solution to Gu = 0 in D. It is a well-known fact in the theory of PDE (cf. [16]) that u is continuous inside D. Next we show that the boundary condition is fulfilled, that is, lim x→y,x∈D u 1 (x) = f (y) (4.14) for every y ∈ ∂D. It is proved in [8] that every point in ∂D is a regular point of D c with respect to Y lim x→y,x∈D v 1 (x) = lim x→y,x∈D E P * x [f (X τ D )] = f (y) for every y ∈ ∂D. As u 1 = v 1 + v 2 , it suffices to show that lim x→y,x∈D v 2 (x) = 0 for y ∈ ∂D. Note that v 2 (x) = E P * x [f (X τ D )(e A µ τ D − 1)]. For any t > 0, write v 2 as v 2 (x) = E P * x [f (X τ D )(e A µ τ D − 1); τ D ≤ t] + E P * x [f (X τ D )(e A µ τ D − 1); τ D > t]. Now |E P * x [f (X τ D )(e A µ τ D − 1); τ D > t]| ≤ f ∞ E P * x [(e A µ τ D + 1); τ D > t] = f ∞ E P * x [e A µ t E P * Xt [e A µ τ D + 1]; τ D > t] ≤ f ∞ E P * x [e A |µ| t (M + 1); τ D > t]. Here in the last inequality, we used the assumption that E x [R τ D ] < ∞ for some x ∈ D and thus by Theorem 4.2, M : = sup x∈D E P * x [exp(A µ τ D ] = sup x∈D E x [R τ D ] < ∞. Let y ∈ ∂D. Since lim x→y,x∈D P * x (τ D > t) = 0 and sup x∈D E P * x [e A |µ| t ] < ∞, we have lim x→y,x∈D E P * x [f (X τ D )(e A µ τ D − 1) : τ D > t] = 0. Thus for every t > 0, lim x→y,x∈D (4.10). On the other hand, since µ is in the Kato class of X, it follows from the Khasminskii's inequality that |v 2 (x)| = lim sup x→y,x∈D |E P * x [f (X τ D )(e A µ τ D − 1); τ D ≤ t]| ≤ f ∞ lim sup x→y,x∈D E P * x [e A |µ| t − 1; τ D ≤ t] ≤ f ∞ lim sup x→y,x∈D (E x [(H t ) 2 ]) 1/2 (E x [(e A |µ| t − 1) 2 ]) 1/2 . By [8], Theorem 3.1, we know that sup x∈D E x [(H t ) 2 ] < ∞, where H is de- fined inlim t→0 sup x∈D E x [(e A |µ| t − 1) 2 ] = 0. Thus we conclude that lim x→y,x∈D v 2 (x) = 0. So (4.14) is established. To prove the uniqueness, let u 1 be any, bounded continuous weak solution of the Dirichlet boundary value problem (4.12). Then for φ ∈ W 1,2 0 (D), E(u 1 , φ) = C(u 1 , φ) + n i=1 D h i (x) ∂u 1 (x) ∂x i φ(x) dx + D u 1 (x)φ(x)µ(dx) (4.15) = n i=1 D h i (x) ∂u 1 (x) ∂x i φ(x) dx + D u 1 (x)φ(x)µ(dx). By [14], Theorem 5.4.2, it follows from (4.15) that the following decomposition holds: u 1 (X t ) − u 1 (X 0 ) = t 0 ∇u 1 (X s ) dM s − n i=1 t 0 h i (X s ) ∂u 1 ∂x i (X s ) ds − t 0 u 1 (X s ) dA µ s TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 27 for t < τ D . Recall that R t = exp t 0 (A −1 h)(X s ) dM s − 1 2 t 0 hA −1 h * (X s ) ds + A µ t satisfies dR t = R t (A −1 h)(X t ) dM t + R t dA µ t . Applying Itô's formula, we get that for t < τ D , d(u 1 (X t )R t ) = R t ∇u 1 (X t ) dM t + u 1 (X t )R t (A −1 h)(X t ) dM t . (4.16) This shows that {u 1 (X t∧τ D )R t∧τ D , t ≥ 0} is a P x -local martingale for every x ∈ D. We claim that {R t∧τ D , t ≥ 0} is uniformly integrable with respect to P x for every x ∈ D. To see this, write R t∧τ D = R τ D 1 {t≥τ D } + R t 1 {t<τ D } . Obviously the first term {R τ D 1 {t≥τ D } } is uniformly integrable. For the second term, by Jensen's inequality, 1 {t<τ D } E x [R τ D |F t ] = 1 {t<τ D } E x [R t (R τ D • θ t )|F t ] = R t 1 {t<τ D } E Xt [R τ D ] ≥ R t 1 {t<τ D } exp − 1 2 sup x∈D E x τ D 0 hA −1 h * (X s ) ds + inf x∈D E x [A µ τ D ] ≥ cR t 1 {t<τ D } for some positive constant c, where we have used the fact that |h| 2 , µ are both in the Kato class. The above inequality implies that {R t 1 {t<τ D } , t ≥ 0} is also P x -uniformly integrable. Therefore, {R t∧τ D , t ≥ 0} is P x -uniformly integrable. Since u is bounded continuous, we conclude that {u 1 (X t∧τ D )R t∧τ D , t ≥ 0} is uniformly integrable, hence a P x -martingale for every x ∈ D. Consequently, E x [u 1 (X t∧τ D )R t∧τ D ] = u 1 (x). Letting t → ∞, we get that u 1 (x) = E x [f (X τ D )R τ D ], which proves the uniqueness. Now we can get to the main results of this section. Define Z t = exp t 0 (A −1 b)(X s ) dM s + t 0 (A −1 b)(X s ) dM s • r t (4.17) − 1 2 t 0 (b − b)A −1 (b − b) * (X s ) ds + t 0 q(X s ) ds . 28 Z.-Q. CHEN AND T. ZHANG Recall that X is the symmetric diffusion with infinitesimal generator 1 2 ∇(A∇) and M is the martingale part of X in (1.9). For domain D ⊂ R n , τ D := inf{t ≥ 0 : X t / ∈ D} is the first exit time from D by diffusion X. The following theorem is a new type of gauge theorem, in comparison with those found in [2,5,9]. 1 D q ∈ K n . Then Z τ D of (4.17) is well defined under P x for every x ∈ D. If E x 0 [Z τ D ] < ∞ for some x 0 ∈ D, then the function x → E x [Z τ D ] is bounded between two positive constants on D. Proof. As before, put M t = t 0 (A −1 b)(X s ) dM s for t ≥ 0. Let R > 0 so that D ⊂ B R := B(0, R). By Lemma 3.2, there exits a bounded function v ∈ W 1,p 0 (B R ) ⊂ W 1,2 0 (B R ) such that M t • r t = − M t + N v t and that v ∈ W 1,2 0 (B R ) satisfies the following equation in the distributional sense: div(A∇v) = −2 div( b) in B R . (4.18) Note that by Sobolev embedding theorem, v ∈ C(R n ) if we extend v = 0 on D c . Thus both M and N v are CAFs of X in the strict sense (cf. [13], Theorem 1), and so is t → M t • r t . Moreover, τ D 0 (A −1 b)(X s ) dM s • r τ D = − τ D 0 (A −1 b)(X s ) dM s + N v τ D = − τ D 0 (A −1 b)(X s ) dM s + v(X τ D ) − v(X 0 ) − M v τ D = − τ D 0 (A −1 b)(X s ) dM s + v(X τ D ) − v(X 0 ) − τ D 0 ∇v(X s ) dM s . TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 29 Thus Z τ D = e v(Xτ D ) e v(X 0 ) exp τ D 0 (A −1 (b − b) − ∇v)(X s ) dM s (4.19) + τ D 0 q − 1 2 (b − b)A −1 (b − b) * (X s ) ds . So Z τ D is well defined under P x for every x ∈ D. Since v is bounded, b − b ∈ L p (D; dx) , ∇v ∈ L p (D; dx) and 1 D q ∈ K n , the theorem follows from Theorem 4.2. Recall that L is the second-order differential operator defined by (1.3). Theorem 4.5. Let D be a bounded Lipschitz domain contained in some ball B R , A be an n × n symmetric positive definitive matrix satisfying the condition (3.1) of Lemma 3.1, |b| + | b| ∈ L p (D; dx) for some p > n, and 1 D q ∈ K n . Let Z be defined in (4.17) and assume that E x [Z τ D ] < ∞ for some x ∈ D. Then for every f ∈ C(∂D), there exists a unique weak solution u to Lu = 0 in D that is continuous on D with u = f on ∂D. Moreover, the solution u admits the following representation: Proof. Let u be defined by the right-hand side of (4.20). Recall that v is the function in W 1,p 0 (B R ) that is continuous on R n if we extend v = 0 off D in the proof of Theorem 4.4. Define u(x) = E x [Z τ D f (X τ D )] for x ∈ D.Z τ D := exp τ D 0 (A −1 (b − b − A∇v))(X s ) dM s − 1 2 τ D 0 (b − b − A∇v)A −1 (b − b − A∇v) * (X s ) ds − τ D 0 b − b, ∇v (X s ) ds + 1 2 τ D 0 (∇v)A(∇v) * (X s ) ds + τ D 0 q(X s ) ds . We have by (4.19) that Z τ D = e v(Xτ D ) e v(X 0 ) Z τ D . Since the function v is bounded and continuous on D, it follows that E x 0 [Z τ D ] < +∞ for some x 0 ∈ D if and only if E x 0 [ Z τ D ] < +∞. Define g := e v u. Then g(x) = E x [ Z τ D (e v f )(X τ D )] for x ∈ D. (4.21) Let L 1 := 1 2 n i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j + n i=1 (b i (x) − b i (x) − (A∇v) i (x)) ∂ ∂x i − b − b, ∇v (x) + 1 2 (∇v)A(∇v) * (x) + q(x). By Theorem 4.3, g is a weak solution to the Dirichlet boundary value problem L 1 g = 0 in D and g = f e v on ∂D. Moreover, g is continuous on D with g = f e v on ∂D. Hence, u = e −v g is bounded and continuous on D with u = f on ∂D. Note that for any ψ ∈ W 1,2 0 (D), Q * (g, ψ) := (−L 1 g, ψ) Recalling the definition of the quadratic form Q from (1.6), we thus have for every φ ∈ C ∞ c (D), = 1 2 n i,j=1 D a ij ∂g ∂x i ∂ψ ∂x j dx − n i=1 D (b i − b i − (A∇v) i ) ∂g ∂x i ψ dxQ(u, φ) = 1 2 n i,j=1 D a ij ∂(ge −v ) ∂x i ∂φ ∂x j dx − n i=1 D b i ∂(ge −v ) ∂x i φ dx − n i=1 D b i ∂φ ∂x i ge −v dx − D qge −v φ dx = 1 2 n i,n i,j=1 D a ij ∂e −v ∂x i ∂φ ∂x j g dx − n i=1 D b i ∂g ∂x i e −v φ dx − n i=1 D b i ∂e −v ∂x i gφ dx − n i=1 D b i ∂φ ∂x i ge −v dx − D qge −v φ dx. Applying (4.22) with ψ = φe −v we obtain 1 2 n i,j=1 R n a ij (x) ∂g ∂x i ∂(φe −v ) ∂x j dx = n i=1 R n (b i (x) − b i (x) − (a∇v) i (x)) ∂g ∂x i φe −v dx + R n q(x)g(x)φe −v dx − R n b − b, ∇v (x)g(x)φe −v dx + 1 2 R n (∇v)A(∇v) * (x)gφe −v dx. Substituting this expression into (4.23), we get after cancellations that Q(u, φ) = − n i=1 R n b i (x) ∂(gφ) ∂x i e −v dx − n i=1 R n (A∇v) i (x) ∂g ∂x i φe −v dx + R n b, ∇v φge −v dx + 1 2 R n (∇v)A(∇v) * gφe −v dx (4.24) − 1 2 n i,j=1 R n a ij (x) ∂g ∂x i ∂e −v ∂x j φ dx + 1 2 n i,j=1 R n a ij (x) ∂e −v ∂x i ∂φ ∂x j g dx. In the sequel, we write div(·) for the divergence in the distribution sense. Now, By virtue of (4.18), − n i=1 R n b i (x) ∂(gφ) ∂x i e −v dx = R n div( be −v )gφ dxR n div( b)e −v gφ dx = − 1 2 R n div(A∇v)e −v gφ dx = 1 2 R n A∇v, ∇(e −v gφ) dx − 1 2 R n A∇v, ∇v e −v gφ dx (4.26) + 1 2 R n A∇v, ∇g e −v φ) dx + 1 2 R n A∇v, ∇φ e −v gφ dx. Combining (4.24)-(4.26), we arrive at Q(u, φ) = − R n (A(x)∇v(x) · ∇g(x))φ(x)e −v(x) dx − 1 2 n i,j=1 R n a ij (x) ∂g ∂x i ∂e −v ∂x j φ dx + 1 2 R n A∇v, ∇g e −v φ dx = 0. This shows that u is a weak solution to Lu = 0 in D. Recall that we have showed earlier that u is continuous on D with u = f on ∂D. It remains to show the uniqueness. Suppose that u is a weak solution to Lu = 0 in D and that u is bounded and continuous on D with u = f on ∂D. Then, for any φ ∈ W 1,2 0 (D), Q(u, φ) = 0. Let v be the solution of (4.18). Put g(x) = e v(x) u(x). Running the above proof backward, we can show that Q * (g, ψ) = 0 for any ψ ∈ W 1,2 0 (D). Hence g is a weak solution to L 1 g = 0 in D that is bounded and continuous on D with g = f e v on ∂D. It follows from Theorem 4.3 that g is given by (4.21). Arguing similarly as at the beginning of the proof, we conclude that u can be expressed as in (4.20). This establishes the uniqueness. Remark 4.6. For b ∈ L p (R n ; dx), by [26], Theorem 4.1, there is a unique bounded weak solution v ∈ W 1,2 0 (B R ) such that div(A∇v) = −2 div( b) in B R . Condition (3.1) of Lemma 3.1 is used to guarantee that ∇v ∈ L p (B R ; dx) for some p > n [see (4.18)], which in turn by the Sobolev embedding theorem implies that v ∈ C(R n ) if we extend v = 0 on D c . Condition (3.1) can be dropped from Theorems 4.4 and 4.5 if we assume directly that ∇v ∈ L p (B R ; dx) for some p > n. Generalized Feynman-Kac transform. In this section, we consider the special case of (1.3) with b = b = −A∇ρ, for some ρ ∈ W 1,2 (R n ) with |∇ρ| 2 ∈ L p (R n ; dx) for some p > n. Note that by Sobolev embedding theorem, ρ ∈ C(R n ). As mentioned in the Introduction, the quadratic form (Q, W 1,2 (R n )) in (1.6) takes the following form: Q(u, v) = 1 2 n i,j=1 R n a ij (x) ∂u ∂x i ∂v ∂x j dx + 1 2 n i,j=1 R n a ij (x) ∂(uv) ∂x i ∂ρ ∂x j dx (5.1) + Remark 5.2. To better illustrate the main ideas of this paper, we have not strived to establish our theorems in the most general possible form. For example, in Theorems 4.4, 4.5 and 5.1, the potential function q can be replaced by a signed Kato class measure µ, just like in Theorems 4.2 and 4.3. an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2009, Vol. 37, No. 3, 1008-1043. This reprint differs from the original in pagination and typographic detail. 1 2 Z.-Q. CHEN AND T. ZHANG n denotes the space of Kato class of measures on R n : when n ≥ 3, a signed measure µ ∈ K n if and only if lim r→0 sup x∈R n B(x,r) |x − y| 2−n |µ|(dy) = 0, TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMS 3 D) and so Lv = 0 in D in the distributional sense. This proves the theorem. Theorem 3 . 3 . 33Assume the Markovian condition (1.7) is satisfied, p > n and that the diffusion matrix A satisfies the condition (3.1) of Lemma 3.1. Assume that | b| ∈ L p (D; dx) , and that 1 D (|b| 2 + |q|) ∈ K n . Let u be the unique weak solution of the Dirichlet boundary value problem(1.8). Then for y ∈ ∂D that is regular for ( 1 2 ∆, D), we have lim x→y,x∈D u(x) = f (y). Lemma 4. 1 . 1Let A ν be the positive CAF of X associated with the smooth measure ν in the strict sense. Then, for sufficiently large α, Theorem 4 . 4 . 44Let D be a bounded Lipschitz domain contained in some ball B R , A be an n × n symmetric positive definitive matrix satisfying the condition (3.1) of Lemma 3.1, |b| + | b| ∈ L p (D; dx) for some p > n, and ∇v φge −v dx. j=1 D TIME-REVERSAL AND ELLIPTIC BOUNDARY VALUE PROBLEMSa ij ∂g ∂x i ∂(e −v φ) ∂x j dx − 1 2 n i,j=1 D a ij ∂g ∂x i ∂e −v ∂x j φ dx (4.23) 31 + 1 2 Acknowledgments. We thank the anonymous referee and K. Kuwae for helpful comments on an earlier version of this paper.for u, v ∈ W 1,2 (R n ). Let X be the symmetric diffusion with infinitesimal generator12 ∇(A∇). When q = 0, it is established in Chen and Zhang[7]that (Q, W 1,2 (R n )) is the quadratic form associated with the generalized Feynman-Kac semigroup {T t , t ≥ 0} defined bywhere N ρ is the CAF of X in the Fukushima's decomposition of(When A is the identity matrix, the above result is proved in[17].) Recall that M is the martingale defined in (1.9) and that M ρ t = t 0 ∇ρ(X s ) dM s . Note that (cf.[3])We thus have from (4.17) that, under the condition of Theorem 4.5, for every f ∈ C(∂D),is the unique weak solution to Lu = 0 in D that is continuous on D with u = f on ∂D. However, we can do better in this case. More specifically, we can drop condition (3.1) of Lemma 3.1 in this case.Then Z t is well defined under P x for every x ∈ R n .is bounded between two positive constants on D.(ii) Suppose that E x 0 [Z τ D ] < ∞ for some x 0 ∈ D and f ∈ C(∂D). Thenis the unique weak solution of Lu = 0 in D that is continuous on D with u = f on ∂D.Proof. (i) Since ∇ρ ∈ L p (R n ; dx) for some p > n, by Sobolev embedding theorem, ρ can be taken to be continuous on R n . By[13], Theorem 1, the following Fukushima decomposition in the strict sense holds:where M ρ is the MAF of X in the strict sense of finite energy and N ρ is the CAF of X in the strict sense of zero energy. It followsSince ρ is bounded on D and ∇ρ ∈ L p (R n ; dx),< ∞, and so the conclusion of (i) follows from Theorem 4.2.(ii) By (5.2),It follows from Theorem 4.3 that v := e ρ u is the unique weak solution for Gv = 0 in D that is continuous on D with v = e ρ f on ∂D. Unwinding it for u similar to that in the proof of Theorem 4.5, we conclude that u ∈ W 1,2 loc (D) is the unique weak solution of Lu = 0 in D, that continuous on D with u = f on ∂D. First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains. A Ancona, J. Anal. Math. 721482989Ancona, A. (1997). 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We present the dynamics of rotating Bose Condensate galactic dark matter halos, made of an ultralight spinless boson. We restrict to the case of adding axisymmetric rigid rotation to initially spherically symmetric structures and show there are three regimes: i) small angular momentum, that basically retains the drawbacks of spherically symmetric halos related to compactness and failure at explaining galactic RCs, ii) an intermediate range of values of angular momentum that allow the existence of long-lived structures with acceptable RC profiles, and iii) high angular momentum, in which the structure is dispersed away by rotation. We also present in detail the new code used to solve the Gross-Pitaevskii Poisson system of equations in three dimensions. PACS numbers: 95.35.+d,98.62.Gq,04.62.+v,

10.1103/physrevd.89.063507

1310.3909

760ea20ad669181cc4485ebe9b0ce7944303c5b1

Rotation curves of rotating galactic BEC dark matter halos 15 Oct 2013 F S Guzmán Department of Physics and Astronomy University of British Columbia 6224 Agricultural RoadV6T1Z1VancouverBCCanada Instituto de Física y Matemáticas Universidad Michoacana de San Nicolás de Hidalgo Edificio C-3, Cd. Universitaria58040Morelia, MichoacánMéxico F D Lora-Clavijo Instituto de Astronomía Universidad Nacional Autónoma de México 70-264, 04510Distrito FederalAPMéxico J J González-Avilés Instituto de Física y Matemáticas Universidad Michoacana de San Nicolás de Hidalgo Edificio C-3, Cd. Universitaria58040Morelia, MichoacánMéxico F J Rivera-Paleo Instituto de Física y Matemáticas Universidad Michoacana de San Nicolás de Hidalgo Edificio C-3, Cd. Universitaria58040Morelia, MichoacánMéxico Rotation curves of rotating galactic BEC dark matter halos 15 Oct 2013(Dated: October 16, 2013)numbers: 9535+d9862Gq0462+v0440-b We present the dynamics of rotating Bose Condensate galactic dark matter halos, made of an ultralight spinless boson. We restrict to the case of adding axisymmetric rigid rotation to initially spherically symmetric structures and show there are three regimes: i) small angular momentum, that basically retains the drawbacks of spherically symmetric halos related to compactness and failure at explaining galactic RCs, ii) an intermediate range of values of angular momentum that allow the existence of long-lived structures with acceptable RC profiles, and iii) high angular momentum, in which the structure is dispersed away by rotation. We also present in detail the new code used to solve the Gross-Pitaevskii Poisson system of equations in three dimensions. PACS numbers: 95.35.+d,98.62.Gq,04.62.+v, I. INTRODUCTION The nature of dark matter is one of the most important quests in theoretical physics nowadays. Candidates are restricted by one or another observation. Nevertheless, the most successful cosmological model is the ΛCDM, which does not assume any particular dark matter particle. An interesting dark matter candidate that escapes to stringent limitations is a spinless ultralight boson. The idea originates at cosmic scale, where a scalar field in effective theories is assumed to play the role of dark matter, and in fact what happens is that such candidate mimics CDM at cosmic scale, and most importantly, the mass power spectrum of structures fits better than CDM by fixing the mass of the scalar field to an ultralight value of m ∼ 10 −22 − 10 −23 eV [1]. Additionally, if such boson exists, the condensation temperature would be of the order of T c ∼TeV for boson masses of m ∼ 10 −22 eV [2], which would allow the early formation of structures. The idea of the scalar field in an affective theory works fine at cosmic scales, however once the structures are assumed to evolve under their own gravitational field, in a low energy and weak gravitational field limit, the Einstein-Klein-Gordon system is replaced by the timedependent Schrödinger-Poisson (SP) system of equations [3][4][5][6]. It happens that the SP system is interpreted as the equations ruling the evolution of a self-gravitating Bose Condensate, in the mean field approximation, where the trap of the condensate is the potential due to the gravitational field produced by the density of the condensate itself [7,8] and Schrödinger equation is the Gross-Pitaevskii equation describing the Bose gas [9]. The coupled system is then called the Gross-Pitaevskii-Poisson (GPP) system of equations. These equations are i ∂Ψ ∂t = − 2 2m∇ 2Ψ +ṼΨ + 2π 2ã m 2 |Ψ| 2Ψ , ∇ 2Ṽ = 4πGm|Ψ| 2 ,(1) where in generalΨ =Ψ(t,x), m is the mass of the boson, V is the gravitational potential acting as the condensate trap andã is the scattering length of the bosons. In spherical symmetry, when the wave function is assumed to depend harmonically in time, the wave function can be written asΨ = e iẼt/ ψ(r), and the system of equations above reduces to a Sturm-Liouville eigenvalue problem for ψ(r), provided boundary conditions on the wave function. Solutions to such eigenvalue problem have been constructed numerically for instance in [7,[10][11][12][13]. There are various aspects of the dynamical behavior of ground state solutions of this type, which could be potentially related to astrophysical phenomena. For instance, the virialization time of structures is short after the turnaround point [14]; BEC structures ruled by the GPP system of equations show a solitonic behavior, which could be consistent with the Bullet Cluster observation [15][16][17]; finally, one of the most appealing properties of these solutions is that ground state solutions are stable under very general perturbations [18], and they are late-time attractors of general initial density fluctuations [7,18]. The size and mass scales of astrophysical systems associated to the equilibrium solutions of the GPP system depend on the mass of the boson considered. In the case of galaxy modeling, for an ultralight boson m ∼ 10 −23 eV, equilibrium configurations are unable to explain the galactic rotation curves (RCs) because they are considerably compact, which in turn implies that RCs are Keplerian at short distances from the center of a galaxy halo. Two possibilities to rescue these equilibrium configurations as galactic halos are: i) that excited state solutions of the GPP system could be the halos, in fact thwy show desirable RCs [10,11], unfortunately these were shown to be unstable and therefore useless as galactic halos [7,10] and ii) a new possibility proposes that galactic halos are a mixture of ground and excited states that unlike the pure excited states are stable and show promising RCs [19]; the question this mixed state configurations has to answer is how, being the critical temperature for condensation high of the order of TeV, at the current universe's temperature there are still a considerable amount of bosons in excited states. An alternative view of BEC dark matter is found in the Thomas-Fermi (TF) limit of the GPP system, where the self-interaction between pairs of boson dominates over the kinetic terms in Schrödinger (Gross-Pitaevskii) equation. The model was presented in [20] and the resulting density profile is of the typeρ BEC (r) =ρ c BEC sin(πr/R) πr/R , whereR is the radius at whichρ BEC (R) = 0, defined as the size of the galaxy, andρ c BEC is the central density of dark matter; with these two parameters, a number of galactic RCs were fitted. Two drawbacks of this model are: 1) the characteristic radius is given byR = 2ã Gm 3 , whereã is the self-interaction parameter between pairs of bosons and m is the boson mass, and therefore, given a boson dark matter candidate with given mass and selfinteraction, the value ofR would be fixed for all galaxies and cannot be a free fitting parameter; a different way of looking at this problem is assuming R can be different for different galaxies, then for a given boson mass m every galaxy will require a different value of a, which in turn means that bosons interact differently in different galaxies; 2) the mass of the boson found in [20] is of the order of meV-eV, which is inconsistent with the boson mass fitting the mass power spectrum, which requires the boson to be ultralight [1]. An attempt to avoid the second inconsistency was presented in [2], where the same model under the TF approximation was used, but this time considering an ultralight boson mass. Then it was shown that not only RCs are acceptable, but also that the model was a serious model at galactic core scales as well. Unfortunately, it was shown that such halos are unstable, with decay time scales of the order of megayears, which is an insufficient lifetime for a halo [21]; furthermore this model, being a particular set of parameters of that in the TF limit model for the ultralight particle, it inherits the overdetermined set of parameters R, a, m of the original model in [20]. Summarizing the status of spherically symmetric BEC halo models with bosons in the same ground state: i) the ground state solutions of the GPP system have good properties of stability and are late-time attractor solutions [7,11,18], however show an unacceptable density profile, and ii) the halos constructed in the TF limit [2,20] have an overdetermined set of parameters and are unstable, and thus are not considered acceptable anymore. Therefore the model needs to be constructed from the beginning at local scales and explore possible solutions. One would like to preserve the ultralight nature of bosonic dark matter that works fine at cosmic scales, construct galactic halos that are stable and preferably latetime attractors in time, with appropriate core density, and with a set of parameters that allows the existence of different halo sizes, avoiding the unphysical overdetermination of parameters (in the case of the TF limit spherical model). We thus explore the possibility that a local parameter, possibly different for each galaxy can help. A good such parameter is angular momentum. Adding angular momentum to a BEC fluctuation may improve the RCs in galaxies, explain the different sizes of galaxies, all this retaining the good properties of the model at cosmic scales. There are some precedents dealing with rotating BEC dark matter halos, for instance in [22], spheroid and ellipsoid analytic solutions to the GPP system with rotation are studied as rotating BEC dark matter halos in various scenarios, and the results particularly focus on the possibility of vortex formation. However these analytic results are limited by the conditions imposed to obtain exact solutions, particularly regimes of domination (selfinteraction and quantum pressure terms). In order to have a more general picture it is necessary to solve the GPP system with rotation under very general conditions. The challenge presented in this way involves the solution of the GPP system out of spherical symmetry, and in order to study general scenarios it is necessary to solve the GPP system numerically. In this sense, some numerical implementations designed to solve the GPP system in 3D have been presented with different purposes, for instance [3][4][5]23]. In this paper we firstly present our numerical code that solves the GPP system in 3D, with various tests such a code must satisfy. Secondly, as a first application of the code, we show the effects of applying angular momentum to spherically symmetric configurations made of an ultralight boson, and particularly focus on the spatial redistribution of bosons, and its direct implication on galactic RCs. In the following section we describe the numerical methods used to solve (1) for a rotating configuration in 3D. In section III we present the RCs for rotating configurations and in section IV we draw some conclusions. II. NUMERICAL METHODS We solve system (1) numerically in cartesian coordinates. It is a coupled system consisting of an evolution equation for Ψ with a potential that is solution of Poisson equation sourced by |Ψ| 2 . The first step before integrating (1) requires the remotion of constants using the following change of variableŝ Ψ = √ 4πG mc 2Ψ ,x = mc x,ŷ = mc ỹ,ẑ = mc z,t = mc 2 t , V =Ṽ mc 2 ,â → c 2 2mGã , so that the numerical coefficients , 2 /m, 2π 2 /m 2 , 4πGm do not appear in (1). Addi-tionally, we set our code units allowing accurate calculations, using the invariance of system (1) under the transformation t = λ 2t , x = λx, y = λŷ, z = λẑ, Ψ =Ψ/λ 2 , V =V /λ 2 , a = λ 2â , for an arbitrary value of the parameter λ [11]. These rescaling reduces the original system (1) to the following one i ∂Ψ ∂t = − 1 2 ∇ 2 Ψ + V Ψ + a|Ψ| 2 Ψ,(2)∇ 2 V = |Ψ| 2 .(3) The strategy we follow to solve the coupled GPP system, consists in considering Schrödinger equation an evolution equation for Ψ, and Poisson equation a constraint that has to be solved every time it is required during the integration of Schrödinger equation. Evolution. We approximate the GPP (2-3) system using finite differences on a uniformly discretized grid on a spatial domain, described with cartesian coordinates. We solve the system on the spatial domain [x min , x max ] × [y min , y max ] × [z min , z max ], using the 3D Fixed Mesh Refinement (FMR) driver Nakode.mx devel- oped by us [24]. We use FMR with various purposes: firstly because we want to keep a good resolution and therefore numerical accuracy in the central parts of the condensate trap, where the density is mostly concentrated; secondly, because part of the matter will be expelled by rotation and it is important to make sure it is not reflected back into the numerical domain and then contaminate the calculations. In order to cover a sufficiently big numerical domain one would require unaffordable computer memory, however with the use of FMR only high resolution is used in the regions where the functions are steeper, in our case in the center of the condensate. Our particular set up considers a halved resolution among consecutive refinement levels and boxes in all cases are centered at the coordinate origin, which coincides with the center of the condensate. In this way, our implementation allows high accuracy at the center and the possibility to place the boundaries far away. We illustrate our FMR set up in Fig. 1, where we show |Ψ| 2 in a domain for one of our production runs. For the evolution, Schrödinger equation is semidiscretized at each point of the numerical domain, and the values of Ψ evolve using the method of lines, with an iterative Crank-Nicholson time integrator and second order accurate spatial stencils to approximate spatial derivatives. We use the Berger-Oliger evolution algorithm in the FMR, which was adapted to Schrödinger equation, which is first order in time and second order in space. Considering that the value of V is required during the time integration, Poisson equation is solved for V : (a) at each of the intermediate steps of the time integration, and (b) on each refinement level required, in order to source Schrödinger equation for the evolution of Ψ. The two dimensional Poisson equation is written in cylindrical coordinates, on the diagonal plane, as follows ∂ 2 V axi ∂r 2 axi + 1 r axi ∂V axi ∂r axi + ∂ 2 V axi ∂z 2 = |Ψ axi | 2 ,(4) where V axi (r axi , z) is the unknown potential on the diagonal plane, and Ψ axi (r axi , z) is the interpolated value of the wave function from the 3D mesh on this plane at a given time. Since the equation is singular along the z-axis r axi = 0, we stagger the plane ∆r axi /2 off the axis. Additionally, we avoid the divergence of the second term in (4) The boundary condition on V axi is monopolar V = −M/ x 2 + y 2 + z 2 at the exterior boundary [r axi = x 2 max + y 2 max , z] ∪ [r axi , z max ] ∪ [r axi , z min ] and a parity condition at the staggered axis V axi (r axi − ∆r axi /2, z) = V axi (r axi + ∆r axi /2, z). Finally, Poisson equation is integrated along the diagonal plane using a Successive Overrelaxation (SOR) algorithm. This solution is calculated for all the refinement levels and at every intermediate time step during the evolution. Each time V axi is interpolated back into the 3D grid, for all the grid points in the domain, based on the fact that the system is axially symmetric. In this way, V is known in 3D and used as source in Schödinger equation. Sponge. It is expected that some of the density of probability will approach the numerical boundary because this is the mechanism of relaxation of the system [7,25]. In order to avoid that such excess of matter reflects back into the numerical domain and therefore contaminate the calculations, we implemented a sponge. This consists in the addition of an imaginary potential, thus we redefine the potential in Schrödinger equation by adding an imaginary part V = V + iV im . The implication of an imaginary potential is that the continuity equation reads ∂ρ/∂t + ∇ · [i/2(Ψ∇Ψ * − ψ * ∇Ψ)] = 2V im |Ψ| 2 , and then, as long as V im < 0 the imaginary potential acts as a sink of particles. The specific profile chosen for V im is V im = − V0 2 [2 + tanh(r − r c )/δ − tanh(r c /δ)] , which is a smooth version of a step function, where V 0 is the amplitude of the imaginary potential, r c is the location and δ is the width of the step. This profile has been used successfully in the past for spherical and axisymmetric cases in 2D [7,11,15,18]. In Fig. 3 the sponge used in the production runs is shown; five refinement levels are used and the sponge covers the domain with the coarsest resolution. Tests. In order to show the code works properly, we show in the appendix two tests of the code, which include the unitarity of the evolution algorithm for an exact solution of Schrödinger equation and the evolution of ground state equilibrium solutions of the GPP problem in spherical symmetry with our 3D code. This shows the accuracy of both, the evolution algorithm and the Poisson equation solver. A. Initial data In order to study the effects produced by rotation we can choose an initial profile of the wave function (and thus of the initial density profile) with rotation. However, we choose to start up our simulations with a wave function Ψ(t = 0, x), which corresponds to a ground state equilibrium configuration constructed following [11] and then rigid rotation is added to such configuration. This technique has been applied to the cases when linear momentum is added to equilibrium configurations for the study of collision of configurations and orbital motion [4,15]. Then, for a spherically symmetric ground state configuration constructed in spherical symmetry following [11], we have the wave function function Ψ 0 . Then we apply a rotation Ψ = e −iL·nθ Ψ 0 . In our particular case we choose a rotation around the z-axis an amount θ = arctan(y/x) with L = L zẑ . In order to parametrize the rigid rotation we choose L z = xp y −yp x to be a constant. We then start up the evolution with this wave function. B. Monitoring the results Total energy. The stability of a configurations depends highly on whether or not a system is bounded. Thus it is important to calculate during the evolution the total energy of the system. Our energy estimate is calculated in terms of the expectation value of the total energy of the system, which has three contributions, kinetic, gravitational and self-interaction energies. The expectation value of these quantities are respectively K = − 1 2 Ψ * ∇ 2 Ψdxdydz, W = 1 2 V |Ψ| 2 dxdydz, I = a |Ψ| 4 dxdydz(5) where the integrals are calculated numerically on the whole numerical domain. Then the total energy of the system is defined by E = K +W +I and can be calculated at every time step.. Mass. Even though the integral of the density of probability is conserved during the evolution in the whole space, we calculate the solution of the GPP system in a finite domain that allows the particles to get off, or equivalently to be absorbed by a sponge once they are near the artificial boundaries of our numerical domain. The useful mass estimate will be the integral of the density of probability in the numerical domain M = |Ψ| 2 dxdydz.(6) In all our simulations in this paper we use a domain consisting of cubic boxes of size [−160kpc, 160kpc] 3 . We estimate the mass and the expectation values of the operators described above within a box of [−40kpc, 40kpc] 3 only, which is our working physical domain. The domain outside this later box serves as a buffer zone that we use to absorb any potential noises in the calculations and contains the sponge as well. Rotation curve. In order to estimate the rotation curve, we place various detectors that measure the tangential velocity v of a test particle. The detectors are located at a set of points along the x-axis, and at each of such points we estimate the velocity of a test particle. For that we assume the test particle describes a circular orbit, as usually considered for RCs. Thus the gravitational force due to the dark matter compensates the centrifugal force according to the usual formula v(r) = 2GM (r)/r, where r is the distance from the coordinate center to the detector, and M (r) is the mass contained within a sphere of radius r. Explicitly, if a particular detector is located at (x, y, z) = (x d , 0, 0) we calculate v(x d ) as v 2 (x d ) = 2G |x d | |Ψ| 2 dxdydz,(7) where the volume integral is calculated in a sphere of radius x d on our 3D cartesian grid. We place a considerable number of these detectors and construct the RC during the evolution. Clearly, the RC profile will strongly depend on |Ψ| 2 , which is expected to be different for different values of L z . III. EVOLUTION OF THE SYSTEM First of all the code units have to be specified. The scale invariance parameter λ is fixed using the spatial coordinates such that λ = mc x x . In order to use a numerical domain with coordinate values in kpc, that is x =x, it suffices to write down the factor mc in kpc. For m = 10 −23 eV/c 2 its value is λ = mc [kpc] = 0.0006399. We fix this scaling parameter for the cases explored in this paper. With this value one recovers the mass in solar massesM = c mG λM , the velocity in km/sṽ = cλv, the energy in JoulesẼ = c 3 4πGm λ 3 E and the potential that transforms with the same formula. Our numerical set up is as follows. Initially, all the cases we analyze show a density distributed within a radius smaller than 5kpc. In order to allow the configuration to relax within a bigger domain, we set production runs for cubic numerical domains with x min = y min = z min = −160 and x max = y max = z max = 160. We add angular momentum with rigid rotation to an originally spherically symmetric equilibrium configuration. These spherical equilibrium configurations have negative total energy and are stable in a very general sense [7,18]. However, the addition of angular momentum changes the energy of the system already at initial time. We present the evolution of the system for various initial values of L z and study in each situation the properties of the resulting configuration. We explore the evolution for two values of a = 0, 0.5 and show the behavior is pretty similar in both cases. A. Case a = 0. This is the case of the free field, in which the bosons do not interact and immediately implies there is no energy of self-interaction I = 0. We present results for the angular momentum L z = 0.6, 0.85, 0.95 which are representative of three different responses of the system. The case with L z = 0.6, starts with negative total energy and therefore it is not expected that the angular momentum distorts significantly the configuration. In the two cases with higher angular momentum, the effect of the addition of angular momentum increases the total energy of the system which results to be positive E = K +W > 0, which in turn implies that the configurations are initially gravitationally unbounded, however in one case the system relaxes and becomes bounded. Explicitly, the three different cases show the following properties. Case L z = 0.6. In this case the initial total energy is negative and the angular momentum applied to the spherical configuration does not suffice to redistribute the density of bosons in such a way that we observe flat RCs. Instead, we notice in Fig. 4 that the gravitational potential dominates over the kinetic energy, confines the configuration and E < 0 all the way. The RCs are Keplerain at scales of the order of a few kpc, as happens for the spherically symmetric ground state solutions of the GPP system. The minimum of the potential oscillates, which is an indication that the bosons remain confined to a small region and as a consequence, the RC starts and remains Keplerian after a few kpc. The mass stabilizes, which is an indication that the system is relaxing. The behavior is pretty much that of a perturbed ground state configuration. Case L z = 0.85. In this case we observe the desired properties of the configuration that are shown in Fig. 5. First we notice the change of sign of the total energy and how it stabilizes around a negative value, which indicates that the system tends toward a bounded state within the scale of 1Gyr. Notice also that the gravitational potential stabilizes. The mass approaches a stable vlaue M ∼ 4 × 10 10 M ⊙ . Finally the RC approaches a very desirable shape, and stabilizes at least during time scales of the order of 1Gyr. With these initial values chosen for the equilibrium configuration, the final nearly equilibrium properties of the system approximate those of a dwarf galaxy. Case L z = 0.95. In this case the angular momentum is so high that redistributes the density of bosons so quickly that they abandon the numerical domain and the configuration is diluted quickly. The physical properties of the configuration are shown in Fig. 6. B. Case a = 0.5. The addition of a self-interaction term contributes with a repulsive term between bosons and changes the inter- play between the kinetic and gravitational terms. This effect is already known in ground state spherically symmetric configurations, in which more mass can be allocated when the self-interaction is stronger [7]. In this case we show that the same three regimes are also possible when repulsive self-interaction (a = 0.5 > 0) is considered. The three values of the angular momentum we use to represent the three regimes are L z = 0.5, 0.9, 1. Since the results are pretty similar to those for the free field case, we present only the total energy and rotation curve in Fig. 7, for the three cases. In the middle we show the results for Lz = 0.9 that shows desirable results for dark matter dominated galaxies. finally in the bottom we show the results for Lz = 1, that corresponds to a configuration washed out by rotation. IV. DISCUSSION AND CONCLUSIONS We explored the effects of rigid rotation of structures formed by Bose Condensates described by the GPP system of equations. There are three regimes: one with an angular momentum insufficient to disperse the distribution of bosons and provide galactic RCs in a dark matter dominated galaxy, in the range of order 5 -20 kpc; a second regime that redistributes the density of bosons in such a way that RCs are pretty comparable to galactic RCs in the range of ∼ 10kpc, where the configurations stabilize and remain bounded; a third regime in which the rotation is so strong that destroys the configuration and disperses the bosons out of the numerical domain. The life time criterion of our results is given by the time window of 1Gyr we used in all cases. Knowing that BEC spherically symmetric halo models are poor at explaining galactic dynamics [21], what we have shown in this paper is that the model for the ultralight spinless boson DM candidate can work at local scales with the addition of angular momentum to halos. In this way, configurations at local scale are parametrized by the boson mass, the self-interaction of the Bose gas and the angular momentum applied to the halo. We have also presented a new code that solves the GPP system of equations in 3D . The tests shown are exhaustive and validate this new numerical tool that will serve for the exploration of the parameter space of rotating BEC halos, including the interplay of self-interaction and rotation, the search of attractor solutions and the study of vortices in a self-gravitating BEC. Acknowledgments We thank L. A. Ureña-López for valuable comments and suggestions on this paper. This research is partly supported by grants CIC-UMSNH-4.9 and CONACyT 106466. The very first test of an algorithm involving Schrödinger equation is the evolution algorithm, which most importantly must be unitary within numerical errors. In order to test this property of the code we choose to evolve the 3D stationary wave function submitted to a harmonic oscillator potential. The code is demanded to evolve the wave function Ψ while the density of probability |Ψ| 2 remains time-independent and its integral N = |Ψ| 2 d 3 x remains constant in time. The exact normalized stationary solution is Ψ HO = 1 π2 nx 2 ny 2 nz n x !n y !n z ! × e −(x 2 +y 2 +z 2 )/2 H nx (x)H ny (y)H nz (z) × e −iEn x,ny ,nz t where the energy is given by E nx,ny,nz = n x + n y + n z + 1 and n x , n y , n z are the number of nodes along each spatial direction. We then start the evolution of the wave function evaluated at t = 0 using our algorithms and the FMR driver. In Fig. 8 we show snapshots of the wave function and the density of probability, showing that even though the wave function evolves, the density of probability remains time-independent. In Fig. 9 we show N in time and that for even nearly fifty cycles of the wave function N looses about 0.02%. We show this figure for two different resolutions, showing that the algorithm converges in the continuum limit to a constant value of N . A discussion on the election of evolution algorithm is in turn. In our implementation we use the MoL instead of the more commonly used implicit Crank-Nicholson algorithm, which in theory is unitary and unconditionally stable for arbitrarily high values of ∆t/∆x 2 . The stability of the integration with MoL -on the other handdepends on the stability of the ode solver used; we use here an iterative Cranck-Nicholson integrator, although we also experienced a similar behavior with a third order Runge-Kutta integrator, and in both cases the stability has shown to be safe as long as ∆t/∆x 2 < 0.5 in all our runs. Our code works also with the fully implicit Crank-Nicholson method [4], nevertheless we decided to use the MoL because when comparing the accuracy with that of the implicit method, it was comparable for the two algorithms when a similar ∆t is used, earning little form the appealing stability properties of the implicit method. B. Test on GPP equilibrium solutions The test of fire of our code consists in the correct evolution of a ground state equilibrium configuration, which incorporates the solution of Poisson equation and therefore includes the accuracy of the elliptic solver imple-mented. The equilibrium configurations are constructed in spherical coordinates according to [7,11], which use the normalization that the central value of the density is ρ(r = 0) = 1. We then interpolate this configuration into the 3D domain, and evolve the whole GPP system according to (2)(3). In the continuum limit the density should remain constant in time and space, however discretization errors must converge to zero. What we show in Fig. 10 is that such errors converge with second order to the result in the continuum limit, that is, the departure of the central value of the density converges to zero with second order. This is consistent with the second order spatial stencils we use to discretize the equations. In order to show that the system has considerable dynamics, we show also in Fig. 10 snapshots of the wave function in time. In Fig. 11 we show that 2K + W + 3I converges to zero for the two cases a = 0, 0.5, which indicates that the systems remain virialized during the evolution. This test confirms that the elliptic solver used for the Poisson equation and the evolution algorithm work fine together. The fact that the departure from ρ(0, 0, 0) = 1, in time with the coarse grid is 2 2 times that with the fine grid indicates the second order convergence of our implementation. (Bottom) we show snapshots of Re(Ψ) and its considerable dynamics, whereas the density of probability converges to static in the continuum limit. We show the 2K + W + 3I for the standardized spherically symmetric ground state equilibrium solution in spherical symmetry. In the top and bottom panels we show the case a = 0 and a = 0.5 respectively. The calculation is made with two resolutions and the second order convergence to zero in both cases indicates the configurations remain virialized in the continuum limit as they should. FIG. 1 : 1We illustrate the grid structure of our FMR driver with the density of probability at initial time for one of our production runs, projected on the xy−plane. We use five refinement levels covering the domain.The different numerical domains covered by each resolution are [−160, 160] 3 , [−80, 80] 3 , [−40, 40] 3 , [−20, 20] 3 and [−10, 10] 3 . The domain covered with each refinement level uses 128 3 points. The highest resolution is kept in the center of the domain, where the density and gravitational potential are steeper. Poisson equation. Solving an elliptic equation every intermediate time step during an evolution problem becomes expensive when there are no symmetries. In order to integrate it efficiently, we integrate Poisson equation only on the diagonal plane across the 3D domain with normal (x +ŷ). This can be done as long as we only analyze axially symmetric configurations. For a given refinement level we show the domain of integration of Poisson equation inFig. 2, defined by the domain r axi ∈ [0, x 2 max + y 2 max ] × z ∈ [z min , z max ] with resolutions ∆r axi = ∆x 2 + ∆y 2 and ∆z. FIG. 2 : 2We show the diagonal plane orthogonal tox +ŷ, where Poisson equation is defined and solved. FIG. 3 : 3We show a projection of Vim on the xy-plane. The region where Vim is non-zero is where the sponge captures the density of probability approaching the boundary. The numerical parameters used in the production runs are V0 = 1, δ = 8 and rc = 128. We use five refinement levels covering boxes with domains [−160, 160] 3 , [−80, 80] 3 , [−40, 40] 3 , [−20, 20] 2 and [−10, 10] 3 respectively. Even if the sponge helps at absorbing the density of probability that approaches the boundaries, our FMR implementation allows the possibility of pulling the boundaries this far, in order to obtain results as free as possible of numerical errors reflected from the boundaries. - FIG. 4 : 4We show the properties of the configuration for Lz = 0.6. The total energy E remains negative all the way. FIG. 5 :FIG. 6 : 56We show the properties of the configuration for Lz = 0.85. The mass stabilizes around the value of a dwarf galaxy, the energy stabilizes around a negative value and the central potential as well. The rotation curve is Keplerian initially and once the bosons are dispersed away by the rotation, it stabilizes with a shape corresponding to a typical one of a dark matter dominated galaxy. We show the properties of the configuration for Lz = 0.95. The total energy remains positive, which is a typical indication of an unbounded system. The increase of the gravitational potential and the decrease of the mass within the domain, indicate the matter is being washed out by rotation. FIG. 7 : 7We show the properties of the configuration for a = 0.5. In the top panel we show the results for Lz = 0.5, in which the total energy E remains negative all the way and the RC is Keplerian after a few kpc. FIG. 8 :FIG. 9 : 89We show 1D snapshots of the evolution for the wave function ΨHO and the density of probability at various times using two refinement levels. We show N as a function of time for about fifty cycles of the wave function. The dissipation introduced by the evolution algorithm can be reduced when increasing the resolution in time, which in turn implies the conservation of N is more accurate. In our evolutions we choose ∆t such that it allows long term evolutions and also keep the unitarity constraint accurate. 10: (Top) We show the central density of the configuration with a = 0 (or equivalently, the infinity norm of the density L∞(ρ)) for two different resolutions of our numerical domain ∆x = ∆y = ∆z = 0.2 and ∆x = ∆y = ∆z = 0.1. FIG. 11: We show the 2K + W + 3I for the standardized spherically symmetric ground state equilibrium solution in spherical symmetry. In the top and bottom panels we show the case a = 0 and a = 0.5 respectively. The calculation is made with two resolutions and the second order convergence to zero in both cases indicates the configurations remain virialized in the continuum limit as they should. . 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